This application relates to the field of electronic transactions and more particularly to the field of distributed public ledgers, securing the contents of sequence of transaction blocks, and the verification of electronic payments.
A public ledger is a tamperproof sequence of data that can be read and augmented by everyone. Shared public ledgers stand to revolutionize the way a modern society operates. They can secure all kinds of traditional transactions—such as payments, asset transfers, and titling—in the exact order in which they occur; and enable totally new transactions—such as cryptocurrencies and smart contracts. They can curb corruption, remove intermediaries, and usher in a new paradigm for trust. The uses of public ledgers to construct payment systems and cryptocurrencies are particular important and appealing.
In Bitcoin and other prior distributed systems, users have public keys (and know the corresponding secret keys) of a digital signature scheme. It is useful to humanize each key, and think of it as a user. Indeed, the public keys in the system are known, but the users behind those keys are not directly known, providing the users with some level of privacy. Money is directly associated to each public key. At each point in time, as deducible for the sequence of blocks so far, each public key owns a given amount of money. A payment from one public key (the “payer”) to another public key (the “payee”) is digitally signed by the payer. Similarly, all transactions are digitally signed. Furthermore, to be valid, a payment must transfer an amount of money that does not exceed the money that, at that time, the payer owns. Payments (and, more generally, transactions) are organized in blocks. A block is valid if all its payments (and transactions) are collectively valid. That is, the total amount of money paid by any payer in a given block does not exceed the amount of money then available to the payer. Bitcoin is a permissionless system. That is, it allows new users to freely join the system. A new user joins the system when he appears as the payee of payment in a (valid) block. Accordingly, in Bitcoin, a user may enhance whatever privacy he enjoys by owning multiple keys, which he may use for different type of payments. In a permissionless system, he can easily increase his number of keys by transferring some money from a key he owns to a new key. Permissionlessness is an important property. Permissionless systems can also operate as permissioned systems, but the viceversa need not be true. In a permissioned system new users cannot automatically join, but must be approved. (A special case of a permissioned system is one in which the set of users is fixed.) Permissionless systems are more applicable and realistic but also more challenging. In Bitcoin and similar distributed systems, users communicate by propagating messages (that is by “gossiping”). Messages are sent to a few, randomly picked, “neighbors”, each of which, in turn, sends them to a few random neighbors, and so forth. To avoid loops, one does not send twice the same message. A distributed (or shared) ledger consists of the sequence of blocks of (valid) transactions generated so far. Typically, the first block, the genesis blocl, is assumed to be common knowledge, by being part of the specification of the system. Shared ledgers differ in the way in which new blocks are “generated”.
As currently implemented, however, public ledgers do not achieve their enormous potential. Centralized public ledgers put all trust on a single entity, who can arbitrarily refuse to publicize payments made by given keys, and are very vulnerable to cyber attacks. Indeed, once the single central authority is compromised, so is the entire system. Decentralized systems, like Bitcoin, are very expensive to run, waste computation and other valuable resources, concentrate power in the hands of new entities (miners), suffer from considerable ambiguity (forks), and have long latency and small thruput. Other decentralized implementations are permissioned, or assume the ability of punishing misbehaving users, or both, and/or trust that some subset of users are immune to cyber attacks for a suitably long time.
It is thus desirable to provide public ledgers and electronic money systems that do not need to trust a central authority, and do not suffer from the inefficiencies and insecurities of known decentralized approaches.
According to the system described herein, in a transaction system in which transactions are organized in blocks, an entity to constructs a new block Br of valid transactions, relative to a sequence of prior blocks, B0, . . . , Brr−1, by having the entity determine a quantity Q from the prior blocks, having the entity use a secret key in order to compute a string S uniquely associated to Q and the entity, having the entity compute from S a quantity T that is at least one of: S itself, a function of S, and hash value of S, having the entity determine whether T possesses a given property, and, if T possesses the given property, having the entity digitally sign Br and make available S and a digitally signed version of Br. The secret key may be a secret signing key corresponding to a public key of the entity and S may be a digital signature of Q by the entity. T may be a binary expansion of a number and may satisfy the property if T is less than a given number p. S may be made available by making S deducible from Br. Each user may have a balance in the transaction system and p may vary for each user according to the balance of each user.
According further to the system described herein, in a transaction system in which transactions are organized in blocks and blocks are approved by a set of digital signatures, an entity approves a new block of transactions, Br, given a sequence of prior blocks, B0, . . . , Br−1, by having the entity determine a quantity Q from the prior blocks, having the entity compute a digital signature S of Q, having the entity compute from S a quantity T that is at least one of: S itself, a function of S, and hash value of S, having the entity determine whether T possesses a given property, and, if T possesses the given property, having the entity make S available to others. T may be a binary expansion of a number and satisfies the given property if T is less than a pre-defined threshold, p, and the entity may also make S available. The entity may have a balance in the transaction system and p may vary according to the balance of the entity. The entity may act as an authorized representative of at least an other entity. The value of p may depend on the balance of the entity and/or a combination of the balance of the entity and a balance of the other entity. The other user may authorize the user with a digital signature. The entity may digitally sign Br only if Br is an output of a Byzantine agreement protocol executed by a given set of entities. A particular one of the entities may belong to the given set of entities if a digital signature of the particular one of the entities has a quantity determined by the prior blocks that satisfies a given property.
According further to the system described herein, in a transaction system in which transactions are organized in a sequence of generated and digitally signed blocks, B0, . . . , Br−1, where each block Br contains some information INFOr that is to be secured and contains securing information Sr, contents of a block are prevented from being undetectably altered by every time that a new block Bi is generated, inserting information INFOi of Bi into a leaf i of a binary tree, merklefying the binary tree to obtain a Merkle tree Ti, and determining the securing information Si of block Bi to include a content Ri of a root of Ti and an authenticating path of contents of the leaf i in Ti. Securing information of Si1 of a preceding block Bi1 may be stored and the securing information Si may be obtained by hashing, in a predetermined sequence, values from a set including at least one of: the values of Si1, the hash of INFOi, and a given value. A first entity may prove to a second entity having the securing information Sz of a block Bz that the information INFOr of the block Br preceding a block Bz is authentic by causing the second entity to receive the authenticating path of INFOi in the Merkle tree Tz.
According further to the system described herein, in a payment system in which users have a balance and transfer money to one another via digitally signed payments and balances of an initial set of users are known, where a first set of user payments is collected into a first digitally signed block, B1, a second set of user payments is collected into a second digitally signed block, B2, becoming available after B1, etc., an entity E provides verified information about a balance ari that a user i has available after all the payments the user i has made and received at a time of an rth block, Br, by computing, from 15 information deducible from information specified in the sequence of block B0, . . . , Br−1, an amount ax for every user x, computing a number, n, of users in the system at the time of an rth block, Br being made available, ordering the users x in a given order, for each user x, if x is the ith user in the given order, storing ax in a leaf i of a binary tree T with at least n leaves, determining Merkle values for the tree T to compute a value R stored at a root of T, producing a digital signature S that authenticates R, and making S available as proof of contents of any leaf i of T by providing contents of every node that is a sibling of a node in a path between leaf 1 and the root of T.
According further to the system described herein, in a payment system in which users have a balance and transfer money to one another via digitally signed payments and balances of an initial set of users are known, where a first set of user payments is collected into a first digitally signed block, B1, a second set of user payments is collected into a second digitally signed block, B2, becoming available after B1, etc., a set of entities E provides information that enables one to verify the balance ai that a user i has available after all the payments the user i has made and received at a time of an rth block, Br, by determining the balance of each user i after the payments of the first r blocks, generating a Merkle-balanced-search-tree Tr, where the balance of each user is a value to be secured of at least one node of Tr, having each member of the set of entities generate a digital signature of information that includes the securing value hvε of the root of Tr, and providing the digital signatures of hvε to prove the balance of at least one of the users after the payments of the first r. The set of entities may consist of one entity. The set of entities are may be selected based on values of digital signatures thereof.
According further to the system described herein, in a payment system in which users have a balance and transfer money to one another via digitally signed payments and balances of an initial set of users are known, where a first set of user payments is collected into a first digitally signed block, B1, a second set of user payments is collected into a second digitally signed block, B2, becoming available after B1, etc., an entity E proves the balance ai that a user i has available after all the payments the user i has made and received at a time of an rth block, Br, by obtaining digital signatures of members of a set of entities of the securing information hvε of the root of a Merkle-balanced-search tree Tr, wherein the balance of each user is an information value of at least one node of Tr and by computing an authentication path and the content of every node that a given search algorithm processes in order to search in Tr for the user i, and providing the authenticating paths and contents and the digital signatures to enable another entity to verify the balance of i.
According further to the system described herein, computer software, provided in a non-transitory computer-readable medium, includes executable code that implements any of the methods described herein.
A shared public ledger should generate a common sequence of blocks of transactions, whose contents are secure and can be easily proved. The present invention thus provides a new way to generate a common sequence of blocks, Algorand, with a new way to secure the contents of a sequence of blocks, Blocktrees, that also enable to easily prove their contents.
Algorand At high level, letting B1, . . . , Br−1 be the first r−1 blocks, in Algorand, the new block, Br, is generated by the following two (mega) steps
We now detail how public keys are selected, how Cr reaches consensus, and how the given number of digital signatures necessary to certify a block is chosen.
Preferably, a public key is selected, as a leader or a member of the committee for block Br, via a secret cryptographic sortition. In essence, a user selects himself, by running his own lottery, so that (a) he cannot cheat (even if he wanted) and select himself with a higher probability that the one envisioned; (b) obtains a proof Pi of been selected, which can be inspected by everyone; (c) he is the only one to know that he has been selected (until he decides to reveal to other that this is the case, by exhibiting his proof Pi).
To enhance the odds that a public key he owns is selected, a user might be tempted to increase the number of the public keys he owns (so called Sybil attacks), which is very easy to do in a permissionless system, as already discussed. To prevent such attacks, the chosen secret cryptographic sortition ensures that the probability of a given public key to be selected is proportional to the money it owns. This way, a user has the same probability of having one of his public keys to be selected whether he keeps all his money associated to a single key or distributes it across several keys. (In particular, a user owning—say—$1M in a single key or owning 1M keys, each with $1, has the same chance of being selected.)
More precisely, to determine whether a public key pk he owns is selected, as a leader or a committee member, a user digitally signs (via the corresponding secret key sk) some information derivable from all prior blocks. In the preferred embodiment, he actually digitally signs a quantity Qr derivable from the last block, Br−1, hash it, and check whether the hash is less that a given (probability) threshold p (which depends on the amount of money owned by the user's public key). If this is the case, this digital signature of the user can be used as a proof that the public key pk has been selected. Indeed, everyone can check the user signature, hash it, and compare it to the given threshold p.
By varying the selection thresholds p, the system controls how many users are selected, for each role. Let us first consider selecting the public keys of a committee Cr. For instance, assume that, in the system, there are 1M keys, each owing the same amount of money, and that we wish CR to have roughly 1K members. Then we can choose p=1/1,000 to be a common selection threshold. In fact, the hash of a signature can be considered as the binary expansion of a random number between 0 and 1, and thus will be less than or equal to 1/1,000 with probability equal to p=1/1,000. Thus, the total expected number of selected users will be 1M/1K=1/1K. As we shall see, we can still achieve to select a given number of committee members when different public keys own different amounts of money.
Let us know consider selecting a single leader r for the rth block. Then, the system may first use secret cryptographic sortition to select—say—100 public keys, and then let the leader be the public key whose (hashed) proof is smaller. More precisely, assume, for simplicity only, that the selected keys are pk1, . . . , pk100, and that their corresponding proofs of selections are P1, . . . , P100. Then, the owner, i, of each selected key, assembles his own block of new valid transactions, Bir, and propagates Pi as well as the properly authenticated block Bir. Then the leader r will be the key whose proof is lexicographically smaller, and the block Br will the block on which the committee Cr reaches consensus as being the block proposed by r.
A big advantage of using secret cryptographic sortition to select block leaders is that an adversary who monitors the activity of the system and is capable of successfully gaining control of any user he wants, cannot easily corrupt the leader r so as to chose the block r proposes. In fact, when the 100 potential leaders secretly select themselves, by running their own lotteries. Thus the adversary does not know who the (expected) 100 public key will be. However, once each i propagates his credential and proposed block, the adversary can quickly figure out who the leader r is. But, at that point, there is little advantage in corrupting him. His block and proof are virally propagating over the network, and cannot be stopped. (Not even a nation state could put back in the bottle a message sent by—say—WikiLeaks.) One last move may still be available to the adversary. Namely, he might corrupt r and, having taken control of him, oblige him to assemble, authenticate, and propagate a different block. Yet, even this opportunity will be closed to the adversary. Indeed, a user keeps his secret signing key that he uses to authenticate his payments and transactions, and to generate his selection proof (when and if he is selected). But he uses a set of different and ephemeral keys to authenticate the block he proposes, when he is selected a leader. More precisely, he selects in advance a set of say 1M public-secret key pairs (pkir, skir), and keeps safe each secret key skir that he might still use. The inventive system guarantees that anyone can recognize that pkir is the only possible public key relative to which i can sign a proposed rth block, anyone can verify a digital signature of i relative to pkir, but only i can generate such a signature, because he is the only one to know the corresponding secret key skir. If i is honest and is selected to propose a block Bir, then i digitally signs it relative to his public key pair (pkir, skir), and destroys skir immediately after. This way, should the adversary succeed to corrupt i, he will not find the necessary secret key skir with which he can force the now corrupted i to authenticate a second and different block. The same advantage exists for committee members. Let us now see how the members of the committee Cr reach consensus on the block propagated by the leader r.
The preferred way for Cr to reach consensus on a new proposed block is via Byzantine agreement (BA). A BA protocol is a communication protocol that enables a fixed set of players, each starting with his own value. to reach consensus on a common value. A player is honest if he follows all his prescribed instructions, and malicious otherwise. Malicious player can controlled and perfectly coordinated by a single entity, the Adversary. So long as only a minority—e.g., ⅓ of the payers are malicious, a BA protocol satisfies two fundamental properties: (1) All honest players output the same value, and (2) if all players start with the same input value, then all honest players output that value.
There are several challenges in using BA in a permissionless distributed-ledger system. A main one is that BA protocol protocols are very slow. (Typically, BA protocols involve at most a few dozen of users.) Algorand overcomes this challenge by means of a new protocol that, in the worst case, takes only 9 steps in expectation. Furthermore, in each of these steps, a participating player need only to propagate a single and short message!
However, a different challenge remains. Since to reach Byzantine agreement on a new block takes more than one step, although the adversary may not be able to attack committee members before they start sending their first messages (since they select themselves via secret cryptographic sortition), he could corrupt all members of the committee Cr after they propagate their first message in the BA protocol. In fact, with their first message, they propagate also their proofs of belonging to the committee. Fortunately, the inventive consensus protocol enjoys an additional and new advantage: it is player replaceable. That is, each of the expected few steps of the protocol can be executed by a different set of players, who need not share any secret information or any hidden internal state with the players selected to perform any prior step. Indeed, Algorand's consensus protocol is a state-less Byzantine agreement (BA) protocol. Accordingly, a different set of users can be selected, via secret cryptographic sortition, to run each step of the new BA protocol.
One last challenge remains. As we have seen, public keys select themselves to have a role in the generation of the block Br by digitally signing a quantity Qr, that is preferably part of the previous block, Br−1, already publicly known, and verifying whether their signature satisfies a special property: namely, its hash must be less than a given selection threshold. The problem is that, since Algorand is permissionless, an adversary may keep on generating many public-secret key pairs, until he finds one whose signature of Qr has the desired property. As he finds such a key, he may bring it into the system, so as to guarantee that one of his keys is the leader in charge of selecting block Br, or most of the committee Cr contains keys that he controls, enabling him to subvert the BA properties. To prevent this possibility, Algorand relies on a two-prong strategy. First of all, it does not allow a newly introduced key to be eligible to be selected right away as a block leader or a committee member. Rather, to have a role in the generation of block Br, a key pk must be around for a while. More precisely, it must appear for the first time in the block Br−k, or in an older block, where k is a sufficiently large integer. (For instance, k=70 suffices, when at least 80% of the money is in honest hands, in each of our detailed embodiments.) Second, he defines Qr inductively, in terms, that is, of the previous quantity, Qr−1. More precisely, in our preferred embodiment, Qr is the hash digital signature of the leader of block Br of the pair (Qr−1, r). This digital signature is actually made explicit part of the block Br−1. The reason this works is as follows. When an adversary, in some block Br−k, decides to bring into the system a new key pk, by making a payment from a public key he controls to pk, the earliest block for which pk can play a role is Br. But we shall prove that, k blocks before Br, no one can predict what Qr−1 might be significantly better than by random guessing. Therefore, an adversary cannot choose pk in any strategic way!
Blocktrees (and StatusTrees) A blockchain guarantees the tamperproofness of its blocks, but also makes operating on its blocks (e.g., proving whether a given payment is part of a given block) quite cumbersome. Shared public ledgers are not synonyms of blockchains, and will in fact benefit from better ways to structure blocks. Algorand works with traditional blockchains, but also with a new way of structuring blocks of information, blocktrees. This inventive way may be of independent interest.
A main advantage of blocktrees is enabling one to efficiently prove the content of an individual past block, without having to exhibit the entire blockchain.
We construct blocktrees via a novel application of Merkle trees.
Furthermore, in case of a distributed payment system, where payments transfers money from a public key to another public key, and payments are organized in a sequence of blocks, we provide efficient methods to prove what is the balance (money on account) available to a given key after the first r blocks of payments. We refer to these methods as statustrees.
Embodiments of the system described herein are explained in more details in accordance with the figures of the drawings, which are briefly described as follows.
The system described herein provides a mechanism for distributing transaction verification and propagation so that no entity is solely responsible for performing calculations to verify and/or propagate transaction information. Instead, each of the participating entities shares in the calculations that are performed to propagate transaction in a verifiable and reliable manner.
Referring to
For concreteness only, we shall assume in our description that all transactions are payments, and focus on describing Algorand only as a money platform. Those skilled in the art will realize that Algorand can handle all kinds of transactions as well.
Algorand has a very flexible design and can be implemented in various, but related, ways. We illustrate its flexibility by detailing two possible embodiments of its general design. From them, those skilled in the art can appreciate how to derive all kinds of other implementations as well.
To facilitate understanding the invention, and allow to internal cross reference of its various parts, we organize its presentation in numbered and titled sections. The first sections are common to both of the detailed embodiments.
Money is becoming increasingly virtual. It has been estimated that about 80% of United States dollars today only exist as ledger entries. Other financial instruments are following suit.
In an ideal world, in which we could count on a universally trusted central entity, immune to all possible cyber attacks, money and other financial transactions could be solely electronic. Unfortunately, we do not live in such a world. Accordingly, decentralized cryptocurrencies, such as Bitcoin, and “smart contract” systems, such as Ethereum, have been proposed. At the heart of these systems is a shared ledger that reliably records a sequence of transactions, as varied as payments and contracts, in a tamperproof way. The technology of choice to guarantee such tamperproofness is the blockehain. Blockchains are behind applications such as cryptocurrencies, financial applications, and the Internet of Things. Several techniques to manage blockchain-based ledgers have been proposed: proof of work, proof of stake, practical Byzantine fault-tolerance, or some combination.
Currently, however, ledgers can be inefficient to manage. For example, Bitcoin's proof-of-work approach requires a vast amount of computation, is wasteful and scales poorly. In addition, it de facto concentrates power in very few hands.
We therefore wish to put forward a new method to implement a public ledger that offers the convenience and efficiency of a centralized system run by a trusted and inviolable authority, without the inefficiencies and weaknesses of current decentralized implementations. We call our approach Algorand, because we use algorithmic randomness to select, based on the ledger constructed so far, a set of verifiers who are in charge of constructing the next block of valid transactions. Naturally, we ensure that such selections are provably immune from manipulations and unpredictable until the last minute, but also that they ultimately are universally clear.
Algorand's approach is quite democratic, in the sense that neither in principle nor de facto it creates different classes of users (as “miners” and “ordinary users” in Bitcoin). In Algorand “all power resides with the set of all users”.
One notable property of Algorand is that its transaction history may fork only with very small probability (e.g., one in a trillion, that is, or even 10−18). Algorand can also address some legal and political concerns.
The Algorand approach applies to blockchains and, more generally, to any method of generating a tamperproof sequence of blocks. We actually put forward a new method—alternative to, and more efficient than, blockchains—that may be of independent interest.
Bitcoin is a very ingenious system and has inspired a great amount of subsequent research. Yet, it is also problematic. Let us summarize its underlying assumption and technical problems—which are actually shared by essentially all cryptocurrencies that, like Bitcoin, are based on proof-of-work.
For this summary, it suffices to recall that, in Bitcoin, a user may own multiple public keys of a digital signature scheme, that money is associated with public keys, and that a payment is a digital signature that transfers some amount of money from one public key to another. Essentially, Bitcoin organizes all processed payments in a chain of blocks, B1, B2, . . . , each consisting of multiple payments, such that, all payments of B1, taken in any order, followed by those of B2, in any order, etc., constitute a sequence of valid payments. Each block is generated, on average, every 10 minutes.
This sequence of blocks is a chain, because it is structured so as to ensure that any change, even in a single block, percolates into all subsequent blocks, making it easier to spot any alteration of the payment history. (As we shall see, this is achieved by including in each block a cryptographic hash of the previous one.) Such block structure is referred to as a blockchain.
Assumption: Honest Majority of Computational Power Bitcoin assumes that no malicious entity (nor a coalition of coordinated malicious entities) controls the majority of the computational power devoted to block generation. Such an entity, in fact, would be able to modify the blockchain, and thus re-write the payment history, as it pleases. In particular, it could make a payment , obtain the benefits paid for, and then “erase” any trace of .
Technical Problem 1: Computational Waste Bitcoin's proof-of-work approach to block generation requires an extraordinary amount of computation. Currently, with just a few hundred thousands public keys in the system, the top 500 most powerful supercomputers can only muster a mere 12.8% percent of the total computational power required from the Bitcoin players. This amount of computation would greatly increase, should significantly more users join the system.
Technical Problem 2: Concentration of Power Today, due to the exorbitant amount of computation required, a user, trying to generate a new block using an ordinary desktop (let alone a cell phone), expects to lose money. Indeed, for computing a new block with an ordinary computer, the expected cost of the necessary electricity to power the computation exceeds the expected reward. Only using pools of specially built computers (that do nothing other than “mine new blocks”), one might expect to make a profit by generating new blocks. Accordingly, today there are, de facto, two disjoint classes of users: ordinary users, who only make payments, and specialized mining pools, that only search for new blocks.
It should therefore not be a surprise that, as of recently, the total computing power for block generation lies within just five pools. In such conditions, the assumption that a majority of the computational power is honest becomes less credible.
Technical Problem 3: Ambiguity In Bitcoin, the blockchain is not necessarily unique. Indeed its latest portion often forks: the blockchain may be—say—B1, . . . , Bk; Bk+1; Bk+2, according to one user, and B1, . . . , Bk, B″k+1; B″k+2; B″k+3 according another user. Only after several blocks have been added to the chain, can one be reasonably sure that the first k+3 blocks will be the same for all users. Thus, one cannot rely right away on the payments contained in the last block of the chain. It is more prudent to wait and see whether the block becomes sufficiently deep in the blockchain and thus sufficiently stable.
Separately, law-enforcement and monetary-policy concerns have also been raised about
The ability to “print money” is one of the very basic powers of a nation state. In principle, therefore, the massive adoption of an independently floating currency may curtail this power. Currently, however, Bitcoin is far from being a threat to governmental monetary policies, and, due to its scalability problems, may never be.
Setting Algorand works in a very tough setting. Briefly,
Main Properties Despite the presence of our powerful adversary, in Algorand
1. A N
Algorand leverages this binary BA protocol to reach agreement, in our different communication model, on each new block. The agreed upon block is then certified, via a prescribed number of digital signature of the proper verifiers, and propagated through the network.
2. S
3. T
To meet this challenge, we purposely construct, and continually update, a separate and carefully defined quantity, Qr, which provably is, not only unpredictable, but also not influentiable, by our powerful Adversary. We may refer to Qr as the rth seed, as it is from Qr that Algorand selects, via secret cryptographic sortition, all the users that will play a special role in the generation of the rth block. The seed Qr will be deducible from the block Br−1.
4. S
To prevent this problem, leaders (and actually verifiers too) secretly learn of their role, but can compute a proper credential, capable of proving to everyone that indeed have that role. When a user privately realizes that he is the leader for the next block, first he secretly assembles his own proposed new block, and then disseminates it (so that can be certified) together with his own credential. This way, though the Adversary will immediately realize who the leader of the next block is, and although he can corrupt him right away, it will be too late for the Adversary to influence the choice of a new block. Indeed, he cannot “call back” the leader's message no more than a powerful government can put back into the bottle a message virally spread by WikiLeaks.
As we shall see, we cannot guarantee leader uniqueness, nor that everyone is sure who the leader is, including the leader himself! But, in Algorand, unambiguous progress will be guaranteed.
5. P
Fortunately we'll prove that protocol BA*, executed by propagating messages in a peer-to-peer fashion, is player-replaceable. This novel requirement means that the protocol correctly and efficiently reaches consensus even if each of its step is execute by a totally new, and randomly and independently selected, set of players. Thus, with millions of users, each small set of players associated to a step of BA* most probably has empty intersection with the next set.
In addition, the sets of players of different steps of BA* will probably have totally different cardinalities. Furthermore, the members of each set do not know who the next set of players will be, and do not secretly pass any internal state.
The replaceable-player property is actually crucial to defeat the dynamic and very powerful Adversary we envisage. We believe that replaceable-player protocols will prove crucial in lots of contexts and applications. In particular, they will be crucial to execute securely small sub-protocols embedded in a larger universe of players with a dynamic adversary, who, being able to corrupt even a small fraction of the total players, has no difficulty in corrupting all the players in the smaller sub-protocol.
An Additional Property/Technique: Lazy Honesty A honest user follows his prescribed instructions, which include being online and run the protocol. Since, Algorand has only modest computation and communication requirement, being online and running the protocol “in the background” is not a major sacrifice. Of course, a few “absences” among honest players, as those due to sudden loss of connectivity or the need of rebooting, are automatically tolerated (because we can always consider such few players to be temporarily malicious). Let us point out, however, that Algorand can be simply adapted so as to work in a new model, in which honest users to be offline most of the time. Our new model can be informally introduced as follows.
Lazy Honesty. Roughly speaking, a user i is lazy-but-honest if (1) he follows all his prescribed instructions, when he is asked to participate to the protocol, and (2) he is asked to participate to the protocol only rarely, and with a suitable advance notice. With such a relaxed notion of honesty, we may be even more confident that honest people will be at hand when we need them, and Algorand guarantee that, when this is the case,
Ideal Hashing. We shall rely on an efficiently computable cryptographic hash function, H, that maps arbitrarily long strings to binary strings of fixed length. Following a long tradition, we model H as a random oracle, essentially a function mapping each possible string s to a randomly and independently selected (and then fixed) binary string, H(s), of the chosen length.
In our described embodiments, H has 256-bit long outputs. Indeed, such length is short enough to make the system efficient and long enough to make the system secure. For instance, we want H to be collision-resilient. That is, it should be hard to find two 5 different strings x and y such that H(x)=H(y). When H is a random oracle with 256-bit long outputs, finding any such pair of strings is indeed difficult. (Trying at random, and relying on the birthday paradox, would require 2256/2=2128 trials.)
Digital Signing. Digital signatures allow users to to authenticate information to each other without sharing any sharing any secret keys. A digital signature scheme consists of three fast algorithms: a probabilistic key generator G, a signing algorithm S, and a verification algorithm V.
Given a security parameter k, a sufficiently high integer, a user i uses G to produce a pair of k-bit keys (i.e., strings): a “public” key pki and a matching “secret” signing key ski. Crucially, a public key does not “betray” its corresponding secret key. That 15 is, even given knowledge of phi, no one other than i is able to compute ski in less than astronomical time.
User i uses ski to digitally sign messages. For each possible message (binary string) m, i first hashes m and then runs algorithm S on inputs II(m) and ski so as to produce the k-bit string
The binary string sigpk
Everyone knowing pki can use it to verify the digital signatures produced by i. Specifically, on inputs (a) the public key pki of a player i, (b) a message m, and (c) a string s, that is, i's alleged digital signature of the message m, the verification algorithm V outputs either YES or NO.
The properties we require from a digital signature scheme are:
Accordingly, to prevent anyone else from signing messages on his behalf, a player i must keep his signing key ski secret (hence the term “secret key”), and to enable anyone to verify the messages he does sign, i has an interest in publicizing his key pki (hence the term “public key”).
Signatures with Message Retrievability In general, a message m is not retrievable from its signature sigi(m). In order to virtually deal with digital signatures that satisfy the conceptually convenient “message retrievability” property (i.e., to guarantee that the signer and the message are easily computable from a signature, we define
SIGpk
Unique Digital Signing. We also consider digital signature schemes (G, S, V) satisfying the following additional property.
s≠s′ and V(pk′, m, s)=V(pk′, m, s′)=1.
Algorand tries to mimic the following payment system, based on an idealized public ledger.
S
0=(pk1, a1), . . . , (pkj, aj),
which is assumed to be common knowledge in the system.
=SIGpk(pk ,pk′, a′, I, H()),
where I represents any additional information deemed useful but not sensitive (e.g., time information and a payment identifier), and any additional information deemed sensitive (e.g., the reason for the payment, possibly the identities of the owners of pk and the pk′, and so on).
We refer to pk (or its owner) as the payer, to each pk′ (or its owner) as a payee, and to a′ as the amount of the payment .
Free Joining Via Payments. Note that users may join the system whenever they want by generating their own public/secret key pairs. Accordingly, the public key pk′ that appears in the payment above may be a newly generated public key that had never “owned” any money before.
L=PAY1,PAY2, . . . ,
Each block PAYr+1 consists of the set of all payments made since the appearance of block PAYr. In the ideal system, a new block appears after a fixed (or finite) amount of time.
In Bitcoin and similar systems, the money owned by a public key pk is segregated into separate amounts, and a payment made by pk must transfer such a segregated amount a in its entirety. If pk wishes to transfer only a fraction a′<a of a to another key, then it must also transfer the balance, the unspent transaction output, to another key, possibly pk itself.
Algorand also works with keys having segregated amounts. However, in order to focus on the novel aspects of Algorand, it is conceptually simpler to stick to our simpler forms of payments and keys having a single amount associated to them.
In the ideal system, an active user continually stores and updates the latest status information, or he would otherwise have to reconstruct it, either from scratch, or from the last time he computed it. (Yet, we later on show how to augment Algorand so as to enable its users to reconstruct the current status in an efficient manner.)
Keys, Users, and Owners Unless otherwise specified, each public key (“key” for short) is long-term and relative to a digital signature scheme with the uniqueness property. A public key i joins the system when another public key j already in the system makes a payment to i.
For color, we personify keys. We refer to a key i as a “he”, say that i is honest, that i sends and receives messages, etc. User is a synonym for key. When we want to distinguish a key from the person to whom it belongs, we respectively use the term “digital key” and “owner”.
Permissionless and Permissioned Systems. A system is permissionless, if a digital key is free to join at any time and an owner can own multiple digital keys; and its permissioned, otherwise.
Unique Representation Each object in Algorand has a unique representation. In particular, each set {(x, y, z, . . . ):x ϵ X, y ϵ Y, z ϵ Z, . . . } is ordered in a pre-specified manner: e.g., first lexicographically in x, then in y, etc.
Same-Speed Clocks There is no global clock: rather, each user has his own clock. User clocks need not be synchronized in any way. We assume, however, that they all have the same speed.
For instance, when it is 12 pm according to the clock of a user i, it may be 2:30 pm according to the clock of another user j, but when it will be 12:01 according to i's clock, it will 2:31 according to j's clock. That is, “one minute is the same (sufficiently, essentially the same) for every user”.
Rounds Algorand is organized in logical units, r=0,1, . . . , called rounds.
We consistently use superscripts to indicate rounds. To indicate that a non-numerical quantity Q (e.g., a string, a public key, a set, a digital signature, etc.) refers to a round r, we simply write Qr. Only when Q is a genuine number (as opposed to a binary string interpretable as a number), do we write Q(r), so that the symbol r could not be interpreted as the exponent of Q.
At (the start of a) round r>0, the set of all public keys is PKr, and the system status is
S
r={(i, ai(r), . . . ):i ϵ PKr},
where ai(r) is the amount of money available to the public key i. Note that PKr is deducible from Sr, and that Sr may also specify other components for each public key i. For round 0, PK0 is the set of initial public keys, and S0 is the initial status. Both PK0 and S0 are assumed to be common knowledge in the system. For simplicity, at the start of round r, so are PK1, . . . , PKr and S1, . . . , Sr.
In a round r, the system status transitions from Sr to Sr+1:symbolically,
Round r: Sr→Sr+1.
Payments In Algorand, the users continually make payments (and disseminate them in the way described in subsection 2.7). A payment of a user i ϵ PKr has the same format and semantics as in the Ideal System. Namely,
=SIGi(i, i′, a, I, H()).
Payment is individually valid at a round r (is a round-r payment, for short) if (1) its amount a is less than or equal to ai(r), and (2) it does not appear in any official payset PAYr′ for r′ <r. (As explained below, the second condition means that has not already become effective.
A set of round-r payments of i is collectively valid if the sum of their amounts is at most ai(r)).
Paysets A round-r payset is a set of round-r payments such that, for each user i, the payments of i in (possibly none) are collectively valid. The set of all round-r paysets is (r). A round-r payset is maximal if no superset of is a round-r payset.
We actually suggest that a payment also specifies a round ρ, =SIGi(p, i, i′, a, I, H()), and cannot be valid at any round outside [ρ, ρ+k], for some fixed non-negative integer k.3 3This simplifies checking whether has become “effective” (i.e., it simplifies determining whether some payset PAYr contains . When k=0, if =SIGi(r, i, i′, a, I, H()), and ∉ PAYr, then i must re-submit .
Official Paysets For every round r, Algorand publicly selects (in a manner described later on) a single (possibly empty) payset, PAYr, the round's official payset. (Essentially, PAYr represents the round-r payments that have “actually” happened.)
As in the Ideal System (and Bitcoin), (1) the only way for a new user j to enter the system is to be the recipient of a payment belonging to the official payset PAYr of a given round r; and (2) PAYr determines the status of the next round, Sr+1, from that of the current round, Sr. Symbolically,
PAYr: Sr→Sr+1.
Specifically,
In sum, as in the Ideal System, each status Sr+1 is deducible from the previous payment history:
PAY0, . . , PAYr.
In Algorand0, the block Br corresponding to a round r specifies: r itself; the set of payments of round r, PAYr; a quantity (Qr−1), to be explained, and the hash of the previous block, H(Br−1). Thus, starting from some fixed block B0, we have a traditional blockchain:
B
1=(1, PAY1,(Q0),H(B0)), B2=(2, PAY2,(Q1),H(B1)), . . .
In Algorand, the authenticity of a block is actually vouched by a separate piece of information, a “block certificate” CERTr, which turns Br into a proven block,
Discussion As we shall see, CERTr consists of a set of digital signatures for H(Br), those of a majority of the members of SVr, together with a proof that each of those members indeed belongs to SVr. We could, of course, include the certificates CERTr in the blocks themselves, but find it conceptually cleaner to keep it separate.)
In Bitcoin each block must satisfy a special property, that is, must “contain a solution of a crypto puzzle”, which makes block generation computationally intensive and forks both inevitable and not rare. By contrast, Algorand's blockchain has two main advantages: it is generated with minimal computation, and it will not fork with overwhelmingly high probability. Each block Bi is safely final as soon as it enters the blockchain.
To analyze the security of Algorand we specify the probability, F, with which we are willing to accept that something goes wrong (e.g., that a verifier set SVr does not have an honest majority). As in the case of the output length of the cryptographic hash function H, also F is a parameter. But, as in that case, we find it useful to set F to a concrete value, so as to get a more intuitive grasp of the fact that it is indeed possible, in Algorand, to enjoy simultaneously sufficient security and sufficient efficiency. To emphasize that F is parameter that can be set as desired, in the first and second embodiments we respectively set
F=10−12 and F=10−18 .
Discussion Note that 10−12 is actually less than one in a trillion, and we believe that such a choice of F is adequate in our application. Let us emphasize that 10−12 is not the probability with which the Adversary can forge the payments of an honest user. All payments are digitally signed, and thus, if the proper digital signatures are used, the probability of forging a payment is far lower than 10−12, and is, in fact, essentially 0. The bad event that we are willing to tolerate with probability F is that Algorand's blockchain forks. Notice that, with our setting of F and one-minute long rounds, a fork is expected to occur in Algorand's blockchain as infrequently as (roughly) once in 1.9 million years. By contrast, in Bitcoin, a forks occurs quite often.
A more demanding person may set F to a lower value. To this end, in our second embodiment we consider setting F to 10−18. Note that, assuming that a block is generated every second, 1018 is the estimated number of seconds taken by the Universe so far: from the Big Bang to present time. Thus, with F=10−18, if a block is generated in a second, one should expect for the age of the Universe to see a fork.
Algorand is designed to be secure in a very adversarial model. Let us explain.
Honest and Malicious Users A user is honest if he follows all his protocol instructions, and is perfectly capable of sending and receiving messages. A user is malicious (i.e., Byzantine, in the parlance of distributed computing) if he can deviate arbitrarily from his prescribed instructions.
The Adversary The Adversary is an efficient (technically polynomial-time) algorithm, personified for color, who can immediately make malicious any user he wants, at any time he wants (subject only to an upperbound to the number of the users he can corrupt).
The Adversary totally controls and perfectly coordinates all malicious users. He takes all actions on their behalf, including receiving and sending all their messages, and can let them deviate from their prescribed instructions in arbitrary ways. Or he can simply isolate a corrupted user sending and receiving messages. Let us clarify that no one else automatically learns that a user i is malicious, although i's maliciousness may transpire by the actions the Adversary has him take.
This powerful adversary however,
Furthermore, his ability to attack honest users is bounded by one of the following assumption.
Honesty Majority of Money We consider a continuum of Honest Majority of Money (IIMM) assumptions: namely, for each non-negative integer k and real h>½,
HHMk>h: the honest users in every round r owned a fraction greater than h of all money in the system at round r−k.
Discussion. Assuming that all malicious users perfectly coordinate their actions (as if controlled by a single entity, the Adversary) is a rather pessimistic hypothesis. Perfect coordination among too many individuals is difficult to achieve. Perhaps coordination only occurs within separate groups of malicious players. But, since one cannot be sure about the level of coordination malicious users may enjoy, we'd better be safe than sorry.
Assuming that the Adversary can secretly, dynamically, and immediately corrupt users is also pessimistic. After all, realistically, taking full control of a user's operations should take some time.
The assumption HMMk>h implies, for instance, that, if a round (on average) is implemented in one minute, then, the majority of the money at a given round will remain in honest hands for at least two hours, if k=120, and at least one week, if k=10,000.
Note that the HMM assumptions and the previous Honest Majority of Computing Power assumptions are related in the sense that, since computing power can be bought with money, if malicious users own most of the money, then they can obtain most of the computing power.
We envisage message propagation—i.e., “peer to peer gossip”4—to be the only means of communication, and assume that every propagated message reaches almost all honest users in a timely fashion. We essentially assume that each message m propagated by honest user reaches, within a given amount of time that depends on the length of m, all honest users. (It actually suffices that m reaches a sufficiently high percentage of the honest users.) 4Essentially, as in Bitcoin, when a user propagates a message m, every active user i receiving m for the first time, randomly and independently selects a suitably small number of active users, his “neighbors”, to whom he forwards m, possibly until he receives an acknowledgement from them. The propagation of m terminates when no user receives m for the first time.
As already emphasized, Byzantine agreement is a key ingredient of Algorand. Indeed, it is through the use of such a BA protocol that Algorand is unaffected by forks. However, to be secure against our powerful Adversary, Algorand must rely on a BA protocol that satisfies the new player-replaceability constraint. In addition, for Algorand to be efficient, such a BA protocol must be very efficient.
BA protocols were first defined for an idealized communication model, synchronous complete networks (SC networks). Such a model allows for a simpler design and analysis of BA protocols. Accordingly, in this section, we introduce a new BA protocol, BA*, for SC networks and ignoring the issue of player replaceability altogether. The protocol BA* is a contribution of separate value. Indeed, it is the most efficient cryptographic BA protocol for SC networks known so far.
To use it within our Algorand protocol, we modify BA* a bit, so as to account for our different communication model and context.
We start by recalling the model in which BA* operates and the notion of a Byzantine agreement.
In a SC network, there is a common clock, ticking at each integral times r=1,2, . . .
At each even time click r, each player i instantaneously and simultaneously sends a single message mi,jr (possibly the empty message) to each player j, including himself. Each mi,jr is correctly received at time click r+1 by player j, together with the identity of the sender i.
Again, in a communication protocol, a player is honest if he follows all his prescribed instructions, and malicious otherwise. All malicious players are totally controlled and perfectly coordinated by the Adversary, who, in particular, immediately receives all messages addressed to malicious players, and chooses the messages they send.
The Adversary can immediately make malicious any honest user he wants at any odd time click he wants, subject only to a possible upperbound t to the number of malicious players. That is, the Adversary “cannot interfere with the messages already sent by an honest user i”, which will be delivered as usual.
The Adversary also has the additional ability to see instantaneously, at each even round, the messages that the currently honest players send, and instantaneously use this information to choose the messages the malicious players send at the same time tick.
The notion of Byzantine agreement might have been first introduced for the binary case, that is, when every initial value consists of a bit. However, it was quickly extended to arbitrary initial values. By a BA protocol, we mean an arbitrary-value one.
Definition 3.1. In a synchronous network, let be a n-player protocol, whose player set is common knowledge among the players, t a positive integer such that n>2t+1. We say that is an arbitrary-value (respectively, binary) (n, t)-Byzantine agreement protocol with soundness σ ∈ (0, 1) if, for every set of values V not containing the special symbol ⊥ (respectively, for V={0, 1}), in an execution in which at most t of the players are malicious and in which every player i starts with an initial value vi ∈ V, every honest player j halts with probability 1, outputting a value outi ∈ V ∪ {⊥} so as to satisfy, with probability at least σ, the following two conditions:
In our BA protocols, a player is required to count how many players sent him a given message in a given step. Accordingly, for each possible value v that might be sent,
#is(v)
(or just #i(v) when s is clear) is the number of players j from which i has received v in step s.
Recalling that a player i receives exactly one message from each player j, if the number of players is n, then, for all i and s, Σv, #is(v)=n.
In this section we present a new binary BA protocol, BBA*, which relies on the honesty of more than two thirds of the players and is very fast: no matter what the malicious players might do, each execution of its main loop not only is trivially executed, but brings the players into agreement with probability 1/3.
In BBA*, each player has his own public key of a digital signature scheme satisfying the unique-signature property. Since this protocol is intended to be run on synchronous complete network, there is no need for a player i to sign each of his messages.
Digital signatures are used to generate a sufficiently common random bit in Step 3. (In Algorand, digital signatures are used to authenticate all other messages as well.)
The protocol requires a minimal set-up: a common random string r, independent of the players' keys. (In Algorand, r is actually replaced by the quantity Qr.)
Protocol BBA* is a 3-step loop, where the players repeatedly exchange Boolean values, and different players may exit this loop at different times. A player i exits this loop by propagating, at some step, either a special value 0* or a special value 1*, thereby instructing all players to “pretend” they respectively receive 0 and 1 from i in all future steps. (Alternatively said: assume that the last message received by a player j from another player i was a bit b. Then, in any step in which he does not receive any message from i, j acts as if i sent him the bit b.)
The protocol uses a counter γ, representing how many times its 3-step loop has been executed. At the start of BBA*, γ=0. (One may think of γ as a global counter, but it 20 is actually increased by each individual player every time that the loop is executed.)
There are n≥3t 1, where t is the maximum possible number of malicious players. A binary string x is identified with the integer whose binary representation (with possible leadings 0s) is x; and 1sb(x) denotes the least significant bit of x.
(C
(C
(C
Theorem 3.1. Whenever n≥3t+1, BBA* is a binary (n,t)-BA protocol with soundness 1.
A proof of Theorem 3.1 can be found in https://people.csail.mit.edu/silvio/Selected-ScientificPapers/DistributedComputation/BYZANTINEAGREEMENTMADETRIVIAL. 15 pdf.
Let us recall, for arbitrary values, a notion of consensus much weaker than Byzantine 20 agreement.
Definition 3.2. Let be a protocol in which the set of all players is common knowledge, and each player i privately knows an arbitrary initial value v′i.
We say that is an (n, t)-graded consensus protocol if, in every execution with n players, at most t of which are malicious, every honest player i halts outputting a value-grade pair (vi, gi), where gi ∈ {0, 1, 2}, so as to satisfy the following three conditions:
The following two-step protocol GC is a graded consensus protocol in the literature. To match the steps of protocol Algorand′1 of section 4.1, we respectively name 2 and 3 the steps of GC. (Indeed, the first step of Algorand′1 is concerned with something else: namely, proposing a new block.)
S
S
O
Since protocol GC is a protocol in the literature, it is known that the following theorem holds.
Theorem 3.2. If n≥3t+1, then GC is a (n,t)-graded broadcast protocol.
We now describe the arbitrary-value BA protocol BA* via the binary BA protocol BBA* and the graded-consensus protocol GC. Below, the initial value of each player i is v′i.
STEPS 1 AND 2. Each player i executes GC, on input v′i, so as to compute a pair (vi, gi).
STEP 3, . . . Each player i executes BBA*—with initial input 0, if gi=2, and 1 otherwise—so as to compute the bit outi.
O
Theorem 3.3. Whenever n≥3t+1, BA* is a (n,t)-BA protocol with soundness 1.
Proof. We first prove Consistency, and then Agreement.
P
P
In case A, all honest players output ⊥ in BA*, and thus Agreement holds. Consider now case B. In this case, in the execution of BBA*, the initial bit of at least one honest player i is 0. (Indeed, if initial bit of all honest players were 1, then, by the Consistency property of BBA*, we would have outj=1 for all honest j.) Accordingly, after the execution of GC, i outputs the pair (v, 2) for some value v. Thus, by property 1 of graded consensus, gj>0 for all honest players j. Accordingly, by property 2 of graded consensus, vj=v for all honest players j. This implies that, at the end of BA*, every honest player j outputs v. Thus, Agreement holds also in case B.
Since both Consistency and Agreement hold, BA* is an arbitrary-value BA protocol. ▪
Protocol BA* works also in gossiping networks, and in fact satisfies the player replaceability property that is crucial for Algorand to be secure in the envisaged very adversarial model.
The Player Replaceability of BBA* and BA* Let us now provide some intuition of why the protocols BA* and BBA* can be adapted to be executed in a network 20 where communication is via peer-to-peer gossiping, satisfy player replaceability. For concreteness, assume that the network has 10M users and that each step x of BBA* (or BA*) is executed by a committee of 10,000 players, who have been randomly selected via secret cryptographic sortition, and thus have credentials proving of being entitled to send messages in step x. Assume that each message sent in a given step specifies the step number, is digitally signed by its sender, and includes the credential proving that its sender is entitled to speak in that step.
First of all, if the percentage h of honest players is sufficiently larger than ⅔ (e.g., 75%), then, with overwhelming probability, the committee selected at each step has the required ⅔ honest majority.
In addition, the fact that the 10,000-strong randomly selected committee changes at each step does not impede the correct working of either BBA* or BA*. Indeed, in either protocol, a player i in step s only reacts to the multiplicity with which, in Step s-1, he has received a given message m. Since we are in a gossiping network, all messages sent in Step s-1 will (immediately, for the purpose of this intuition) reach all users, including those selected to play in step s. Furthermore because all messages sent in step s-1 specify the step number and include the credential that the sender was indeed authorized to speak in step s-1. Accordingly, whether he happened to have been selected also in step s-1 or not, a user i selected to play in step s is perfectly capable of correctly counting the multiplicity with which he has received a correct step s-1 message. It does not at all matter whether he has been playing all steps so far or not. All users are in “in the same boat” and thus can be replaced easily by other users.
As discussed, at a very high level, a round of Algorand ideally proceeds as follows. First, a randomly selected user, the leader, proposes and circulates a new block. (This process includes initially selecting a few potential leaders and then ensuring that, at least a good fraction of the time, a single common leader emerges.) Second, a randomly selected committee of users is selected, and reaches Byzantine agreement on the block proposed by the leader. (This process includes that each step of the BA protocol is run by a separately selected committee.) The agreed upon block is then digitally signed by a given threshold (TH) of committee members. These digital signatures are propagated so that everyone is assured of which is the new block. (This includes circulating the credential of the signers, and authenticating just the hash of the new block, ensuring that everyone is guaranteed to learn the block, once its hash is made clear.)
In the next two sections, we present two embodiments of the basic Algorand design, Algorand′1 and Algorand′2, that respectively work under a proper majority-of-honest-users assumption. In Section 8 we show how to adopts these embodiments to work under a honest-majority-of-money assumption.
Algorand′1 only envisages that >⅔ of the committee members are honest. In addition, in Algorand′1, the number of steps for reaching Byzantine agreement is capped at a suitably high number, so that agreement is guaranteed to be reached with overwhelming probability within a fixed number of steps (but potentially requiring longer time than the steps of Algorand′2). In the remote case in which agreement is not yet reached by the last step, the committee agrees on the empty block, which is always valid.
Algorand′2 envisages that the number of honest members in a committee is always greater than or equal to a fixed threshold tH (which guarantees that, with overwhelming 5 probability, at least ⅔ of the committee members are honest). In addition, Algorand′2 allows Byzantine agreement to be reached in an arbitrary number of steps (but potentially in a shorter time than Algorand′1).
Those skilled in the art will realize that many variants of these basic embodiments can be derived. In particular, it is easy, given Algorand′2, to modify Algorand′1 so as to enable to reach Byzantine agreement in an arbitrary number of steps.
Both embodiments share the following common core, notations, notions, and parameters.
Objectives Ideally, for each round r, Algorand should satisfy the following properties:
1. Perfect Correctness. All honest users agree on the same block Br.
2. Completeness 1. With probability 1, the block Br has been chosen by a honest user.
(Indeed a malicious user may always choose a block whose payset contains the payments of just his “friends”.)
Of course, guaranteeing perfect correctness alone is trivial: everyone always chooses the official payset PAYr to be empty. But in this case, the system would have completeness 0. Unfortunately, guaranteeing both perfect correctness and completeness 1 is not easy in the presence of malicious users. Algorand thus adopts a more realistic objective. Informally, letting h denote the percentage of users who are honest, h>⅔, the goal of Algorand is
Guaranteeing, with overwhelming probability, perfect correctness and completeness close to h.
Privileging correctness over completeness seems a reasonable choice: payments not processed in one round can be processed in the next, but one should avoid forks, if possible.
Led Byzantine Agreement Disregarding excessive time and communication for a moment, perfect Correctness could be guaranteed as follows. At the start of round r, each user i proposes his own candidate block Bir. Then, all users reach Byzantine agreement on just one of the candidate blocks. As per our introduction, the BA protocol employed requires a ⅔ honest majority and is player replaceable. Each of its step can be executed by a small and randomly selected set of verifiers, who do not share any inner variables.
Unfortunately, this approach does not quite work. This is so, because the candidate blocks proposed by the honest users are most likely totally different from each other. Indeed, each honest user sees different payments. Thus, although the sets of payments seen by different honest users may have a lot of overlap, it is unlikely that all honest users will construct a propose the same block. Accordingly, the consistency agreement of the BA protocol is never binding, only the agreement one is, and thus agreement may always been reached on ⊥ rather than on a good block.
Algorand′ avoids this problem as follows. First, a leader for round r, r, is selected. Then, propagates his own candidate block, . Finally, the users reach agreement on the block they actually receive from . Because, whenever is honest, Perfect Correctness and Completeness 1 both hold, Algorand′ ensures that is honest with probability close to h.
Leader Selection In Algorand's, the rth block is of the form
B
r=(r, PAYr,(Qr−1), H(Br−1).
As already mentioned in the introduction, the quantity Qr−1 is carefully constructed so as to be essentially non-manipulatable by our very powerful Adversary. (Later on in this section, we shall provide some intuition about why this is the case.) At the start of a round r, all users know the blockchain so far, B0, . . . , Br−1, from which they deduce the set of users of every prior round: that is, PK1, . . . , PKr−1. A potential leader of round r is a user i such that
.H (SIGi (r,1, Qr−1))≤p.
Let us explain. Note that, since the quantity Qr−1 is deducible from block Br−1, because of the message retrievability property of the underlying digital signature scheme. Furthermore, the underlying signature scheme satisfies the uniqueness property. Thus, SIGi(r,1,Qr−1) is a binary string uniquely associated to i and r. Accordingly, since H is a random oracle, H (SIGi(r,1,Qr−1)) is a random 256-bit long string uniquely associated to i and r. The symbol “.” in front of H (SIGi(r,1,Qr−1)) is the decimal (in our case, 5 binary) point, so that ri .H (SIGi(r,1,Qr−1)) is the binary expansion of a random 256-bit number between 0 and 1 uniquely associated to i and r. Thus the probability that ri is less than or equal to p is essentially p.
The probability p is chosen so that, with overwhelming (i.e., 1−F) probability, at least one potential verifier is honest. (If fact, p is chosen to be the smallest such probability.)
Note that, since i is the only one capable of computing his own signatures, he alone can determine whether he is a potential verifier of round 1. However, by revealing his own credential, σirSIGi(r,1,Qr−1), i can prove to anyone to be a potential verifier of round r.
The leader r is defined to be the potential leader whose hashed credential is smaller that the hashed credential of all other potential leader j: that is, H()≤H(σjr,s).
Note that, since a malicious may not reveal his credential, the correct leader of round r may never be known, and that, barring improbable ties, is indeed the only leader of round r.
Let us finally bring up a last but important detail: a user i can be a potential leader (and thus the leader) of a round r only if he belonged to the system for at least k rounds. This guarantees the non-manipulatability of Qr and all future Q-quantities. In fact, one of the potential leaders will actually determine Qr.
Verifier Selection Each step s>1 of round r is executed by a small set of verifiers, SVr,s. Again, each verifier i ∈ SVr,s is randomly selected among the users already in the system k rounds before r, and again via the special quantity Qr−1. Specifically, i ∈ PKr−k is a verifier in SVr,s, if
.H(SIGi(r,s, Qr−1))≤p′.
Once more, only i knows whether he belongs to SVr,s, but, if this is the case, he could prove it by exhibiting his credential σir,s H(SIGi(r, s, Qr−1)). A verifier i ∈ SVr,s sends a message, mir,s, in step s of round r, and this message includes his credential σir,s, so as to enable the verifiers f the nest step to recognize that mir,s is a legitimate step-s message.
The probability p′ is chosen so as to ensure that, in SVr,s, letting # good be the number of honest users and # bad the number of malicious users, with overwhelming probability the following two conditions hold.
For embodiment Algorand′1:
(1) # good>2. # bad and
(2) # good+4. # bad <2n, where n is the expected cardinality of SVr,s.
For embodiment Algorand′2:
(1) # good>tH and
(2) # good+2 # bad<2tH, where tH is a specified threshold.
These conditions imply that, with sufficiently high probability, (a) in the last step of the BA protocol, there will be at least given number of honest players to digitally sign the new block Br, (b) only one block per round may have the necessary number of signatures, and (c) the used BA protocol has (at each step) the required ⅔ honest majority.
Clarifying Block Generation If the round-r leader is honest, then the corresponding block is of the form
B
r=(r, PAYr, (Qr−1), H(Br−1))
where the payset PAYr is maximal. (recall that all paysets are, by definition, collectively valid.)
Else (i.e., if is malicious), Br has one of the following two possible forms:
B
r=(r, PAYr, SIGi(Qr−1), H(Br−1)) and Br=Bεr(r, 0, Qr−1, H(Br−1)).
In the first form, PAYr is a (non-necessarily maximal) payset and it may be PAYr=0; and i is a potential leader of round r. (However, i may not be the leader . This may indeed happen if if keeps secret his credential and does not reveal himself.)
The second form arises when, in the round-r execution of the BA protocol, all honest players output the default value, which is the empty block Bεr in our application. (By definition, the possible outputs of a BA protocol include a default value, generically denoted by ⊥. See section 3.2.) Note that, although the paysets are empty in both cases, Br=(r, 0, SIGi(Qr−1); H(Br−1)) and Bεr are syntactically different blocks and arise in two different situations: respectively, “all went smoothly enough in the execution of the BA protocol”, and “something went wrong in the BA protocol, and the default value was output”.
Let us now intuitively describe how the generation of block Br proceeds in round r of Algorand′. In the first step, each eligible player, that is, each player i ∈ PKr−k, checks whether he is a potential leader. If this is the case, then i is asked, using of all the payments he has seen so far, and the current blockchain, B0, . . . , Br−1, to secretly prepare a maximal payment set, PAYir, and secretly assembles his candidate block, Br=(r, PAYir, SIGi(Qr−1), H (Br−1)). That, is, not only does he include in Bir, as its second component, the just prepared payset, but also, as its third component, his own signature of Qr−1, the third component of the last block, Br−1. Finally, he propagates his round-r-step-1 message, mir,1, which includes (a) his candidate block Bir, (b) his proper signature of his candidate block (i.e., his signature of the hash of Bir, and (c) his own credential σir,1, proving that he is indeed a potential verifier of round r.
(Note that, until an honest i produces his message mir,1, the Adversary has no clue that i is a potential verifier. Should he wish to corrupt honest potential leaders, the Adversary might as well corrupt random honest players. However, once he sees mir,1, since it contains i's credential, the Adversary knows and could corrupt i, but cannot prevent mir,1, which is virally propagated, from reaching all users in the system.)
In the second step, each selected verifier j ∈ SVr,2 tries to identify the leader of the round. Specifically, j takes the step-1 credentials, , σi
Recall that each considered credential is a digital signature of Qr−1, that SIGi(r,1, Qr−1) is uniquely determined by i and Qr−1, that H is random oracle, and thus that each H(SIGi (r, 1, Qr−1) is a random 256-bit long string unique to each potential leader i of round r.
From this we can conclude that, if the 256-bit string Qr−1 were itself randomly and independently selected, than so would be the hashed credentials of all potential leaders of round r. In fact, all potential leaders are well defined, and so are their credentials (whether actually computed or not). Further, the set of potential leaders of round r is a random subset of the users of round r−k, and an honest potential leader i always properly constructs and propagates his message mir, which contains i's credential. Thus, since the percentage of honest users is h, no matter what the malicious potential leaders might do (e.g., reveal or conceal their own credentials), the minimum hashed potential-leader credential belongs to a honest user, who is necessarily identified by everyone to be the leader r of the round r. Accordingly, if the 256-bit string Qr−1 were itself randomly and independently selected, with probability exactly h (a) the leader r is honest and (b) j=r for all honest step-2 verifiers j.
In reality, the hashed credential are, yes, randomly selected, but depend on Qr−1, which is not randomly and independently selected. A careful analysis, however, guarantees that Qr−1 is sufficiently non-manipulatable to guarantee that the leader of a round is honest with probability h′ sufficiently close to h: namely, h′>h2(1+h−h2). For instance, if h=80%, then h′>0.7424.
Having identified the leader of the round (which they correctly do when the leader r is honest), the task of the step-2 verifiers is to start executing BA* using as initial values what they believe to be the block of the leader. Actually, in order to minimize the amount of communication required, a verifier j ∈ SVr,2 does not use, as his input value v′j to the Byzantine protocol, the block Bj that he has actually received from j (the user j believes to be the leader), but the the leader, but the hash of that block, that is, v′j=H(Bi). Thus, upon termination of the BA protocol, the verifiers of the last step do not compute the desired round-r block Br, but compute (authenticate and propagate) H(Br). Accordingly, since H(Br) is digitally signed by sufficiently many verifiers of the last step of the BA protocol, the users in the system will realize that H(Br) is the hash of the new block. However, they must also retrieve (or wait for, since the execution is quite asynchronous) the block Br itself, which the protocol ensures that is indeed available, no matter what the Adversary might do.
Asynchrony and Timing Algorand′1 and Algorand′2 have a significant degree of asynchrony. This is so because the Adversary has large latitude in scheduling the delivery of the messages being propagated. In addition, whether the total number of steps in a round is capped or not, there is the variance contribute by the number of steps actually taken.
As soon as he learns the certificates of B0, . . . , Br−1, a user i computes Qr−1 and starts working on round r, checking whether he is a potential leader, or a verifier in some step s of round r.
Assuming that i must act at step s, in light of the discussed asynchrony, i relies on various strategies to ensure that he has sufficient information before he acts.
For instance, he might wait to receive at least a given number of messages from the verifiers of the previous step (as in Algorand′1), or wait for a sufficient time to ensure that he receives the messages of sufficiently many verifiers of the previous step (as in Algorand′2).
The Seed Qr and the Look-Back Parameter κ Recall that, ideally, the quantities Qr should random and independent, although it will suffice for them to be sufficiently non-manipulatable by the Adversary.
At a first glance, we could choose Qr−1 to coincide with H (PAYr−1). An elementary analysis reveals, however, that malicious users may take advantage of this selection mechanism.5 Some additional effort shows that myriads of other alternatives, based 5We are at the start of round r−1. Thus, Qr−2=PAYr−2 is publicly known, and the Adversary privately knows who are the potential leaders he controls. Assume that the Adversary controls 10% of the users, and that, with very high probability, a malicious user w is the potential leader of round r−1. That is, assume that H (SIGw(r−2,1,Qr−2)) is so small that it is highly improbable an honest potential leader will actually be the leader of round r−1. (Recall that, since we choose potential leaders via a secret cryptographic sortition mechanism, the Adversary does not know who the honest potential leaders are.) The Adversary, therefore, is in the enviable position of choosing the payset PAY′ he wants, and have it become the official payset of round r 1. However, he can do more. He can also ensure that, with high probability, (*) one of his malicious users will be the leader also of round r, so that he can freely select what PAYr will be. (And so on. At least for a long while, that is, as long as these high-probability events really occur.) To guarantee (*), the Adversary acts as follows. Let PAY′ be the payset the Adversary prefers for round r−1. Then, he computes H(PAY′) and checks whether, for some already malicious player z, SIGz(r,1, H(PAY′)) is particularly small, that is, small enough that with very high probability z will be the leader of round r. If this is the case, then he instructs w to choose his candidate block to be Bir−1=(r−1, PAY′, H(Br−2). Else, he has two other malicious users x and y to keep on generating a new payment ′, from one to the other, until, for some malicious user z (or even for some fixed user z) H (SIGz(PAY′ ∪ {})) is particularly small too. This experiment will stop quite quickly. And when it does the Adversary asks w to propose the candidate block Bir−1=(r−1, PAY′ ∪ {}, H (Br−1)). on traditional block quantities are easily exploitable by the Adversary to ensure that malicious leaders are very frequent. We instead specifically and inductively define our brand new quantity Qr so as to be able to prove that it is non-manipulatable by the Adversary. Namely,
Qr H((Qr−1), r), if Br is not the empty block, and Qr H (Qr−1, r) otherwise.
The intuition of why this construction of Qr works is as follows. Assume for a moment that Qr−1 is truly randomly and independently selected. Then, will so be Qr? When r is honest the answer is (roughly speaking) yes. This is so because
H((·), r):{0,1}256→{0,1}256
is a random function. When is malicious, however, Qr is no longer univocally defined from Qr−1 and r. There are at least two separate values for Qr. One continues to be Qr H(SIG
The options for Qr may even be more numerous for the Adversary who controls a malicious r. For instance, let x, y, and z be three malicious potential leaders of round r such that
H (σxr,1)<H (σyr,1)<H (σzr,1)
and H (σzr,1) is particulary small. That is, so small that there is a good chance that H (σzr,1) is smaller of the hashed credential of every honest potential leader. Then, by asking x to hide his credential, the Adversary has a good chance of having y become the leader of round r−1. This implies that he has another option for Qr: namely, H (SIGy(Qr−1),r). Similarly, the Adversary may ask both x and y of withholding their credentials, so as to have z become the leader of round r−1 and gaining another option for Qr: namely, H (SIGz(Qr−1), r).
Of course, however, each of these and other options has a non-zero chance to fail, because the Adversary cannot predict the hash of the digital signatures of the honest potential users.
A careful, Markov-chain-like analysis shows that, no matter what options the Adversary chooses to make at round r−1, as long as he cannot inject new users in the system, he cannot decrease the probability of an honest user to be the leader of round r+40 much below h. This is the reason for which we demand that the potential leaders of round r are users already existing in round r−k. It is a way to ensure that, at round r−k, the Adversary cannot alter by much the probability that an honest user become the leader of round r. In fact, no matter what users he may add to the system in rounds r−k through r, they are ineligible to become potential leaders (and a fortiori the leader) of round r. Thus the look-back parameter k ultimately is a security parameter. (Although, as we shall see in section 7, it can also be a kind of “convenience parameter” as well.)
Ephemeral Keys Although the execution of our protocol cannot generate a fork, except with negligible probability, the Adversary could generate a fork, at the rth block, after the legitimate block r has been generated.
Roughly, once Br has been generated, the Adversary has learned who the verifiers of each step of round r are. Thus, he could therefore corrupt all of them and oblige them to certify a new block . Since this fake block might be propagated only after the legitimate one, users that have been paying attention would not be fooled.7 Nonetheless, would be syntactically correct and we want to prevent from being manufactured. 7Consider corrupting the news anchor of a major TV network, and producing and broadcasting today a newsreel showing secretary Clinton winning the last presidential election. Most of us would recognize it as a hoax. But someone getting out of a coma might be fooled.
We do so by means of a new rule. Essentially, the members of the verifier set SVr,s of a step s of round r use ephemeral public keys pkir,s to digitally sign their messages. These keys are single-use-only and their corresponding secret keys skir,s are destroyed once used. This way, if a verifier is corrupted later on, the Adversary cannot force him to sign anything else he did not originally sign. Naturally, we must ensure that it is impossible for the Adversary to compute a new key and convince an honest user that it is the right ephemeral key of verifier i ∈ SVr,s to use in step s.
For each round r and step s>1, SVr,s {i ∈ PKr−k: .H(SIGi(r,s,Qr−1))≤p}. Each user i ∈ PKr−k privately computes his signature using his long-term key and decides whether i ∈ SVr,s or not. If i ∈ SVr,s, then SIGi(r, s,Qr−1) is i's (r, s)—credential, compactly denoted by σir,s.
For the first step of round r, SVr,1 and σir,1 are similarly defined, with p replaced by p1. The verifiers in SVr,1 are potential leaders.
User i ∈ SVr,1 is the leader of round r, denoted by r, if H(σir,1)≤H(σjr,1) for all potential leaders j ∈ SVr,1. Whenever the hashes of two players' credentials are compared, in the unlikely event of ties, the protocol always breaks ties lexicographically according to the (long-term public keys of the) potential leaders.
By definition, the hash value of player r's credential is also the smallest among all users in PKr−k. Note that a potential leader cannot privately decide whether he is the leader or not, without seeing the other potential leaders' credentials.
Since the hash values are uniform at random, when SVr,1 is non-empty, r always exists and is honest with probability at least h. The parameter n1 is large enough so as to ensure that each SVr,1 is non-empty with overwhelming probability.
A non-empty block is of the form Br=(r, PAYr, (Qr−1), H(Br−1)), and an empty block is of the form B∈r=(r,0,Qr−1, H(Br−1)).
Note that a non-empty block may still contain an empty payset PAYr, if no payment occurs in this round or if the leader is malicious. However, a non-empty block implies that the identity of r, his credential and (Qr−1) have all been timely revealed. The protocol guarantees that, if the leader is honest, then the block will be non-empty with overwhelming probability.
Seed Qr.
If Br is non-empty, then QrH((Qr−1), r), otherwise Qr H(Qr−1,r).
Relationships among various parameters.
Example Choices of Important Parameters.
The outputs of H are 256-bit long.
Initialization of the Protocol.
The protocol starts at time 0 with r=0. Since there does not exist “B−1”or “CERT−1”, syntactically B−1 is a public parameter with its third component specifying Q−1, and all users know B−1 at time 0.
In this section, we construct a version of Algorand′ working under the following assumption.
H
In Section 8, we show how to replace the above assumption with the desired Honest Majority of Money assumption.
Notations
and there are at most m/3 trials. If all trials fail then Lr m/3. Lr will be used to upper-bound the time needed to generate block Br.
the number of signatures needed in the ending conditions of the protocol.
Relationships Among Various Parameters.
|HSVr,s|>2|MSVr,s| and |HSVr,s|+4|MSVr,s|<2n.
Example Choices of Important Parameters.
As already mentioned, we wish that a verifier i ∈ SVr,s digitally signs his message mir,s of step s in round r, relative to an ephemeral public key pkir,s, using an ephemeral secrete key skir,s that he promptly destroys after using. We thus need an efficient method to ensure that every user can verify that pkir,s is indeed the key to use to verify i's signature of mir,s. We do so by a (to the best of our knowledge) new use of identity-based signature schemes.
At a high level, in such a scheme, a central authority A generates a public master key, PMK, and a corresponding secret master key, SMK. Given the identity, U, of a player U, A computes, via SMK, a secret signature key skU relative to the public key U, and privately gives skU to U. (Indeed, in an identity-based digital signature scheme, the public key of a user U is U itself!) This way, if A destroys SMK after computing the secret keys of the users he wants to enable to produce digital signatures, and does not keep any computed secret key, then U is the only one who can digitally sign messages relative to the public key U. Thus, anyone who knows “U's name”, automatically knows U's public key, and thus can verify U's signatures (possibly using also the public master key PMK).
In our application, the authority A is user i, and the set of all possible users U coincides with the round-step pair (r, s) in—say—S={i}×{r′, . . . , r′+106}×{1, . . . , m+3}, where r′ is a given round, and m+3 the upperbound to the number of steps that may occur within a round. This way, pkir,s (i, r, s), so that everyone seeing i's signature
can, with overwhelming probability, immediately verify it for the first million rounds r following r′.
In other words, i first generates PMK and SMK. Then, he publicizes that PMK is i's master public key for any round r ∈ [r′, r′+106], and uses SMK to privately produce and store the secret key skir,s for each triple (i, r, s) ∈ S. This done, he destroys SMK. If he determines that he is not part of SVr,s, then i may leave skir,s alone (as the protocol does not require that he aunthenticates any message in Step s of round r). Else, i first uses skir,s to digitally sign his message mir,s, and then destroys skir,s.
Note that i can publicize his first public master key when he first enters the system. That is, the same payment that brings i into the system (at a round r′ or at a round close to r′), may also specify, at i's request, that i's public master key for any round r ∈ [r′, r′+106] is PMK e.g., by including a pair of the form (PMK, [r′, r′+106]).
Also note that, since m+3 is the maximum number of steps in a round, assuming that a round takes a minute, the stash of ephemeral keys so produced will last i for almost two years. At the same time, these ephemeral secret keys will not take i too long to produce. Using an elliptic-curve based system with 32B keys, each secret key is computed in a few microseconds. Thus, if m+3=180, then all 180M secret keys can be computed in less than one hour.
When the current round is getting close to r′+106, to handle the next million rounds, i generates a new (PMK′, SMK′) pair, and informs what his next stash of ephemeral keys is by—for example—having SIGi(PMK′,[r′+106+1, r′+2·106+1]) enter a new block, either as a separate “transaction” or as some additional information that is part of a payment. By so doing, i informs everyone that he/she should use PMK′ to verify i's ephemeral signatures in the next million rounds. And so on.
(Note that, following this basic approach, other ways for implementing ephemeral keys without using identity-based signatures are certainly possible. For instance, via Merkle trees.10) 10In this method, i generates a public-secret key pair (pkir,s, skir,s) for each round-step pair (r, s) in—say—{r′, . . . , r′+106}×{1, . . . , m+3}. Then he orders these public keys in a canonical way, stores the jth public key in the jth leaf of a Merkle tree, and computes the root value Ri, which he publicizes. When he wants to sign a message relative to key pkir,s, i not only provides the actual signature, but also the authenticating path for pkir,s relative to Ri. Notice that this authenticating path also proves that pkir,s is stored in the jth leaf. Form this idea, the rest of the details can be easily filled.
Other ways for implementing ephemeral keys are certainly possible—e.g., via Merkle trees.
5.3 Matching the Steps of Algorand′1 with those of BA*
As we said, a round in Algorand′1 has at most m+3 steps.
STEP 1. In this step, each potential leader i computes and propagates his candidate block Bir, together with his own credential, σir,1.
(H(Bir)).
(H(Bir)), he could have propagated
(H(Bir)). However, in our analysis we need to have explicit access to
(H(Bir)).
STEPS 2. In this step, each verifier i sets ir to be the potential leader whose hashed credential is the smallest, and Bir to be the block proposed by ir. Since, for the sake of efficiency, we wish to agree on H(Br), rather than directly on Br, i propagates the message he would have propagated in the first step of BA* with initial value v′i=H(Bir). That is, he propagates v′i, after ephemerally signing it, of course. (Namely, after signing it relative to the right ephemeral public key, which in this case is pkir,2.) Of course too, i also transmits his own credential.
Since the first step of BA* consists of the first step of the graded consensus protocol GC, Step 2 of Algorand′ corresponds to the first step of GC.
STEPS 3. In this step, each verifier i ∈ SVr,2 executes the second step of BA*. That is, he sends the same message he would have sent in the second step of GC. Again, i's message is ephemerally signed and accompanied by i's credential. (From now on, we shall omit saying that a verifier ephemerally signs his message and also propagates his credential.)
STEP 4. In this step, every verifier i ∈ SVr,4 computes the output of GC, (vi, gi), and ephemerally signs and sends the same message he would have sent in the third step of BA*, that is, in the first step of BBA*, with initial bit 0 if gi=2, and 1 otherwise.
STEP s=5, . . . , m+2. Such a step, if ever reached, corresponds to step s−1 of BA*, and thus to step s−3 of BBA*.
Since our propagation model is sufficiently asynchronous, we must account for the possibility that, in the middle of such a step s, a verifier i ∈ SVr,s is reached by information proving him that block Br has already been chosen. In this case, i stops his own execution of round r of Algorand′, and starts executing his round-(r+1) instructions.
Accordingly, the instructions of a verifier i ∈ SVr,s, in addition to the instructions corresponding to Step s−3 of BBA*, include checking whether the execution of BBA* has halted in a prior Step s′. Since BBA* can only halt is a Coin-Fixed-to-0 Step or in a Coin-Fixed-to-1 step, the instructions distinguish whether
A (Ending Condition 0): s′−2≡0 mod 3, or
B (Ending Condition 1): s′−2≡1 mod 3.
In fact, in case A, the block Br is non-empty, and thus additional instructions are necessary to ensure that i properly reconstructs Br, together with its proper certificate CERTr. In case B, the block Br is empty, and thus i is instructed to set Br=Bεr=(r,0,Qr−1,H(Br−1)), and to compute CERTr.
If, during his execution of step s, i does not see any evidence that the block Br has already been generated, then he sends the same message he would have sent in step s−3 of BBA*.
STEP m+3. If, during step m+3, i ∈ SVr,m+3 sees that the block Br was already generated in a prior step s′, then he proceeds just as explained above.
Else, rather then sending the same message he would have sent in step m of BBA*, i is instructed, based on the information in his possession, to compute Br and its corresponding certificate CERTr.
Recall, in fact, that we upperbound by m|3 the total number of steps of a round.
Recall that, in each step s of a round r, a verifier i ∈ SVr,s uses his long-term public-secret key pair to produce his credential, σir,sSIGi(r,s,Qr−1), as well as SIGi(Qr−1) in case s=1. Verifier i uses his ephemeral secret key skir,s to sign his (r, s)-message mir,s. For simplicity, when r and s are clear, we write esigi(x) rather than
(x) to denote i's proper ephemeral signature of a value x in step s of round r, and write ESIGi(x) instead of
(x) to denote (i,x, esigi(x)).
Instructions for every user i ∈ PKr−k: User i starts his own Step 1 of round r as soon as he knows Br−1.
Remark. In practice, to shorten the global execution of Step 1, it is important that the (r, 1)-messages are selectively propagated. That is, for every user i in the system, for the first (r, 1)-message that he ever receives and successfully verifies,11 player i propagates it as usual. For all the other (r, 1)-messages that player i receives and successfully verifies, he propagates it only if the hash value of the credential it contains is the smallest among the hash values of the credentials contained in all (r, 1)-messages he has received and successfully verified so far. Furthermore, as suggested by Georgios Vlachos, it is useful that each potential leader i also propagates his credential σir,1 separately: those small messages travel faster than blocks, ensure timely propagation of the mjr,1's where the contained credentials have small hash values, while make those with large hash values disappear quickly. 11That is, all the signatures are correct and both the block and its hash are valid although i does not check whether the included payset is maximal for its proposer or not.
Instructions for every user i ∈ PKr−k: User i starts his own Step 2 of round r as soon as he knows Br−1.
Instructions for every user i ∈ PKr−k: User i starts his own Step 3 of round r as soon as he knows Br−1.
Instructions for every user i ∈ PKr−k: User i starts his own Step 4 of round r as soon as he knows Br−1.
Step s, 5≤s≤m+2, s−2≡0 mod 3: A Coin-Fixed-To-0 Step of BBA*
Instructions for every user i ∈ PKr−k: User i starts his own Step s of round r as soon as he knows Br−1.
+1 valid message mjr,s′−1=(ESIGj(0), ESIGj(v), σjr,s′−1),17 and 17Such a message from player j is counted even if player i has also received a message from j signing for 1. Similar things for Ending Condition 1. As shown in the analysis, this is done to ensure that all honest users know Br within time λ from each other.
Else, he sets bi0.
He computes the message mir,s (ESIGi(bi),ESIGi(vi), σir,s), destroys his ephemeral secret key skir,s, and then propagates mir,s.
Step s, 6<s<m+2, s−2≡1 mod 3: A Coin-Fixed-To-1 Step of BBA*
Instructions for every user i ∈ PKr−k: User i starts his own Step s of round r as soon as he knows Br−1.
Step s, 7≤s≤m+2, s−2≡2 mod 3: A Coin-Genuinely-Flipped Step of BBA*
Instructions for every user i ∈ PKr−k: User i starts his own Step s of round r as soon as he knows Br−1.
20With overwhelming probability BBA* has ended before this step, and we specify this step for completeness.
Instructions for every user i ∈ PKr−k: User i starts his own Step m+3 of round r as soon as he knows Br−1.
Otherwise, at the end of the wait, user i does the following.
He sets outi 1 and BrBεr.
He computes the message mir,m+3=(ESIGi(outi), ESIGi(H (Br)), πir,m+3), destroys his ephemeral secret key skir,m+3, and then propagates mir,m+3 to certify Br.21 21A certificate from Step m+3 does not have to include ESIGi(outi). We include it for uniformity only: the certificates now have a uniform format no matter in which step they are generated.
Instructions for every user i in the system: User i starts his own round r as soon as he knows Br−1, and waits for block information as follows.
We introduce the following notations for each round r≥0, used in the analysis.
Note that T0=0 by the initialization of the protocol. For each s≥1 and i ∈ SVr,s, recall that αir,s and βir,s are respectively the starting time and the ending time of player i's step s. Moreover, recall that ts=(2s−3)λ+Λ for each 2≤s≤m+3. In addition, let I0{0} and t10.
Finally, recall that Lr≤m/3 is a random variable representing the number of Bernoulli trials needed to see a 1, when each trial is 1 with probability
and there are at most m/3 trials. If all trials fail then Lrm/3.
In the analysis we ignore computation time, as it is in fact negligible relative to the 5 time needed to propagate messages. In any case, by using slightly larger λ and Λ, the computation time can be incorporated into the analysis directly. Most of the statements below hold “with overwhelming probability,” and we may not repeatedly emphasize this fact in the analysis.
Theorem 5.1. The following properties hold with overwhelming probability for each round r≥0:
Block-Generation and True Latency. The time to generate block Br is defined to be Tr+1−Tr. That is, it is defined to be the difference between the first time some honest user learns Br and the first time some honest user learns Br−1. When the round-r leader is honest, Property 2 our main theorem guarantees that the exact time to generate Br is 8λ+Λ time, no matter what the precise value of h>⅔ may be. When the leader is malicious, Property 3 implies that the expected time to generate Br is upperbounded by
again no matter the precise value of h.22 However, the expected time to generate Br depends on the precise value of h. Indeed, by Property 4, ph=h2(1+h−h2) and the leader is honest with probability at least ph, thus
For instance, if h=80%, then [Tr+1−Tr]≤12.7λ+Λ.
Proof of Theorem 5.1. We prove Properties 1-3 by induction: assuming they hold for round r−1 (without loss of generality, they automatically hold for “round-1” when r=0), we prove them for round r.
Since Br−1 is uniquely defined by the inductive hypothesis, the set SVr,s is uniquely defined for each step s of round r. By the choice of n1, SVr,1≠0 with overwhelming probability. We now state the following two lemmas, proved in Sections 5.7 and 5.8. Throughout the induction and in the proofs of the two lemmas, the analysis for round 0 is almost the same as the inductive step, and we will highlight the differences when they occur.
Lemma 5.2. [Completeness Lemma] Assuming Properties 1-3 hold for round r−1, when the leader r is honest, with overwhelming probability,
Lemma 5.3./[Soundness Lemma] Assuming Properties 1-3 hold for round r−1, when the leader r is malicious, with overwhelming probability, all honest users agree on the same block Br, Tr+1≤Tr+(6Lr+10)λ+Λ and all honest users know Br in the time interval Ir+1.
Properties 1-3 hold by applying Lemmas 5.2 and 5.3 to r=0 and to the inductive step. Finally, we restate Property 4 as the following lemma, proved in Section 5.9.
Lemma 5.4. Given Properties 1-3 for each round before r, ph=h2(1+h−h2) for Lr, and the leader r is honest with probability at least ph.
Combining the above three lemmas together, Theorem 5.1 holds.
The lemma below states several important properties about round r given the inductive hypothesis, and will be used in the proofs of the above three lemmas.
Lemma 5.5. Assume Properties 1-3 hold for round r−1. For each step s≥1 of round r and each honest verifier i ∈ HSVr,s, we have that
Proof. Property (a) follows directly from the inductive hypothesis, as player i knows Br−1 in the time interval Ir and starts his own step s right away. Property (b) follows directly from (a): since player i has waited an amount of time ts before acting, βir,s=αir,s+ts. Note that αir,s=0 for r=0.
We now prove Property (c). If s=2, then by Property (b), for all verifiers j ∈ HSVr,1 we have
βir,s=αir,s+ts≥Tr+ts=Tr+λ+Λ≥βjr,1+Λ.
Since each verifier j ∈ HSVr,1 sends his message at time βjr,1 and the message reaches all honest users in at most A time, by time βir,s player i has received the messages sent by all verifiers in HSVr,1 as desired.
If s>2, then ts=ts−1+2λ. By Property (b), for all steps s′<s and all verifiers j ∈ HSVr,s′,
βir,s=αir,s+ts≥Tr+ts=Tr+ts−1+2λ≥Tr+ts′+2λ=Tr+λ+ts′+λ≥βjr,s′+λ.
Since each verifier j ∈ HSVr,s′ sends his message at time βjr,s′ and the message reaches all honest users in at most λ time, by time βir,s player i has received all messages sent by all honest verifiers in HSVr,s′ for all s′<s. Thus Property (c) holds.
Finally, we prove Property (d). Note that the verifiers j ∈ SVr,s−1 sign at most two things in Step s−1 using their ephemeral secret keys: a value vj of the same length as the output of the hash function, and also a bit bj ∈{0, 1} if s−1≥4. That is why in the statement of the lemma we require that v and v′ have the same length: many verifiers may have signed both a hash value v and a bit b, thus both pass the ⅔ threshold.
Assume for the sake of contradiction that there exist the desired verifiers i, i′ and values v, v′. Note that some malicious verifiers in MSVr,s−1 may have signed both v and v′, but each honest verifier in HSVr,s−1 has signed at most one of them. By Property (c), both i and i′ have received all messages sent by all honest verifiers in HSVr,s−1.
Let HSVr,s−1(v) be the set of honest (r, s−1)-verifiers who have signed v, MSVir,s−1 the set of malicious (r, s−1)-verifiers from whom i has received a valid message, and MSVir,s−1(v) the subset of MSVir,s−1 from whom i has received a valid message signing v. By the requirements for i and v, we have
We first show
|MSVir,s−1(v)|≤|HSVr,s−1(v)|. (2)
Assuming otherwise, by the relationships among the parameters, with overwhelming probability |HSVr,s−1|>2|MSVr,s−1|≥2|MSVir,s−1|, thus
contradicting Inequality 1.
Next, by Inequality 1 we have
2|HSVr,s−1|+2|MSVir,s−1|<3|HSVr,s−1(v)|+3|MSVir,s−1(v)|≤3|HSVr,s−1(v)|+2|MSVir,s−1|+|MSVir,s−1(v)|.
Combining with Inequality 2,
2|HSVr,s−1|<3|HSVr,s−1(v)|+|MSVir,s−1(v)|≤4|HSVr,s−1(v)|,
which implies
Similarly, by the requirements for i′ and v′, we have
Since an honest verifier j ∈ HSVr,s−1 destroys his ephemeral secret key skjr,s−1 before propagating his message, the Adversary cannot forge j's signature for a value that j did not sign, after learning that j is a verifier. Thus, the two inequalities above imply |HSVr,s−1|≥|HSVr,s−1(v)|+|HSVr,s−1(v′)|>|HSVr,s−1|, a contradiction. Accordingly, the desired i, i′, v, v′ do not exist, and Property (d) holds. ▪
Lemma 5.2. [Completeness Lemma, restated] Assuming Properties 1-3 hold for round r−1, when the leader r is honest, with overwhelming probability,
Proof. By the inductive hypothesis and Lemma 5.5, for each step s and verifier i ∈ HSVr,s, αir,s ∈ Ir. Below we analyze the protocol step by step.
Step 1. By definition, every honest verifier i ∈ HSVr,1 propagates the desired message mir,1 at time βir,1=αir,1, where mir,1=(Bir, esigi(H(Bir)), σir,1), Bir=(r, PAYir, SIGi(Qr−1), H (Br−1)), and PAYir is a maximal payset among all payments that i has seen by time αir,1.
Step 2. Arbitrarily fix an honest verifier i ∈ HSVr,2. By Lemma 5.5, when player i is done waiting at time βir,2=αir,2+t2, he has received all messages sent by verifiers in HSVr−1, including . By the definition of r, there does not exist another player in PKr−k whose credential's hash value is smaller than H(). Of course, the Adversary can corrupt r after seeing that H() is very small, but by that time player r has destroyed his ephemeral key and the message has been propagated. Thus verifier i sets his own leader to be player r. Accordingly, at time βir,2, verifier i propagates mir,2=(ESIGi(v′i), σir,2) where v′i=H(B⊇
Step 3. Arbitrarily fix an honest verifier i ∈ HSVr,3. By Lemma 5.5, when player i is done waiting at time βir,3=αir,3+t3, he has received all messages sent by verifiers in HSVr,2.
By the relationships among the parameters, with overwhelming probability |HSVr,2|>2|MSVr,2|. Moreover, no honest verifier would sign contradicting messages, and the Adversary cannot forge a signature of an honest verifier after the latter has destroyed his corresponding ephemeral secret key. Thus more than ⅔ of all the valid (r, 2)-messages i has received are from honest verifiers and of the form mjr,2=(ESIGj(H ()), σjr,2), with no contradiction.
Accordingly, at time βir,3 player i propagates mir,3=(ESIGi(v′), σir,3), where v′=H().
Step 4. Arbitrarily fix an honest verifier i ∈ HSVr,4. By Lemma 5.5, player i has received all messages sent by verifiers in HSVr,3 when he is done waiting at time βir,4=αir,4+t4. Similar to Step 3, more than ⅔ of all the valid (r, 3)-messages i has received are from honest verifiers and of the form mjr,3=(ESIGj(H ()), σjr,3).
Accordingly, player i sets vi=H(), gi=2 and bi=0. At time βir,79 =αir,Λ+t4 he propagates mir,4=(ESIGi(0), ESIGi(H()), σir,4).
Step 5. Arbitrarily fix an honest verifier i ∈ HSVr,5. By Lemma 5.5, player i would have received all messages sent by the verifiers in HSVr,4 if he has waited till time αir,5+t5. Note that |HSVr,4|≥tH.23 Also note that all verifiers in HSVr,4 have signed for H(). 23Strictly speaking, this happens with very high probability but not necessarily overwhelming However, this probability slightly effects the running time of the protocol, but does not affect its correctness. When h=80%, then |HSVr,4|≥tH with probability 1-10−8. If this event does not occur, then the protocol will continue for another 3 steps. As the probability that this does not occur in two steps is negligible, the protocol will finish at Step 8. In expectation, then, the number of steps needed is almost 5.
As |MSVr,4|<tH, there does not exist any v′≠H() that could have been signed by tH verifiers in SVr,4 (who would necessarily be malicious), so player i does not stop before he has received tH valid messages mjr,4=(ESIGj(0), ESIGj(H()), σjr,4). Let T be the time when the latter event happens. Some of those messages may be from malicious players, but because |MSVr,4<tH, at least one of them is from an honest verifier in HSVr,4 and is sent after time Tr|t4. Accordingly, T≥Tr|t4>Tr|λ|Λ≥(|Λ, and by time T player i has also received the message . By the construction of the protocol, player i stops at time βir,5=T without propagating anything; sets Br=;
and sets his own CERTr to be the set of (r, 4)-messages for 0 and H() that he has received.
Step s>5. Similarly, for any step s>5 and any verifier i ∈ HSVr,s, player i would have received all messages sent by the verifiers in HSVr,4 if he has waited till time αir,s+ts. By the same analysis, player i stops without propagating anything, setting Br= (and setting his own CERTr properly). Of course, the malicious verifiers may not stop and may propagate arbitrary messages, but because |MSVr,s|<tH, by induction no other v′could be signed by tH verifiers in any step 4≤s′<s, thus the honest verifiers only stop because they have received tH valid (r, 4)-messages for 0 and H(Bl
Reconstruction of the Round-r Block. The analysis of Step 5 applies to a generic honest user i almost without any change. Indeed, player i starts his own round r in the interval Ir and will only stop at a time T when he has received tH valid (r, 4)-messages for H(Bl
It only remains to show that all honest users finish their round r within the time interval Ir+1. By the analysis of Step 5, every honest verifier i ∈ HSVr,5 knows Br on or before αir,5+t5≤Tr+λ+t5=Tr+8λ+Λ. Since Tr+1 is the time when the first honest user ir knows Br, we have
Tr+1<Tr|8λ|Λ
as desired. Moreover, when player ir knows Br, he has already helped propagating the messages in his CERTr. Note that all those messages will be received by all honest users within time λ, even if player ir were the first player to propagate them. Moreover, following the analysis above we have Tr+1≥Tr+t4≥+Λ, thus all honest users have received by time Tr+1+λ. Accordingly, all honest users know Br in the time interval Ir+1=[Tr+1, Tr+1+λ].
Finally, for r=0 we actually have T1≤t4+λ=6λ+Λ. Combining everything together, Lemma 5.2 holds. ▪
Lemma 5.3. [Soundness Lemma, restated] Assuming Properties 1-3 hold for round r −1, when the leader r is malicious, with overwhelming probability, all honest users agree on the same block Br, Tr+1≤Tr+(6Lr+10)λ+Λ and all honest users know Br in the time interval Ir+1.
Proof. We consider the two parts of the protocol, GC and BBA*, separately.
GC. By the inductive hypothesis and by Lemma 5.5, for any step s ∈ {2, 3, 4} and any honest verifier i ∈ HSVr,ss, when player i acts at time βir,s=αir,s+ts, he has received all messages sent by all the honest verifiers in steps s′<s. We distinguish two possible cases for step 4.
Case 1. No verifier i ∈ HSVr,4 sets gi=2.
In this case, by definition bi=1 for all verifiers i ∈ HSVr,4. That is, they start with an agreement on 1 in the binary BA protocol. They may not have an agreement on their vi's, but this does not matter as we will see in the binary BA.
Case 2. There exists a verifier î ∈ HSVr,4 such that gi=2.
Indeed, since player î is honest and sets gi=2, more than ⅔ of all the valid messages mjr,3 he has received are for the same value v′≠⊥, and he has set vî=v′. By Property (d) in Lemma 5.5, for any other honest (r, 4)-verifier i, it cannot be that more than ⅔ of all the valid messages mjr,3 that i′ has received are for the same value v″≠v′. Accordingly, if i sets gi=2, it must be that i has seen >⅔ majority for v′ as well and set vi=v′, as desired.
Now consider an arbitrary verifier i ∈ HSVr,4 with gi<2. Similar to the analysis of Property (d) in Lemma 5.5, because player î has seen >⅔ majority for v′, more than 1/2|HSVr,3 honest (r, 3)-verifiers have signed v′. Because i has received all messages by honest (r, 3)-verifiers by time βir,4=αir,4+tr, he has in particular received more than 1/2|HSVr,3| messages from them for v′. Because HSVr,3|>2|MSVr,3|, i has seen >⅓ majority for v′. Accordingly, player i sets gi=1, and Property (1) holds. Does player i necessarily set vi=v′? Assume there exists a different value v″≠⊥ such that player i has also seen >⅓ majority for v″. Some of those messages may be from malicious verifiers, but at least one of them is from some honest verifier j ∈ HSVr,3: indeed, because |HSVr,3|>2|MSVr,3| and i has received all messages from HSVr,3, the set of malicious verifiers from whom i has received a valid (r, 3)-message counts for <⅓ of all the valid messages he has received.
By definition, player j must have seen >⅔ majority for v″ among all the valid (r, 2)-messages he has received. However, we already have that some other honest (r, 3)-verifiers have seen >⅔ majority for v′ (because they signed v′). By Property (d) of Lemma 5.5, this cannot happen and such a value v″ does not exist. Thus player i must have set vi=v′ as desired, and Property (2) holds. Finally, given that some honest (r, 3)-verifiers have seen >⅔ majority for v′, some (actually, more than half of) honest (r, 2)-verifiers have signed for v′ and propagated their messages. By the construction of the protocol, those honest (r, 2)-verifiers must have received a valid message from some player ∈ SVr,1 with v′=H(), thus Property (3) holds.
BBA*. We again distinguish two cases.
Case 1. All verifiers i ∈ HSVr,Λ have bi=1.
This happens following Case 1 of GC. As MSVr,4|<tH, in this case no verifier in SVr,5 could collect or generate tH valid (r, 4)-messages for bit 0. Thus, no honest verifier in HSVr,5 would stop because he knows a non-empty block Br.
Moreover, although there are at least tH valid (r, 4)-messages for bit 1, s′=5 does not satisfy s′−2≡1 mod 3, thus no honest verifier in HSVr,5 would stop because he knows Br=B∈r.
Instead, every verifier i ∈ HSVr,5 acts at time βir,5=αir,5+t5, by when he has received all messages sent by HSVr,4 following Lemma 5.5. Thus player i has seen >⅔ majority for 1 and sets bi=1.
In Step 6 which is a Coin-Fixed-To-1 step, although s′=5 satisfies s′−2≡0 mod 3, there do not exist tH valid (r, 4)-messages for bit 0, thus no verifier in HSVr,6 would stop because he knows a non-empty block Br. However, with s′=6, s′−2≡1 mod 3 and there do exist |HSVr,5|≥tH valid (r, 5)-messages for bit 1 from HSVr,5. For every verifier i ∈ HSVr,6, following Lemma 5.5, on or before time αir,6+t6 player i has received all messages from HSVr,5, thus i stops without propagating anything and sets Br=B∈r. His CERTr is the set of tH valid (r, 5)-messages mjr,5=(ESIGj(1), ESIGj(vj), σjr,5) received by him when he stops.
Next, let player i be either an honest verifier in a step s>6 or a generic honest user (i.e., non-verifier). Similar to the proof of Lemma 5.2, player i sets Br=B∈r and sets his own CERTr to be the set of tH valid (r, 5)-messages mjr,5=(ESIGj(1), ESIGj(vj), σjr,5) he has received.
Finally, similar to Lemma 5.2,
and all honest users know Br in the time interval Ir+1 because the first honest user i who knows Br has helped propagating the (r, 5)-messages in his CERTr.
Case 2. There exists a verifier î ∈ HSVr,Λ with bi=0.
This happens following Case 2 of GC and is the more complex case. By the analysis of GC, in this case there exists a valid message such that vi=H() for all i ∈ HSVr,4. Note that the verifiers in HSVr,4 may not have an agreement on their bi's.
For any step s ∈ {5, . . . , m+3} and verifier i ∈ HSVr,s, by Lemma 5.5 player i would have received all messages sent by all honest verifiers in HSVr,4 ∪ . . . ∪ HSVr,s−1 if he has waited for time ts.
We now consider the following event E: there exists a step s* ≥5 such that, for the first time in the binary BA, some player i* ∈ SVr,s* (whether malicious or honest) should stop without propagating anything. We use “should stop” to emphasize the fact that, if player i* is malicious, then he may pretend that he should not stop according to the protocol and propagate messages of the Adversary's choice.
Moreover, by the construction of the protocol, either (E.a) i* is able to collect or generate at least tH valid messages mjr,s′−1=(ESIGj(0), ESIGj(v), σjr,s′−1) for the same v and s′, with 5≤s′≤s* and s′−2≡0 mod 3; or
(E.b) i* is able to collect or generate at least tH valid messages mjr,s′−1=(ESIGj(1), ESIGj(vj), σjr,s′−1) for the same s′, with 6≤s′≤s* and s′−2≡1 mod 3.
Because the honest (r, s′−1)-messages are received by all honest (r, s′)-verifiers before they are done waiting in Step s′, and because the Adversary receives everything no later than the honest users, without loss of generality we have s′=s* and player i* is malicious. Note that we did not require the value v in E.a to be the hash of a valid block: as it will become clear in the analysis, v=H() in this sub-event. Below we first analyze Case 2 following event E, and then show that the value of s* is essentially distributed accordingly to Lr (thus event E happens before Step m+3 with overwhelming probability given the relationships for parameters). To begin with, for any step 5≤s<s*, every honest verifier i ∈ HSVr,s has waited time ts and set vi to be the majority vote of the valid (r, s−1)-messages he has received. Since player i has received all honest (r, s−1)-messages following Lemma 5.5, since all honest verifiers in HSVr,4 have signed H() following Case 2 of GC, and since |HSVr,s−1|>2|MSVr,s−1| for each s, by induction we have that player i has set
v
i
=H().
The same holds for every honest verifier i ∈ HSVr,s* who does not stop without propagating anything. Now we consider Step s* and distinguish four subcases.
Case 2.1.a. Event E.a happens and there exists an honest verifier i′ ∈ HSVr,s* who should also stop without propagating anything.
|HSV
r,s*−1|+4|MSVr,s*−1|<2n,
which comes from the relationships for the parameters. Accordingly, i does not see >⅔ majority for 1, and he sets bi=0 because Step s* is a Coin-Fixed-To-0 step. As we have seen, vi=H(). Thus i propagates (ESIGi(0), ESIGi(H()), σir,s) as we wanted to show.
T
r+1≤αi′r,s*+ts*≤Tr+λ+ts*.
Case 2.1.b. Event E.b happens and there exists an honest verifier i′ ∈ HSVr,s* who should also stop without propagating anything.
T
r+1≤αi′r,s*+ts*≤Tr+λ+ts*.
Case 2.2.a. Event E.a happens and there does not exist an honest verifier i′ ∈ HSVr,s* who should also stop without propagating anything.
T
r+1
≤T
r
+λ+t
s*+1.
T
r+1
≤T
r
+λ+t
s*+2.
Otherwise, all honest verifiers in Step s*+2 have received all the (r, s*+1)-messages for 0 and H() from HSVr,s*+1 after waiting time ts*+2, which leads to >⅔ majority. Thus all of them propagate their messages for 0 and H() accordingly: that is they do not “flip a coin” in this case. Again, note that they do not stop without propagating, because Step s*+2 is not a Coin-Fixed-To-0step.
T
r+1
≤T
r
+λ+t
s*+3.
Case 2.2.b. Event E.b happens and there does not exist an honest verifier i′ ∈ HSVr,s* who should also stop without propagating anything.
Thus, every verifier i ∈ HSVr,s* sets bi=1 and propagates mir,s*=(ESIGi(1), ESIGi(vi), σir,s*) at time αir,s*+ts*. Similar to Case 2.2.a, in at most 3 more steps (i.e., the protocol reaches Step s*+3, which is another Coin-Fixed-To-1 step), all honest users know Br=B∈r within the time interval Ir+1. Moreover, Tr+1 may be ≤Tr+λ+ts*+1, or ≤Tr+λ+ts*+2, or ≤Tr+λ+ts*+3, depending on when is the first time an honest verifier is able to stop without propagating.
Combining the four sub-cases, we have that all honest users know Br within the time interval Ir+1 with
T
r+1
≤T
r
+λ+t
s* in Cases 2.1.a and 2.1.b, and
T
r+1
≤T
r
+λ+t
s*+3 in Cases 2.2.a and 2.2.b.
It remains to upper-bound s* and thus Tr+1 for Case 2, and we do so by considering how many times the Coin-Genuinely-Flipped steps are actually executed in the protocol: that is, some honest verifiers actually have flipped a coin.
In particular, arbitrarily fix a Coin-Genuinely-Flipped step s′ (i.e., 7<s′<m|2 and s′−2≡2 mod 3), and let ′ arg min i ∈ SVr,s′−1 H(σjr,s′−1). For now let us assume s′<s*, because otherwise no honest verifier actually flips a coin in Step s′, according to previous discussions.
By the definition of SVr,s′−1, the hash value of the credential of ′ is also the smallest among all users in PKr−k. Since the hash function is a random oracle, ideally player ′ is honest with probability at least h. As we will show later, even if the Adversary tries his best to predict the output of the random oracle and tilt the probability, player ′ is still honest with probability at least ph=h2(1+h−h2). Below we consider the case when that indeed happens: that is, ′ ∈ HSVr,s′−1.
Note that every honest verifier i ∈ HSVr,s′ has received all messages from HSVr,s′−1 by time αir,s′+ts′. If player i needs to flip a coin (i.e., he has not seen >⅔ majority for the same bit b ∈ {0, 1}), then he sets bi=1sb(H()). If there exists another honest verifier i′ ∈ HSVr,s′ who has seen >⅔ majority for a bit b ∈ {0, 1}, then by Property (d) of Lemma 5.5, no honest verifier in HSVr,s′ would have seen >⅔ majority for a bit b′≠b. Since 1sb(H())=b with probability ½, all honest verifiers in HSVr,s′ reach an agreement on b with probability ½. Of course, if such a verifier i′ does not exist, then all honest verifiers in HSVr,s′ agree on the bit 1sb(H()) with probability 1.
Combining the probability for ′ ∈ HSVr,s′−1, we have that the honest verifiers in HSVr,s′ reach an agreement on a bit b ∈ {0, 1} with probability at least
Moreover, by induction on the majority vote as before, all honest verifiers in HSVr,s′ have their vi's set to be H(). Thus, once an agreement on b is reached in Step s′, Tr+1 is
either ≤Tr+λ+ts′+1 or ≤Tr+λ+ts′+2,
depending on whether b=0 or b=1, following the analysis of Cases 2.1.a and 2.1.b. In particular, no further Coin-Genuinely-Flipped step will be executed: that is, the verifiers in such steps still check that they are the verifiers and thus wait, but they will all stop without propagating anything. Accordingly, before Step s*, the number of times the Coin-Genuinely-Flipped steps are executed is distributed according to the random variable Lr. Letting Step s′ be the last Coin-Genuinely-Flipped step according to Lr, by the construction of the protocol we have
s′=4+3Lr.
When should the Adversary make Step s* happen if he wants to delay Tr+1 as much as possible? We can even assume that the Adversary knows the realization of Lr in advance. If s*>s′ then it is useless, because the honest verifiers have already reached an agreement in Step s′. To be sure, in this case s* would be s′+1 or s′+2, again depending on whether b=0 or b=1. However, this is actually Cases 2.1.a and 2.1.b, and the resulting Tr+1 is exactly the same as in that case. More precisely,
T
r+1
≤T
r
+λ+t
s*
≤T
r
+λ+t
s′+2.
If s*<s′−3−that is, s* is before the second last Coin Genuinely Flipped step—then by the analysis of Cases 2.2.a and 2.2.b,
T
r+1
≤T
r
+λ+t
s*+3
<T
r
+λ+t
s′,
That is, the Adversary is actually making the agreement on Br happen faster. If s*=s′−2 or s′−1 that is, the Coin-Fixed-To-0 step or the Coin-Fixed-To-1 step immediately before Step s′—then by the analysis of the four sub-cases, the honest verifiers in Step s′ do not get to flip coins anymore, because they have either stopped without propagating, or have seen >⅔ majority for the same bit b. Therefore we have
T
r+1
≤T
r
+λ+t
s*+3
≤T
r
+λ+t
s′+2.
In sum, no matter what s* is, we have
as we wanted to show. The worst case is when s*=s′−1 and Case 2.2.b happens. Combining Cases 1 and 2 of the binary BA protocol, Lemma 5.3 holds.
It remains to prove Lemma 5.4. Recall that the verifiers in round r are taken from PKr−k and are chosen according to the quantity Qr−1. The reason for introducing the look-back parameter k is to make sure that, back at round r−k, when the Adversary is able to 10 add new malicious users to PKr−k, he cannot predict the quantity Qr−1 except with negligible probability. Note that the hash function is a random oracle and Qr−1 is one of its inputs when selecting verifiers for round r. Thus, no matter how malicious users are added to PKr−k, from the Adversary's point of view each one of them is still selected to be a verifier in a step of round r with the required probability p (or p1 for Step 1). More precisely, we have the following lemma.
Lemma 5.6. With k=0(log1/2 F), for each round r, with overwhelming probability the Adversary did not query Qr−1 to the random oracle back at round r−k.
Proof. We proceed by induction. Assume that for each round γ<r, the Adversary did not query Qγ−1 to the random oracle back at round γ−k.25 Consider the following mental game played by the Adversary at round r−k, trying to predict Qr−1. 25As k is a small integer, without loss of generality one can assume that the first k rounds of the protocol are run under a safe environment and the inductive hypothesis holds for those rounds.
In Step 1 of each round γ=r−k , . . . , r−1, given a specific Qγ−1 not queried to the random oracle, by ordering the players i ∈ PKγ−k according to the hash values H(SIGi(γ,1, Qγ−1)) increasingly, we obtain a random permutation over PKγ−k. By definition, the leader γ is the first user in the permutation and is honest with probability h. Moreover, when PKγ−k is large enough, for any integer x≥1, the probability that the first x users in the permutation are all malicious but the (x|1)st is honest is (1−h)xh.
If γ is honest, then Qγ=H((Qγ−1), γ). As the Adversary cannot forge the signature of γ, Qγ is distributed uniformly at random from the Adversary's point of view and, except with exponentially small probability,26 was not queried to H at round r−k. Since each Qγ+1, Qγ+2, . . . , Qr−1 respectively is the output of H with Qγ, Qγ+1, . . . , Qr−2 as one of the inputs, they all look random to the Adversary and the Adversary could not have queried Qr−1 to H at round r−k. 26That is, exponential in the length of the output of H. Note that this probability is way smaller than F.
Accordingly, the only case where the Adversary can predict Qr−1 with good probability at round r−k is when all the leaders r−k, . . . , r−1 are malicious. Again consider a round γ ∈ {r−k . . . ,r−1} and the random permutation over PKγ−k induced by the corresponding hash values. If for some x≥2, the first x−1 users in the permutation are all malicious and the x-th is honest, then the Adversary has x possible choices for Qγ: either of the form H(SIGi(Qγ−1,γ)), where i is one of the first x−1 malicious users, by making player i the actually leader of round γ; or H (Qγ−1 , γ), by forcing Bγ=B∈γ. Otherwise, the leader of round γ will be the first honest user in the permutation and Qr−1 becomes unpredictable to the Adversary.
Which of the above x options of Qγ should the Adversary pursue? To help the
Adversary answer this question, in the mental game we actually make him more powerful than he actually is, as follows. First of all, in reality, the Adversary cannot compute the hash of a honest user's signature, thus cannot decide, for each Qγ, the number x(Qγ) of malicious users at the beginning of the random permutation in round γ+1 induced by Qγ. In the mental game, we give him the numbers x(Qγ) for free. Second of all, in reality, having the first x users in the permutation all being malicious does not necessarily mean they can all be made into the leader, because the hash values of their signatures must also be less than p1. We have ignored this constraint in the mental game, giving the Adversary even more advantages.
It is easy to see that in the mental game, the optimal option for the Adversary, denoted by {circumflex over (Q)}γ, is the one that produces the longest sequence of malicious users at the beginning 5 of the random permutation in round γ+1. Indeed, given a specific Qγ, the protocol does not depend on Qγ−1 anymore and the Adversary can solely focus on the new permutation in round γ+1, which has the same distribution for the number of malicious users at the beginning. Accordingly, in each round γ, the above mentioned {circumflex over (Q)}γ gives him the largest number of options for Qγ+1 and thus maximizes the probability that the consecutive leaders are all malicious.
Therefore, in the mental game the Adversary is following a Markov Chain from round r−k to round r−1, with the state space being {0} ∪ {x:x≥2}. State 0 represents the fact that the first user in the random permutation in the current round γ is honest, thus the Adversary fails the game for predicting Qr−1; and each state x≥2 represents the fact that the first x−1 users in the permutation are malicious and the x-th is honest, thus the Adversary has x options for Qγ. The transition probabilities P(x, y) are as follows.
Note that state 0 is the unique absorbing state in the transition matrix P, and every other state x has a positive probability of going to 0. We are interested in upper-bounding the number k of rounds needed for the Markov Chain to converge to 0 with overwhelming 25 probability: that is, no matter which state the chain starts at, with overwhelming probability the Adversary loses the game and fails to predict Qr−1 at round r−k.
Consider the transition matrix P(2) P·P after two rounds. It is easy to see that P(2)(0,0)=1 and P(2)(0, x)=0 for any x≥2. For any x≥2 and y≥2, as P(0, y)=0, we have
Below we compute the limit of
as h goes to 1—that is,
When h is sufficiently close to 1,27 we have 27For example, h=80% as suggested by the specific choices of parameters.
for any x≥2 and y≥2. By induction, for any k>2, P(k)Pk is such that
As P(x, y)≤1, after 1−log2F rounds, the transition probability into any state y≥2 is negligible, starting with any state x≥2. Although there are many such states y, it is easy to see that
Therefore each row x of the transition matrix P decreases as a geometric sequence with rate
when y is large enough, and the same holds for P(k). Accordingly, when k is large enough but still on the order of log1/2F, Σy>2 P(k)(X, y)<F for any x≥2. That is, with overwhelming probability the Adversary loses the game and fails to predict Qr−1 at round r−k. For h ∈ (⅔, 1], a more complex analysis shows that there exists a constant C slightly larger than ½, such that it suffices to take k=0(logCF). Thus Lemma 5.6 holds. ▪
Lemma 5.4. (restated) Given Properties 1-3 for each round before r, ph=h2(1+h−h2) for Lr, and the leader r is honest with probability at least ph.
Proof. Following Lemma 5.6, the Adversary cannot predict Qr−1 back at round r−k except with negligible probability. Note that this does not mean the probability of an honest leader is h for each round. Indeed, given Qr−1, depending on how many malicious users are at the beginning of the random permutation of PKr−k, the Adversary may have more than one options for Qr and thus can increase the probability of a malicious leader in round r+1 again we are giving him some unrealistic advantages as in Lemma 5.6, so as to simplify the analysis.
However, for each Qr−1 that was not queried to H by the Adversary back at round r−k, for any x≥1, with probability (1−h)x−1h the first honest user occurs at position x in the resulting random permutation of PKr−k. When x=1, the probability of an honest 20 leader in round r+1 is indeed h; while when x=2, the Adversary has two options for Qr and the resulting probability is h2. Only by considering these two cases, we have that the probability of an honest leader in round r+1 is at least h·h (1−h)h·h2=h2(1+h−h2) as desired.
Note that the above probability only considers the randomness in the protocol from round r−k to round r. When all the randomness from round 0 to round r is taken into consideration, Qr−1 is even less predictable to the Adversary and the probability of an honest leader in round r+1 is at least h2(1+h−h2). Replacing r+1 with r and shifts everything back by one round, the leader r is honest with probability at least h2(1+h−h2), as desired.
Similarly, in each Coin-Genuinely-Flipped step s, the “leader” of that step—that is the verifier in SVr,s whose credential has the smallest hash value, is honest with probability at least h2(1+h−h2). Thus ph=h2(1+h−h2) for Lr and Lemma 5.4 holds. ▪
Algorand′2
In this section, we construct a version of Algorand′ working under the following assumption.
H
In Section 8, we show how to replace the above assumption with the desired Honest Majority of Money assumption.
6.1 Additional Notations and Parameters for Algorand′2
Notations
Lr will be used to upper-bound the time needed to generate block Br.
Parameters
|HSVr,s|>tH and |HSVr,s|+2|MSVr,s|<2tH.
Specific Choices of Important Parameters.
Recall that a verifier i ∈ SVr,s digitally signs his message mir,s of step s in round r, relative to an ephemeral public key pkir,s, using an ephemeral secrete key skir,s that he promptly destroys after using. When the number of possible steps that a round may take is capped by a given integer μ, we have already seen how to practically handle ephemeral keys. For example, as we have explained in Algorand′1 (where μ=m+3), to handle all his possible ephemeral keys, from a round r′ to a round r′+106, i generates a pair (PMK, SMK), where PMK public master key of an identity based signature scheme, and SMK its corresponding secret master key. User i publicizes PMK and uses SMK 20 to generate the secret key of each possible ephemeral public key (and destroys SMK after having done so). The set of i's ephemeral public keys for the relevant rounds is S={i}×{r′, . . . , r′+106}×{1, . . . , μ}. (As discussed, as the round r′+106 approaches, i “refreshes” his pair (PMK, SMK).)
In practice, if μ is large enough, a round of Algorand′2 will not take more than μ steps. In principle, however, there is the remote possibility that, for some round r the number of steps actually taken will exceed μ. When this happens, i would be unable to sign his message mir,s for any step s>μ, because he has prepared in advance only μ secret keys for round r. Moreover, he could not prepare and publicize a new stash of ephemeral keys, as discussed before. In fact, to do so, he would need to insert a new public master key PMK′ in a new block. But, should round r take more and more steps, no new blocks would be generated.
However, solutions exist. For instance, i may use the last ephemeral key of round r, pkir,μ, as follows. He generates another stash of key-pairs for round r—e.g., by (1) generating another master key pair (
Of course, this approach can be repeated, as many times as necessary, should round r continue for more and more steps! The last ephemeral secret key is used to authenticate a new master public key, and thus another stash of ephemeral keys for round r. And so on.
6.3 The Actual Protocol Algorand′2
Recall again that, in each step s of a round r, a verifier i ∈ SVr,s uses his long-term public-secret key pair to produce his credential, σir,sSIGi(r, s, Qr−1), as well as SIGi(Qr−1) in case s=1. Verifier i uses his ephemeral key pair, (pkir,s, skir,s), to sign any other message m that may be required. For simplicity, we write esigi(m), rather than
(m), to denote i's proper ephemeral signature of m in this step, and write ESIGi(m) instead of
(m) (i, m, esigi(m)).
Instructions for every user i ∈ PKr−k: User i starts his own Step 1 of round r as soon as he has CERTr−1, which allows i to unambiguously compute H (Br−1) and Qr−1.
To shorten the global execution of Step 1 and the whole round, it is important that the (r, 1)-messages are selectively propagated. That is, for every user j in the system,
Note that, under selective propagation it is useful that each potential leader i propagates his credential σir,1 separately from mir,1:31 those small messages travel faster than blocks, ensure timely propagation of the mir,1's where the contained credentials have small hash values, while make those with large hash values disappear quickly. 31We thank Georgios Vlachos for suggesting this.
Instructions for every user i ∈ PKr−k: User i starts his own Step 2 of round r as soon as he has CERTr−1.
Instructions for every user i ∈ PKr−k: User i starts his own Step 3 of round r as soon as he has CERTr−1.
Instructions for every user i ∈ PKr−k: User i starts his own Step 4 of round r as soon as he finishes his own Step 3.
User i waits a maximum amount of time 2λ.37 While waiting, i acts as follows. 37Thus, the maximum total amount of time since i starts his Step 1 of round r could be tΛt3+2λ=5λ−Λ.
valid messages mjr,j=(ESIGj(v′), σjr,3), then he sets vi v′ and gi 1.39 39It can be proved that the v′ in this case, if exists, must be unique.
bi 0 if gi=2, and bi 1 otherwise.
Step s, 5≤s≤m+2, s−2≡0 mod 3: A Coin-Fixed-To-0 Step of BBA*
Instructions for every user i ∈ PKr−k: User i starts his own Step s of round r as soon as he finishes his own Step s−1.
User i waits a maximum amount of time 2λ.40 While waiting, i acts as follows. 40Thus, the maximum total amount of time since i starts his Step 1 of round r could be tsts−1+2λ=(2s−3)λ+Λ.
Ending Condition 0: If at any point there exists a string v≠⊥ and a step s′ such that
m
j
r,s′−1=(ESIGj(0), ESIGj(v), σjr,s′−1),41 and
41Such a message from player j is counted even if player i has also received a message from j signing for 1. Similar things for Ending Condition 1. As shown in the analysis, this is to ensure that all honest users know CERTr within time λ from each other.
Otherwise, when time 2λ runs out, i sets bi0.
Step s, 6≤s≤m+2, s−2≡1 mod 3: A Coin-Fixed-To-1 Step of BBA*
Instructions for every user i ∈ PKr−K: User i starts his own Step s of round r as soon as he finishes his own Step s−1.
User i waits a maximum amount of time 2λ. While waiting, i acts as follows.
Step s, 7≤s≤m+2, s−2≡2 mod 3: A Coin-Genuinely-Flipped Step of BBA*
Instructions for every user i ∈ PKr−k: User i starts his own Step s of round r as soon as he finishes his own step s−1.
User i waits a maximum amount of time 2λ. While waiting, i acts as follows.
Otherwise, when time 2λ runs out, letting SVir,s−1 be the set of (r, s−1)-verifiers from whom he has received a valid message mjr,s−1, i sets bi1sb(minj∈SV
Remark. In principle, as considered in subsection 6.2, the protocol may take arbitrarily 5 many steps in some round. Should this happens, as discussed, a user i ∈ SVr,s with s>μ has exhausted his stash of pre-generated ephemeral keys and has to authenticate his (r, s)-message mir,s by a “cascade” of ephemeral keys. Thus i's message becomes a bit longer and transmitting these longer messages will take a bit more time. Accordingly, after so many steps of a given round, the value of the parameter λ will automatically increase slightly. (But it reverts to the original λ once a new block is produced and a new round starts.)
Instructions for every user i in the system: User i starts his own round r as soon as he has CERTr−1.
6.4 Analysis of Algorand′1
The analysis of Algorand′2 is easily derived from that of Algorand′1. Essentially, in Algorand′2, with overwhelming probability, (a) all honest users agree on the same block Br; the leader of a new block is honest with probability at least ph=h2(1+h−h2).
7 Handling Offline Honest users
As we said, a honest user follows all his prescribed instructions, which include that of being online and running the protocol. This is not a major burden in Algorand, since the computation and bandwidth required from a honest user are quite modest. Yet, let us point out that Algorand can be easily modified so as to work in two models, in which honest users are allowed to be offline in great numbers.
Before discussing these two models, let us point out that, if the percentage of honest players were 95%, Algorand could still be run setting all parameters assuming instead that h=80%. Accordingly, Algorand would continue to work properly even if at most half of the honest players chose to go offline (indeed, a major case of “absenteeism”). In fact, at any point in time, at least 80% of the players online would be honest.
From Continual Participation to Lazy Honesty As we saw, Algorand′1 and Algorand′2 choose the look-back parameter k. Let us now show that choosing k properly large enables one to remove the Continual Participation requirement. This requirement ensures a crucial property: namely, that the underlying BA protocol BBA* has a proper honest majority. Let us now explain how lazy honesty provides an alternative and attractive way to satisfy this property.
Recall that a user i is lazy-but-honest if (1) he follows all his prescribed instructions, when he is asked to participate to the protocol, and (2) he is asked to participate to the protocol only very rarely—e.g., once a week—with suitable advance notice, and potentially receiving significant rewards when he participates.
To allow Algorand to work with such players, it just suffices to “choose the verifiers of the current round among the users already in the system in a much earlier round.”Indeed, recall that the verifiers for a round r are chosen from users in round r−k, and the selections are made based on the quantity Qr−1. Note that a week consists of roughly 10,000 minutes, and assume that a round takes roughly (e.g., on average) 5 minutes, so a week has roughly 2,000 rounds. Assume that, at some point of time, a user i wishes to plan his time and know whether he is going to be a verifier in the coming week. The protocol now chooses the verifiers for a round r from users in round r−k−2, 000, and the selections are based on Qr−2,001. At round r, player i already knows the values Qr−2,000, . . ., Qr−1, since they are actually part of the blockchain. Then, for each M between 1 and 2,000, i is a verifier in a step s of round r+M if and only if
.H(SIGi(r+M s, Qr+M−2,001))≤p.
Thus, to check whether he is going to be called to act as a verifier in the next 2,000 rounds, i must compute πiM,s=SIGi(r+M, s, Qr+M−2,001) for M 32 1 to 2, 000 and for each step s, and check whether .H(σiM,s)≤p for some of them. If computing a digital signature takes a millisecond, then this entire operation will take him about 1 minute of computation. If he is not selected as a verifier in any of these rounds, then he can go off-line with an “honest conscience”. Had he continuously participated, he would have essentially taken 0 steps in the next 2,000 rounds anyway! If, instead, he is selected to be a verifier in one of these rounds, then he readies himself (e.g., by obtaining all the information necessary) to act as an honest verifier at the proper round.
By so acting, a lazy-but-honest potential verifier i only misses participating to the propagation of messages. But message propagation is typically robust. Moreover, the payers and the payees of recently propagated payments are expected to be online to watch what happens to their payments, and thus they will participate to message propagation, if they are honest.
8 Protocol Algorand′ with Honest Majority of Money
We now, finally, show how to replace the Honest Majority of Users assumption with the much more meaningful Honest Majority of Money assumption. The basic idea is (in a proof-of-stake flavor) “to select a user i ∈ PKr−k to belong to SVr,s with a weight (i.e., decision power) proportional to the amount of money owned by i.”45 45We should say PKr−k−2,000 so as to replace continual participation. For simplicity, since one may wish to require continual participation anyway, we use PKr−k as before, so as to carry one less parameter.
By our HMM assumption, we can choose whether that amount should be owned at round r−k or at (the start of) round r. Assuming that we do not mind continual participation, we opt for the latter choice. (To remove continual participation, we would have opted for the former choice. Better said, for the amount of money owned at round r−k−2, 000.)
There are many ways to implement this idea. The simplest way would be to have each key hold at most 1 unit of money and then select at random n users i from PKr−k such that ai(r)=1.
The Next Simplest Implementation
The next simplest implementation may be to demand that each public key owns a maximum amount of money M, for some fixed M. The value M is small enough compared with the total amount of money in the system, such that the probability a key belongs to the verifier set of more than one step in say k rounds is negligible. Then, a key i ∈ PKr−k, owning an amount of money ai(r) in round r, is chosen to belong to SVr,s if
And all proceeds as before.
A More Complex Implementation
The last implementation “forced a rich participant in the system to own many keys”.
An alternative implementation, described below, generalizes the notion of status and consider each user i to consist of K+1 copies (i, v), each of which is independently selected to be a verifier, and will own his own ephemeral key (pki,vr,s, ski,vr,s) in a step s of a round r. The value K depends on the amount of money ai(r) owned by i in round r.
Let us now see how such a system works in greater detail.
Number of Copies Let n be the targeted expected cardinality of each verifier set, and let ai(r) be the amount of money owned by a user i at round r. Let Ar be the total amount of money owned by the users in PKr−k at round r, that is,
If i is an user in PKr−k, then i′s copies are (i, 1), . . . , (i, K+1), where
EXAMPLE. Let n=1,000, Ar=109, and ai(r)=3.7 millions. Then,
Verifiers and Credentials Let i be a user in PKr−k with K+1 copies.
For each v=1, . . . , K, copy (i, v) belongs to SVr,s automatically. That is, i's credential is σi,vr,sSIGi((i, v), r, s, Qr−1), but the corresponding condition becomes .H(σi,vr,s)≤1, which is always true.
For copy (i, K+1), for each Step s of round r, i checks whether
If so, copy (i, K+1) belongs to SVr,s. To prove it, i propagates the credential
σi,K+1r,1=SIGi((i,K+1), r,s,Qr−1).
EXAMPLE. As in the previous example, let n=1K, ai(r)=3.7M, Ar=1B, and i has 4 copies: (i, 1), . . . , (i, 4). Then, the first 3 copies belong to SVr,s automatically. For the 4th one, conceptually, Algorand′ independently rolls a biased coin, whose probability of Heads is 0.7. Copy (i, 4) is selected if and only if the coin toss is Heads.
(Of course, this biased coin flip is implemented by hashing, signing, and comparing as we have done all along in this application—so as to enable i to prove his result.)
Business as Usual Having explained how verifiers are selected and how their credentials are computed at each step of a round r, the execution of a round is similar to that already explained.
9 Forks
Having reduced the probability of forks to 10−12 or 10−18, it is practically unnecessary to handle them in the remote chance that they occur. Algorand, however, can also employ various fork resolution procedures, with or without proof of work.
10 New Structures for Blocks and Status Information
This section proposes a better way to structure blocks, that continues to guarantee the tamperproofness of all blocks, but also enables more efficient way to prove the content of an individual block, and more generally, more efficient ways to use blocks to compute specific quantities of interest without having to examine the entire blockchain.
The Tamperproofness of Blockchains Recall that, to form a block chain, a block Br has the following high-level structure:
B
r=(r, INFOr, H(Br−1)) .46
46Recall that we use superscripts to indicate rounds in Algorand. This section, however, is dedicated to blockchains in general, and thus the rth block may not correspond to the rth round in the sense of Algorand. That is, above “r” is just the block number, and it is included in block Br for clarity. Also note that the above general structure of a block is conceptual. For instance, in Bitcoin, Br may include the digital signature of the block constructor, the one who has solved the corresponding computational riddle. In Algorand , the authentication of Br—that is, a matching certificate CERTx—may be separately provided . However, it could also be provided as an integral part of Br. In this latter case, since there may be many valid certificates, the leader r of round r may also include, in its message to all round-r verifiers, a valid certificate for the output of the previous round, so that agreement will also be reached on what the certificate of each round r is.
Above, INFOr is the information that one wishes to secure within the rth block: in the case of Algorand, INFOr includes PAYr, the signature of the block leader of the quantity Qr−1, etc.
A well-known fundamental property of a blockchain is that it makes the content of each of its blocks tamperproof. That is, no one can alter a past block without also changing the last block. We quickly recall this property below.
Let the latest block be Blast, and assume that one replaces Br with a different (“crooked”) block . Then, since H is collision resilient, H() is, with overwhelming probability, different from H(Br). Accordingly, no matter how one chooses the information in the block in the blockchain, namely =(r+1, , H()), it is the case that, due again to the collision resiliency of H,
≠Br+1.
This inequality propagates. That is, differs from Br+2; and so on; so that, ultimately,
≠Blast.
Inefficient Verifiability of Individual Blocks in Blockchains Consider a person X who does not know the entire blockchain, but knows that Bz is a correct block in it. Then, the above fundamental property enables one to prove to such a person that any individual block Br, where r is smaller than z, is also correct. Namely, one provides, as a “proof”, all the intermediate blocks Br+1, . . . , Bz−1 to X, who then uses H to regenerate the blockchain from r onwards, until he reconstructs the zth block and checks whether or not it coincides with the block Bz that he knows. If this is the case, then X is convinced of the correctness of Br. Indeed, anyone capable of finding such a seemingly legitimate proof must also have found a collision in the hash function H, which is practically impossible to find.
To see the usefulness of such verifiability, consider the following example. Assume that the blockchain is generated by a payment system, such as Bitcoin or Algorand, and let X be a judge in a court case in which the defendant wishes to prove to X that he had indeed made a disputed payment P to the plaintiff two years earlier. Since it it reasonable to assume that the judge can obtain the correct last block in the chain, or at least a sufficiently recent correct block, Bz, “all” the defendant has to do is to provide the proof Br+1, . . . , Bz−1 to the judge, who then verifies it as explained. The problem, of course, is that such a proof may be quite long.
Blocktrees Since the ability to prove efficiently the exact content of a past individual block is quite fundamental, we develop new block structures. In these structures, like in blockchains, the integrity of an entire block sequence is compactly guaranteed by a much shorter value v. This value is not the last block in the sequence. Yet, the fundamental property of blockchains is maintained: any change in one of the blocks will cause a change in v.
The advantage of the new structures is that, given v, and letting n be the number of blocks currently in the sequence, the content of each individual block can be proved very efficiently. For instance, in blocktrees, a specific embodiment of our new block structures, if the total number of blocks ever produced is n, then each block can be proved by providing just 32·┌log n┐ bytes of information.
This is indeed a very compact proof. In a system generating one block per minute, then, after 2 millennia of operations, ┌log n┐<30. Thus, less than 1KB—1,000 bytes—suffice to prove the content of each individual block. (Less than 2KB suffice after two billion years, and 4KB suffice essentially forever.) Moreover, such a proof is very efficiently verified.
Efficient Status A different efficiency problem affects Bitcoin and, more generally, payment systems based on blockchains. Namely, to reconstruct the status of the system (i.e., which keys owns what at a given time), one has to obtain the entire payment history (up to that time), something that may be hard to do when the numbers of payments made is very high.
We shall use blocktrees in order to attain also such desiderata
We construct blocktrees by properly modifying a much older notion recalled below.
10.1 Merkle Trees
Merkle trees are a way to authenticate n already known values, v0, . . . , vn−i, by means of a single value v, so that the authenticity of each value vi can be individually and efficiently verified.
For simplicity, assume that n is a power of 2, n=2k, so that each value is uniquely identified by a separate k-bit string, s. Then, a Merkle tree T is conceptually constructed by storing specific values in a full binary tree of depth k, whose nodes have been uniquely named using the binary strings of length≤k.
The root is named ε, the empty string. If an internal node is named s, then the left child of s is named s0 (i.e., the string obtaining by concatenating s with 0), and the right child is named s1. Then, identifying each integer i ∈ {0, . . . , n−1}, with its binary k-bit expansion, with possible leading 0s, to construct the Merkle tree T, one stores each value vi in leaf i. After this, he rnerklefies the tree, that is, he stores in all other nodes of T in a bottom up fashion (i.e., by first choosing the contents of all nodes of depth k−1, then those of all nodes of depth k−2, and so on) as follows. If vs0 and vs1 are 10 respectively stored in the left and right child of node s, then he stores the 256-bit value vsH(vs0, vs1) in node s. At the end of this process, the root will contain the 256-bit value vε.
A Merkle tree of depth 3 and 8 leaves is shown in
Assume now that vε is known or digitally signed, and let us show how each original value vi can be authenticated relative to vε.
Consider the (shortest) path P that, starting from a node x, reaches the root. Then, the authenticating path of the content vx of x is the sequence of the contents of the siblings of the nodes in P, where the sibling of a node s0 is node s1, and viceversa. Accordingly, an authenticating path of a leaf value in a tree of depth k consists of k−1 values. For example, in the Merkle tree of
One can use a full binary tree with 2k leaves to have a Merkle tree that authenticates n<2k values v0, . . . , vn−1, by storing each vi in the ith leave, and a special string e (for empty) in all remaining leaves, and then filling the rest of the nodes in the usual manner. The resulting Merkle tree has (de facto) have n leaves (since all other leaves are empty).47 47One can also assume that the hash function H is such that H(e, e) e, so that one can “trim” all the nodes in the Merkle tree below any node whose content is ε.
To verify (the authenticity of) vi, given its authenticating path, relative to the root value vε, one reconstructs the contents of the nodes in the path from leaf i to the root, and then checks whether the last reconstructed value is indeed vε. That is, if the authenticating path is x1, . . . , xk−1, then one first H-hashes together vi and x1, in the right order—i.e., computes y2=H(vi, x1), if the last bit of i is 0, and y2=H(x1, vi) 10 otherwise. Then, one H-hashes together y2 and x2, in the right order. And so on, until one computes a value yk and compares it with v. The value vi is authenticated if and only if yk=vε.
The reason why such verification works is, once again, that H is collision resilient. Indeed, changing even a single bit of the value originally stored in a leaf or a node also changes, with overwhelming probability, the value stored in the parent. This change percolates all the way up, causing the value at the root to be different from the known value vε.
10.2 Blocktrees
As we have seen, Merkle trees efficiently authenticate arbitrary, and arbitrarily many, known values by means of a single value: the value vε stored at the root. Indeed, in order to authenticate k values v0, . . . , vk−1 by the single root content of a Merkle tree, one must first know v0, . . . , vk−1 in order to store them in the first k leaves of the tree, store e in other proper nodes, and then compute the content of all other nodes in the tree, including the root value.
Merkle trees have been used in Bitcoin to authenticate the payments of a given block. Indeed, when constructing a given block, one has already chosen the payments to put in the block.
However, using Merkle trees to authenticate a growing blockchain is more challenging, because one does not know in advance which blocks to authenticate.
Yet, we show how to use Merkle trees, used in a novel way, to obtain new block structures enabling the efficient provability of individual blocks. Let us illustrate our preferred such structures, blocktrees.
Blocktree Guarantees Blocktrees secure the information contained in each of a sequence of blocks: B0, B1, . . . This important property is not achieved, as in a blockchain, by also storing in each block the hash of the previous one. However, each block also stores some short securing information, with the guarantee that any change made in the content of a block Bi preceding a given block Br will cause the securing information of Br to change too. This guarantee of blocktree is equivalent to that offered by blockchains. The main advantage is that the new securing information allows one to prove, to someone who knows the securing information of block Br, the exact content of any block Bi, without having to process all the blocks between Bi and Br. This is a major advantage, because the number of blocks may be (or may become) very very large.
Block Structure in Blocktrees In a blocktree, a block Bi has the following form:
B
r=(r,INFOr,Sr)
where INFOr represents the information in block Br that needs to be secure, and Sr the securing information of Br.
In our preferred embodiment of block trees, the securing information Sr information is actually quite compact. It consists of a sequence of ┌log r┐ 256-bit strings, that is, ┌log r┐ strings of 32 bytes each. Notice that, in most practical applications ┌log r┐<40, because 240 is larger than a quadrillion.
Below, we specify the information Sr just for blocktrees. Those skilled in the art will realize that it can be easily generalized to a variety of other block structures with essentially identical guarantees, all of which within the scope of the invention.
Block Generation in Blocktrees For brevity, let us set INFOr=vr. Conceptually, we start with a full binary tree T of depth k such that 2k upper-bounds the number of possible values vr. Blocks are generated in order, because so are the values v0, v1, . . .
When a new value vi is generated, it is, conceptually speaking, stored in leaf i of T, and then various strings are computed and stored in the nodes of T, so as to construct a Merkle tree Ti. One of these strings is the distinguished string e. (When appearing in a node x of Ti, string e signifies that no descendant of x belongs to Ti, and we assume that H(e, e) e.)
When the first value, v0, is generated and stored in leaf 0, T0 coincides with (the so filled) node 0 of T. In fact, such T0 is an elementary Merkle tree. Its depth is ┌log(0+1)┐=0, its root is R0=0, and it stores v0 in its first depth-0 leaf (and in fact in its only leaf and node).
When the i+1st value, vi, has been generated and stored in leaf i of T (possibly replacing the string e already there), the Merkle tree Ti is constructed as follows from the previous Merkle tree Ti−1. (By inductive hypothesis, Ti−1 has depth is ┌log i┐; root Ri−1; and i depth- ┌log(i+1)┐ leaves, respectively storing the values v0, . . . , vi−1.)
Let Ri=Ri−1, if leaf i is a descendant of Ri−1, and let Ri be the parent of Ri−1 otherwise. Let P be the (shortest) path, in T, from leaf i to node Ri. For every node j in P, store the special string e in its sibling j′, if j′ is empty. Finally, for each node s in P, in order from leaf i (excluded) to node Ri (included), store in s the value vs=H(vs0, vs1), if vs0 and vs1 respectively are the values stored in the left and right child of s. It is easy to see that the subtree of T rooted at Ri, storing the so computed values in its nodes, is 15 a Merkle tree. This Merkle tree is Ti.
The construction of the first 8 consecutive Merkle trees, when the initially empty full binary tree T has depth 3, is synthesized in
With this in mind, we generate a new block Bi as follows. After choosing the information INFOi that we want to secure in the ith block, we store the value vi=INFOi into leaf i of T; construct the Merkle tree Ti; and set
S
i=(Ri, authi),
where Ri is the root of Ti and authi is the authenticating path of vi in Ti. Then, the new block is
B
i=(i, INFOi, Si).
Notice that Si indeed consists of ┌log i┐ strings. To ensure that each string in authi, and thus every string in Si, is actually 256-bit long, rather than storing vi in leaf i, we may store H(vi) instead.
Security and Provability with Blocktrees Assume that someone knows a block Br and wishes to correctly learn the information INFOi of a previous block Bi. Note that, when constructing the sequence of Merkle trees T0, T1, . . . , each tree contains the previous one as a subtree. In fact, the leaves of each tree Tx are the first x+1 leaves of each subsequent tree Ty, because the contents of previous leaves are left alone, and new values are inserted in the first leaf on the right of the last filled one. Thus, INFOi is the content of the ith leaf of the Merkle tree Tr, whose rth leaf contains INFOr and whose root value Rr is the first component of Sr, the security information of block Br.
Note that someone who knows Br also knows Sr and Rr. Accordingly, to prove to such a “person” that INFOi is the information of block Bi, it suffices for one to provide him with the authenticating path of INFOi in the Merkle tree Tr. In fact, we have already seen that such authenticating information can be easily checked, but not easily faked!
Note such authenticating path consists of d=┌log r┐ values (because Tr has depth d+1), each consisting of 32 Bytes (because H produces 32-byte outputs). Thus, proving the content of Bi relative to Br is very easy. As mentioned, in most practical applications d<40.
Block Constructibility from Readily Available Information To construct the structural information Si that is part of block Bi, it would seem that one would need information from all over the Merkle tree Ti. After all, INFOi and thus the value vi stored in leaf i, are readily available, but the authenticating path of vi, authi, comprises contents of nodes of previous trees, which in principle may not be readily available. If one had to obtain the entire Ti−1 in order to construct Si, then constructing a new block Bi might not be too efficient.
However, note that, very much in the spirit of block chains, each Bi is trivially computable from the previous block Bi−1 and the chosen information INFOi. Indeed, each string in Si is one of
(a) H(INFOi),
(b) the fixed string e,
(c) a string in Si−1, and
(d) a string obtained by hashing in a predetermined manner strings of the above types.
In sum, in a blocktree-based system, block generation is very efficient.
10.3 Efficient Management of Status Information
Blocktrees improve the secure handling of blocks in all kinds of applications, including payment systems, such as Algorand. In such systems, however, there is another aspect that would greatly benefit from improvement: the status information.
The Need for Efficient Status Information Recall that the official status at round r, Sr, consists of a list of tuples, specifying, for each current public key x, the amount owned by x, and possibly additional information: Sr=. . . , (x, ax(r), . . . ), . . . The amount of money owned by the keys in the system changes dynamically, and we need to keep track of it as efficiently as possible.
So far, as in Bitcoin (although its status information is quite different), the current status is not authenticated, but is deducible from the authenticated history of payments. Blocktrees do not guarantee the ability to prove the status Sr at a round r. In fact, via blocktrees, the payset PAYi of a block Bi preceding Br could be efficiently proven, relative to Br. However, to prove the status Sr one should, in general, provably obtain the paysets of all the blocks preceding Br. It is therefore needed an efficient method, for a prover P to prove Sr to other users or entities who do not know Sr, or who know it, but would like to receive a tangible evidence of its current value.
In particular, such a method would make it easy and secure, for a new user, or a user who has been off-line for a while, to catch up with the current system status.
Let us now provide such a method: statustrees.
Statustrees: A First Method A statustree STr is a special information structure that enables P to efficiently prove the value of status Sr at the end of round r−1.
In fact, STr enables P to efficiently prove, for any possible user i ∈ PKr, the exact amount air that i owns at the start of round r, without having to prove the status of all users (let alone provide the entire block sequence B0, . . . , Br−1). In fact, it may be important for some user to correctly learn the status of just a few users—e.g., those from which he may consider receiving a payment, to make sure they have the money before—say—start negotiating with them. It may be even be useful for a user i to receive a proof of his own value of air—e.g., in order to get a loan, be taken seriously in a negotiation, or putting a purchase order.
Assume first that P knows the entire block sequence B0, . . . , Br−1 and is widely trusted, at least by a group of users and entities. In this case, P
Then, any time that someone asks P for the status of some user x ∈ PKr, P provides him with the authenticating path in Tr of the value stored in leaf x, if x stores the status information of i.
Importantly, after computing and digitally signing Rr, P no longer needs to be around in order to answer queries about the status of some user. Indeed, any other entity P′, who also knows Tr could answer the queries of an entity V about x. In fact, P′ could provide V with the same digitally signed root value Rr, the status information about x, and the same authenticating path for the latter information that P would have provided to V. Thus, if V trusts P, then he can verify the answer provided by P′ even if he does not trusts P′. In fact, since the reported authenticating path guarantees that x's status is correct relative to the root Rr of the Merkle tree Tr, and since Rr is digitally signed by P, V can verify the authenticity of the status information about x. As long as he trusts P, V does not need to trust P′.
Thus, P may publicize (or make available to another entity P′) his digital signature of Rr, and let others (P′) answer status question in a provable manner.
We now wish to highlight the following properties of this first method.
The inability to easily provide credible answers to queries about not (at least yet) existing users may not be a concern. (E.g., because the total number of users is not too large.) In this case, our first method is just fine. However, should it be a concern, we thus put forward a different method.
Statustrees: A Second Method Assuming again that P knows the entire block sequence B0, . . . , Brr−1. Then, in a second method, P
Note that this second method continues to enable another (not necessarily trusted) entity P′ to credibly answer questions about the status of individual users, but “reverses” the news relative to properties 1 and 2. That is,
1′. (“Bad News”) It is not easy to use Tr−1 as before to easily construct Tr.
2′. (“Good News”) P or P′ can prove that a user x ∈ PKr indeed does not belong to PKr.
For instance, consider
(a) h1=H(v010, v011),
(b) h2=H(v00, h1),
(c) h3=H(h1, v1),
and then checks whether h3 indeed coincides the the root value vε.
Note that, since H(x,y)≠H(y,x),
hashing (a) proves that v010 is stored in the 0-child of whatever node stores h1;
hashing (b) proves that h1 is stored in the 1-child of whatever node stores h2; and
hashing (c) proves that h2 is stored in the 0-child of whatever node stores h3.
Thus, since we have checked that h3 is stored at the root, V010 must be stored in the leaf 010, which is indeed the case.
Accordingly, if V trusts P to have stored, for each user i ∈ PKr, the status in formation of i in the ith leaf of Tr, then, seeing a proof that one leaf contains the status information of a user i′<i, and a proof that the next leaf contains the status information of a user i″>i, then V may safely conclude that i ∈ PKr.
(Also note that such authentication paths may share many values, and thus is not necessary to transmit to V both authentication paths in their entirety.)
Property 1′ may not be a concern. (E.g., because P is perfectly capable of constructing Tr from scratch.) If this is the case, then the second method is just fine.
Else, we need a method enabling P to both easily construct Tr from Tr−1 and easily prove that a user x ∈ PKr indeed does not belong to PKr.
We now provide this method.
Statustrees: A Third Method Recall that a search tree is a data structure that, conceptually, dynamically stores values from an ordered set in the nodes of a binary (for simplicity only!) tree. Typically, in such a tree, a node contains a single value (or is empty). The operations dynamically supported by such a data structure include inserting, deleting, and searching for a given value v. (If the value v is not present, one can determine that too—e.g., because the returned answer is ⊥.)
Initially, a search tree consists only of an empty root. If more values get inserted than deleted, then the tree grows. The depth of a binary tree with n nodes is at best log n, which is the case when the tree is full, and at most n−1, which is the case when the tree is a path. The depth of a search tree is not guaranteed to be small, however. In the worst case, inserting n specific nodes in a specific order, may cause the search tree to 10 consist of depth n.
A “balanced” search tree guarantees that, if the number of currently stored values is n, then the depth of the tree is short, that is, logarithmic in n. In a balanced search tree with n nodes, each of the three operations mentioned above can be performed in O(log n) elementary steps, such as comparing two values, swapping the contents of two nodes, or 15 looking up the content of the parent/right son/left son of a node.
Several examples of balanced search trees are known by now. Particularly well known examples are AVL trees and B-trees. In some examples, values may be stored only in the leaves of the tree (while all other nodes contain “directional information” enabling one to locate a given value, if indeed it is stored in the tree). In other examples, values can be stored in any node. Below, we assume this more general case, and, for simplicity only, that the balance search tree operations are well known and deterministic algorithms.
Assume for a moment that prover P, having obtained the status information of all users in PKr, wishes to construct a balanced statustree Tr of a round r from scratch. Then, he acts as follows.
P constructs a balanced search tree Tr, with n=PKr| nodes, storing the users in PKr.
Note that this operation takes at most O(n log n) elementary steps, because, in the worst case, each insertion takes O(log n) elementary steps. (In fact, P may want to gain some additional efficiency by ordering the users in PKr prior to inserting them in Tr.)
We call the tree Tr so obtained a Merkle-balanced-search-tree. Notice that such a Tr is a balanced search tree, and thus the search/insert/delete algorithms work on Tr. At the same time, Tr is a generalized Merkle tree, but a Merkle tree nonetheless. An ordinary Merkle tree stores the information values, that is, the values that need to be secured, only in the leaves, and stores in the internal nodes only securing values, that is, the hash values used to “secure” the tree. The Merkle tree Tr stores, in each node x, both an information value, denoted by vx, and a securing value, denoted by hvx.
A proof, relative to the root securing value Rr, of what the information value yx actually is, comprises hvx0, hvx1, H(vx), and the authenticating path consisting (in a bottom-up order) of the value hvy for every node y that is a sibling of a node in the path from x to the root of Tr.
Let us now show that P need not to compute Tr from scratch. Indeed, assume that
P has available the Merkle-balanced-search-tree Tr−1 of the previous round. Then, after obtaining the new block Br, P acts as follows for each payment 0 in the payset PAYr of Br.
If modifies the amount owned by an existing user, then P updates the status information of that user, in the node x in which it is stored, and then re-merklifies the tree. This simply entails recomputing the hvy values along the path from x to the root, and thus at most ┌log n┐ hashes, if n is the number of users.
If brings in a new user i with an initial amount of money ai, then P inserts the status information (i, air) in Tr. Since Tr is a balanced search tree, this entails only O(log n) elementary steps and affects at most logarithmically many nodes. After that, P re-merklifies the tree.
After constructing Tr, P publicizes his digital signature of Rr, or gives it to P′ who handles queries from various verifiers V, who may or may not trust P′.
To answer a query of V about a user i, P′ acts as follows.
Accordingly, V may use P's digital signature of Rr and the received proofs to check that the content of every node x provided by P′ is correct. Thus, V de facto runs the same search algorithm, in the Merkle tree Tr, to correctly retrieve the status information of user i, if i ∈ PKr. Else, if i ∈ PKr (i.e., if the search algorithm returns the symbol ⊥/the string e), then V is convinced that i was not a user in PKr.
Realizing a Trusted P in Algorand When there is not an individual prover P trusted by many people, it becomes important to “construct” one.
Note that, in Algorand, one is assured that, at every step s of a round r the (weighted) majority of the verifiers in SVr,s are honest. Accordingly, we now ensure that verifiers are capable of carrying out the task of P!
More precisely,
Note that the new official block may be empty, Br=Bεr. Such empty block is of the form
B
ε
r=(r, PAYr, SIGi(Qr−1), Rr−1, H (Br−1))
corresponding to the fact that, when the new block is the empty block, the status information of all users does not change.
In sum, the digital signature of P of Rr. is replaced by the certificate of Br. Since Rr is authenticated anyway, even an untrusted prover P′ could prove the status information of every user i at round r, relative to Rr, as before.
Note that everyone, who timely learn new blocks as they are generated, can, whenever the adopted statustree is of the “easily updatable” type, more easily construct Tr from the previous Tr−1.
Representative Verifiers The probability of a user i to be selected as a member of SVr,s can also be based on (e.g., again, proportionally to) the money that other users “vote” to i. A user U may wish to retain control of all payments he makes and receives payments as usual. However, he may wish to delegate to another user i the right and duties to act as a leader and/or a verifier. In this case, and for this purpose only, such a user U may indicate that he wishes to make i his representative for leader/verifier purposes. User U may in fact delegate his leader/verifier duties to one or more users i. Let us assume, for a moment and without loss of generality, that U chooses to have a single representative, i, for leader and verifier purposes.
There are several (preferably digitally signed) ways for U to elect i as his representative at a round r. For instance, he can do so via a payment P to i (e.g., using P's non-sensitive field I) or via a payment P′ to another user, so as not to introduce (without excluding it) any additional machinery. Once such a payment P or P′ is inserted in the payset PAYr of a block Br, the community at large realizes U's choice, and it becomes effective for i to act as U's representative.
If U chooses i as his representative at a round r, then this choice supersedes any prior one made by U, and i preferably remains U's representative until U makes a different choice.48 A user U may also elect i as his representative from the very moment he enters the system. For instance, if U plans to enter the system via a payment from another user, U may ask that user to include in a signature od U indicating that U chooses i as his representative. 48Of course, here and elsewhere, ambiguity and ties are broken in some pre-specified way. For instance, if PAYr contains one signature of U indicating that U votes all his money to potential verifier i, and another signature indicting that i votes all his money to a different potential verifier j, then U's choice of representative is ignored, or, alternatively, U de facto votes for i if the corresponding signed statement precedes the one corresponding to j in—say—the lexicographic order.
The money so voted to i by U (and possibly other users) and that directly owned by i can be treated equally. For instance, if the verifiers were to be selected from the users in PKx, according to the money owned at round x, then U (and all other users who have chosen to “vote” their money to a representative) would have considered to have 0 money for the purpose of this selection, while the money according to which i would be selected would be aix+VMix, where aix is the money that i personally own in round x, and VMix is the total amount of money “voted” to i (by all users) at round x. For instance, if U is the only user voting his money to i in that round, then VMix=aUx. If the set of users voting their money to i in round x were S, then VMix=Σj∈Sajx. Of course, one way to select i according to the money owned by him and voted to him at round x consists of selecting i, via secret cryptographic sortition, with probability
It also possible to treat the money directly owned by i differently from the money voted to i. For instance, i could be chosen according to the amount of money aix+e·VMix, where c is a given coefficient. For instance, when c=0.5, the money directly owned by i counts twice as much that voted to him.
It is also possible for a user U to vote to i only part of the money he owns. For instance, U may choose to vote to i only ¾ of his money, and have the balance counted for having himself selected for SVr,s. In this case, U contributes only 0.75aUx to VMix; and U is selected in SVr,s with probability 0.25aUx.
There may be rewards offered to a user i as a consequence of his being selected to belong in SVr,s, and i can share them with the users who voted part of their money to him. For instance, i may receive a reward R if he becomes the leader of a block. In this case, may correspond to U part of this reward. For instance, if the money voted by U to i is m, then the fraction of R paid i to U may be
Users' ability to choose and change representatives may help keeping representatives honest. For instance, if U votes his money to i, but suspects that, whenever i is the leader of a block Br, the payset of PAYr is too often empty or “meager”, U may change his representative.
An ordinary user U may also specify multiple representatives, voting to each one of them a different percentage of his money. Of course, a pre-specified procedure is used to prevent U from voting more money than he actually has.
Another way to choose a representative is for a user U to generate a separate public-secret digital signature pair (pk′U, sk′U), and transfer to pk′U (e.g., via a digital signature that enters a block in the blockchain) the power of representing U for leader/verifier selection purposes. That is, pk′U cannot make payments on U's behalf (in fact, pk′U may never have directly money), but can act as a leader or a verifier on U's behalf, and can be selected as a leader/verifier according to the money that U at the relevant round x. For instance, pk′Umay be selected with probability
20 (This assumes that U delegates all his leader/verifier rights/authorities, but of course, as discussed above, U can delegate only part of his rights/authorities.) To delegates another user or entity i to serve in his stead as a leader/verifier, U gives sk′U to i. This makes it very clear how to split between U and i (in whatever determined proportions) any rewards that pk′U may “earn”, because it is pk′U itself to be directly selected, rather than having U contribute to the selection probability of i, as discussed earlier.
Note that it is not necessary that U generates (pk′U, sk′U) himself. For instance, i can generate (pk′U, sk′U), and give pk′U to U to sign, if U wishes to choose i as his representative.
In particular, a user U may choose a bank i as his representative, in any of the approaches discussed above. One advantage of choosing representatives is that the latter may have much faster and secure communication networks than typical users. If all ordinary users chose (or were required to choose) a proper representative, the generation of a block would be much sped up. The block may then be propagated on the network to which ordinary users have access. Else, if i represents U, the i may give the new blocks directly to U, or give U evidence that the payments he cares about have entered the blockchain.
Potential Verifiers So far, in Algorand, each user i can be selected as a leader of a verifier in some round r. However, it is easily realizable by those skilled in the art that the above is not a restriction. Indeed, Algorand can have a set users, who can join at any time in a permissionless way and make and receive payments (more generally transactions) as usual, and a special class of potential verifiers, from whom round leaders and verifiers are selected. These two sets can be overlapping (in which case at least some potential verifier can also make and receive payments), or separate (in which case a potential verifier can only, if selected, act as a leader or a verifier). In a sense, in Algorand as described in the previous sections, each user is a potential verifier. Algorand may also have users or potential verifiers i who have two separate amount of money, only one of which counts for i to be selected as a leader or a verifier.
The class of potential verifiers can be made permissioned. In this case, the probability of selecting a verifier/leader i among a given set S of potential verifier need not depend on the amount of money i owns. For instance, i can be selected from S via cryptographic sortition with uniform probability.
Alternatively, all potential verifiers can be always selected (and/or only one of then is selected as a round leader). In this case, the can use the protocol BA* to reach agreement on a new block, or more generally, they may use an efficient and preferably player-replaceable protocol.
12 Permissioned Algorand
In this section we discuss a permissioned version of Algorand that balances privacy and traceability, enables a new class of incentives, and provides a new, and non-controlling, role for banks or other external entities. This permissioned version relies on the classical notion recalled below.
Assume that the public key ply of a party R, who has the authority to register users in the system, is publicly known. Then, after identifying a user i and verifying that a public key pki really belongs to i, R may issue a digital certificate, Ci, guaranteeing that, not only is pki a legitimate public key in the system, but also that some suitable additional information infoi holds about i. In this case a certificate has essentially the following form:
C
i=SIGR(R, pki, infoi).
The information specified in infoi may be quite varied. For instance, it may specify i's role in the system and/or the date of issuance of the certificate. It might also specify an expiration date, that is, the date after which one should no longer rely on Ci. If no such a date is specified, then Ci is non-expiring. Certificates have long been used, in different applications and in different ways.
In Algorand, one may require that each user i ∈ PKr has digital certificate Ci issued by a suitable entity R (e.g., by a bank in a pre-specified set). When making a round-r payment to another user j, i may forward Ci and/or Cj along with , or include one or both of them in itself: that is, symbolically,
=SIGi(r,r′, i, j, a, Ci,Cj, H()).
Assuming the latter practice,
At the same time, if such a payment belongs to PAYr, then its corresponding money transfer is executed unconditionally.
A Role for Non-Expiring Certificates Once the certificate Ci of a user i expires, or prior to its expiration, as long as i is “in good standing”, R may issue a new certificate with a later expiration date.
In this case, therefore, R has significant control over i′s money. It cannot confiscate it for its personal use (because doing so would require being able to forge i's digital signature), but he can prevent i from spending it. In fact, in a round r following the expiration date of Ci, no payment of i may enter PAYr.
Note that this power of R corresponds to the power of a proper entity E e.g., the government—to freeze a user's traditional bank account. In fact, in a traditional setting, such E might even appropriate i′s money.
A main attraction of cryptocurrencies is exactly the impossibility, for any entity E, to separate a user from his money. Let us thus emphasize that this impossibility continues to hold in a certificate-based version of Algorand in which all certificates are non-expiring. Indeed, a user i wishing to make a payment to another user j in a round r can always include the non-expiring certificates Ci and Cj in a round-r payment to j, and will appear in PAYr, if the leader of the round is honest. In sum,
The payer and the payee of a payment made via a traditional check are readily identifiable by every one holding the check. Accordingly, checks are not ideal for money laundering or other illegal activities. Digital certificates, issued by a proper registration agent, could be used in Algorand to ensure that only some given entities can identify the owner i of a given public key pki, and only under proper circumstances, without preventing i from making the payments he wants. Let us give just a quick example.
There may be multiple registration authorities in the system. For concreteness only, let them be approved banks, whose public keys are universally known (or have been certified, possibly via a certificate chain) by another higher authority whose public key is universally known, and let G be a special entity, referred to as the government.
To join the system as the owner of a digital public key, i needs to obtain a certificate Ci from one of the approved banks. To obtain it, after generating a public-secret signature pair (pki, ski), i asks an approved bank B to issue a certificate for pki. In order to issue such a certificate, B is required to identify i, so as to produce some identifying information IDi.49 Then, B computes H(IDi), and (preferably) makes it a separate field of the certificate. For instance, ignoring additional information, B computes and gives i 49Going “overboard”, such Ii may include i′s name and address, a picture of i, the digital signature of i's consent—if it was digital—, or i can be photographed together with his signed consent, and the photo digitized for inclusion in Ii. For its own protection and that of i as well, the bank may also obtain and keep a signature of i testifying that Ii is indeed correct.
C
i=SIGB(B,pki, H(IDi)).
Since H is a random oracle, no one can recover the owner's identity from Ci. Only the bank knows that pki's owner is i. However, if the government wishes to investigate—say—the payer of a payment =SIGpk, (r, pki′, pki′, a, Ci, Ci′, I, H ()), it retrieves from both the relevant certificate Ci and the bank B that has issued Ci, and then asks or compels B, with proper authorization (e.g., a court order), to produce the correct identifying information IDi of i.
Note that the bank cannot reveal an identifying piece of information ID′i that is different from that originally inserted in the certificate, because H is collision-resilient. Alternatively, instead of H(IDi, it suffices to use any “commitment” to IDi, in the parlance of cryptography.
One particular alternative to store, in Ci, an encryption of IDi, E(IDi, preferably using a private- or public-key encryption scheme E, that is uniquely decodable. In particular, IDi may be encrypted with a key known only to the bank B: in symbols, E(IDi)=EB(IDi). This way, the government needs B's help to recover the identifying information IDi.
Alternatively, E may be a public-key encryption scheme, and IDi may be encrypted with a public key of the government, who is the only one to know the corresponding decryption key. In symbols, E(IDi)=EG(IDi). In this case the goverment needs not to have B's help to recover IDi. In fact, the identities of the payers and the payees of all payments are transparent to G. However, no one else may learn IDi from EG(IDi), besides the government and the bank be that has compute EG(IDi) from IDi. Moreover, if the encryption EG(IDi) is probabilistic, then even someone who has correctly guessed who the owner i of pki may be would be unable to confirm his guess.
Let us stress that neither B nor G, once the certificate Ci has been issued, has any control over i's digital signatures, because only i knows the secret key ski corresponding to pki. In addition, neither B or G can understand the sensitive information of the payment of i, because only H() is part of . Finally, neither B nor G is involved in processing the payment , only the randomly selected verifiers are.
In sum, the discussed law-enforcement concern about Bitcoin-like systems is put to rest, without totally sacrificing the privacy of the users. In fact, except for B and G, i continues to enjoy the same (pseudo) anonymity he enjoys in Bitcoin, and similar systems, relative to any one else: banks, merchants, users, etc.
Finally, by using more sophisticated cryptographic techniques, it is also possible to increase user privacy while maintaining his traceability under the appropriate circumstances.50 50For instance, focusing on the last scenario, once i has obtained from bank B a certificate Ci=SIGB(B,pki, EG(IDi)) , it is also possible for i to obtain another certificate C′i=SIGB(B,pk′i,E′G(IDi)), for a different public pk′i, for which B no longer knows that IDi is the
Key certification in Algorand is a new role for the banks. A role that a bank B can easily perform for its customers. Indeed, B already knows them very well, and typically interacts with them from time to time. By certifying a key i of one of its customers, B performs a simple but valuable (and possibly financially rewarded) service.
Another role a bank B may have in Algorand is that, properly authorized by a customer i, B can transfer money from i's traditional bank accounts to a digital key the digital pki that i owns in Algorand. (In fact, B and all other banks do so “at the same exchange rate”, Algorand may be used as a very distributed, convenient, and self-regulated payment system, based on a national currency.) One way for a bank B to transfer money to pki is to make a payment to pki from a digital key pkB that B owns in Algorand. In fact, banks may, more easily than their customers, convert national currency into Algorand at a public exchange.51 51Going a step further, the Government may also be allowed to print money in Algorand, and may transfer it, in particular, to banks, or, if the banks are sufficiently regulated, it may allow them to generate Algorand money within certain parameters.
Finally, since banks are often trusted, a user i, retaining full control of the payments he makes, may wish to delegate a bank B to act on his behalf as a verifier, sharing with B his verification incentives.52 52“Representative” mechanisms for doing just that are described in Appendix??. That section is devoted to implementations of Algornand with a specially constructed communication network. However, the delegation mechanisms described there may be adopted no matter what the underlying network may be.
Whether or not the registration authorities are banks, and whether or not law-enforcement concerns are addressed, a permissioned deployment of Algorand enables one to identify (e.g., within its certificate Ci) that a given key pki belongs to a merchant.
A merchant, who currently accepts credit card payments, has already accepted his having to pay transaction fees to the credit card companies. Thus, he may prefer paying a 1% fee in Algorand to paying the typically higher transaction fee in a credit card system (in addition to preferring to be paid within minutes, rather than days, and in much less disputable ways.)
Accordingly, a certificate-based Algorand may ensure that
For instance, letting A′ be the total amount paid to retails in the payments of PAYr, one decryption of E′G(IDi). could compute a maximum potential reward R′ (e.g., R′=1%A′). The leader and the verifiers, however, will not collectively receive the entire amount R′, but only a fraction of R′ that grows with the total number (or the total amount, or a combination thereof) of all payments in PAYr. For example, keeping things very simple, if the total number of payments in PAYr is m, then the actual reward that will be distributed will be R′(1-1/m). As before, this can be done automatically by deducting a fraction 1%(1-1/m) from the amount paid to each retailer, and partitioning this deducted amount among the leader and verifiers according to a chosen formula.
Let us discuss some possible variants to Algorand.
So far, Algorand selects the leader r and the verifier set SVr of a round r automatically, from quantities depending on previous rounds, making sure that SVr has a prescribed honest majority. We wish to point out, however, alternative ways to select the verifiers and the leader.
One such way, of course, is via a cryptographic protocol run by all users. This approach, however, may be slow when the number of users in the system is high. Let us thus consider two classes of alternative mechanisms: chained , nature-based, and trusted-party. (One may also mix these mechanisms with those already discussed.)
Inductively assuming that each SVr has an honest majority, we could have SVr itself (or more generally some of the verifiers of the rounds up to r) select the verifier set and/or the leader of round r. For instance, they could do so via multi-party secure computation. Assuming that the initial verifier set is chosen so as to have an honest majority, we rely on boot-strapping: that is, the honest majority of each verifier set implies the honest majority of the next one. Since a verifier set is small with respect to the set of all users, his members can implement this selection very quickly.
Again, it suffices for each round to select a sufficiently long random string, from which 10 the verifier set and the leader are deterministically derived.
The verifier set SVr and the leader r of a given round r can be selected, from a prescribed set of users PVr−k, in a pre-determined manner from a random value vr. associated to the round r. In particular, vr. may be a natural and public random value. By this we mean that it is the widely available result of a random process that is hardly controllable by any given individual. For instance, vr. may consist of the temperatures of various cities at a given time (e.g., at the start of round r, or at a given time of the previous round), or the numbers of stock of given security traded at a given time at a given stock exchange, and so on.
Since natural and public random value may not be sufficiently long, rather than setting
we may instead set
elongating H(vr) in a suitable peseudo-random fashion, is needed, as already discussed.
An alternative approach to selecting SVr involves one or more distinguished entities, the trustees, selected so as to guarantee that at least one of them is honest. The trustees may not get involved with building the payset PAYr, but may choose the verifier set SVr and/or the leader r.
The simplest trustee-based mechanisms, of course, are the single-trustee ones. When there is only one trustee, T, he is necessarily honest. Accordingly, he can trivially select, digitally sign, and make available SVr (or a sufficiently random string Sr from which SVr is derived) at round r.
This a simple mechanism, however, puts so much trust on T. To trust him to a lesser extent, T may make available a single string, sr, uniquely determined by the round r, that only he can produce: for instance, sr=SIGT(r). Then, every user can compute the random string H(SIGT(vr)), from which SVr is derived.
This way, T does not have the power to control the set SVr. Essentially, he has a single strategic decision at his disposal: making SIGT(r) available or not. Accordingly, it is easier to check whether T is acting honestly, and thus to ensure that he does so, with proper incentives or punishments.
The problem of this approach is (the lack of) unpredictability. Indeed, T may compute SIGT(r) way in advance, and secretly reveal it to someone, who thus knows the verifier set of a future round, and has sufficient time to attack or corrupt its members.
To avoid this problem, we may rely on secure hardware. Essentially, we may have T be a tamper-proof device, having a public key posted “outside” and a matching secret key locked “inside”, together with the program that outputs the proper digital signatures at the proper rounds. This approach, of course, requires trusting that the program deployed inside the secure hardware has no secret instructions to divulge future signatures in advance.
A different approach is using a natural public random value vr associated to each round r. For instance, T may be asked to make available SIGT(vr). This way, since the value vr of future rounds r is not known to anyone, T has no digital signature to divulge in advance.
The only thing that T may still divulge, however, is its own secret signing key. To counter this potential problem we can rely on k trustees. If they could be chosen so as to ensure that a suitable majority of them are honest, then they can certainly use multi-party secure computation to choose SVr at each round r. More simply, and with less trust, at each round r, we may have each trustee i make available a single string, uniquely associated to r and that only i can produce, and then compute SVr for all such strings. For instance, each trustee i could make available the string SIGi(r), so that one can compute the random string sr=H(SIGi(r), . . . , SIGk(r)), from which the verifier set SVr is derived. In this approach we might rely on incentives and punishments to ensure that each digital signature SIGi(r) is produced, and rely on the honesty of even a single trustee i to ensure that the sequence s1, s2, . . . remains unpredictable.
The tricky part, of course, is making a required string “available”. If we relied on propagation protocols then a malicious trustee may start propagating it deliberately late in order to generate confusion. So, trustee-based mechanisms must rely on the existence of a “guaranteed broadcast channel”, that is, a way to send messages so that, if one user receives a message m, then he is guaranteed that everyone else receives the same m.
Finally, rather than using secure computation at each round, one can use a secure computation pre-processing step. This step is taken at the start of the system, by a set of trustees, selected so as to have an honest majority. This step, possibly by multiple stages of computation, produces a public value pv and a secret value vi for each trustee i. While this initial computation may take some time, the computation required at each round r could be trivial. For instance, for each round r, each trustee i, using his secret value vi, produces and propagates a (preferably digitally signed) single reconstruction string sir, such that, given any set of strings Sr that contains a majority of the correct reconstruction strings, anyone can unambiguously construct SVr (or a random value from which SVr is derived). The danger of this approach, of course, is that a fixed set of trustees can be more easily attacked or corrupted.
Algorand can also benefit from more sophisticated cryptographic tools. In particular,
This allows to turn multiple related propagations into a single propagation. In essence, assume that, during a propagation protocol, a user has received SIG1,2,3(D) together with a record {1, 2, 3} as well as SIG4,5(D) together with a record {4, 5}. Then he might as well propagate SIG1,2,3,4,5(D) and the record {1, 2, 3, 4, 5}.
2. Tree-Hash-and-Sign. When a signature authenticates multiple pieces of data, it may be useful to be able to extract just a signature of a single piece of data, rather than having to keep or send the entire list of signed items. For instance, a player may wish to keep an authenticated record of a given payment P ∈ PAYr rather than the entire authenticated PAYr. To this end, we can first generate a Merkle tree storing each payment P ∈ PAYr in a separate leaf, and then digitally sign the root. This signature, together with item P and its authenticating path, is an alternative signature of essentially P alone.
3. Certified Email. One advantage of the latter way of proceeding is that a player can send his payment to by certified email,53 preferably in a sender-anonymous way, so as to obtain a receipt that may help punish if it purposely decides not to include some of those payments in . 53E.g., by the light-weight certified email of U.S. Pat. No. 5,666,420.
Software implementations of the system described herein may include executable code that is stored in a computer readable medium and executed by one or more processors. The computer readable medium may be non-transitory and include a computer hard drive, ROM, RAM, flash memory, portable computer storage media such as a CD-ROM, a DVD-ROM, a flash drive, an SD card and/or other drive with, for example, a universal serial bus (USB) interface, and/or any other appropriate tangible or non-transitory computer readable medium or computer memory on which executable code may be stored and executed by a processor. The system described herein may be used in connection with any appropriate operating system.
Other embodiments of the invention will be apparent to those skilled in the art from a consideration of the specification or practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.
This application claims priority to U.S. Prov. App. No. 62/331,654, filed May 4, 2016, and entitled “ALGORAND: A VERY EFFICIENT MONEY PLATFORM”, and to U.S. Prov. App. No. 62/333,340, filed May 9, 2016, and entitled “ALGORAND: A VERY EFFICIENT MONEY PLATFORM”, and to U.S. Prov. App. No. 62/343,369, filed May 31, 2016, and entitled “ALGORAND: THE EFFICIENT BLOCK CHAIN”, and to U.S. Prov. App. No. 62/344,667, filed Jun. 2, 2016, and entitled “ALGORAND THE EFFICIENT BLOCK CHAIN”, and to U.S. Prov. App. No. 62/346,775, filed Jun. 7, 2016, and entitled “ALGORAND THE EFFICIENT BLOCK CHAIN”, and to U.S. Prov. App. No. 62/351,011, filed Jun. 16, 2016, and entitled “ALGORAND THE EFFICIENT AND DEMOCRATIC LEDGER”, and to U.S. Prov. App. No. 62/353,482, filed Jun. 22, 2016, and entitled “ALGORAND THE EFFICIENT AND DEMOCRATIC LEDGER”, and to U.S. Prov. App. No. 62/354,195, filed Jun. 24, 2016, and entitled “ALGORAND THE EFFICIENT AND DEMOCRATIC LEDGER”, and to U.S. Prov. App. No. 62/363,970, filed Jul. 19, 2016, entitled “ALGORAND THE EFFICIENT AND DEMOCRATIC LEDGER”, and to U.S. Prov. App. No. 62/369,447, filed Aug. 1, 2016, entitled “ALGORAND THE EFFICIENT AND DEMOCRATIC LEDGER”, and to U.S. Prov. App. No. 62/378,753, filed Aug. 24, 2016, entitled “ALGORAND THE EFFICIENT PUBLIC LEDGER”, and to U.S. Prov. App. No. 62/383,299, filed Sep. 2, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/394,091, filed Sep. 13, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/400,361, filed Sep. 27, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/403,403, filed Oct. 3, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/410,721, filed Oct. 20, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/416,959, filed Nov. 3, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/422,883, filed Nov. 16, 2016, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/455,444, filed Feb. 6, 2017, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/458,746, filed Feb. 14, 2017, entitled “ALGORAND”, and to U.S. Prov. App. No. 62/459,652, filed Feb. 16, 2017, entitled “ALGORAND”, which are all incorporated by reference herein.
Filing Document | Filing Date | Country | Kind |
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PCT/US2017/031037 | 5/4/2017 | WO | 00 |
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62331654 | May 2016 | US | |
62333340 | May 2016 | US | |
62343369 | May 2016 | US | |
62344667 | Jun 2016 | US | |
62346775 | Jun 2016 | US | |
62351011 | Jun 2016 | US | |
62353482 | Jun 2016 | US | |
62354195 | Jun 2016 | US | |
62363970 | Jul 2016 | US | |
62369447 | Aug 2016 | US | |
62378753 | Aug 2016 | US | |
62383299 | Sep 2016 | US | |
62394091 | Sep 2016 | US | |
62400361 | Sep 2016 | US | |
62403403 | Oct 2016 | US | |
62410721 | Oct 2016 | US | |
62416959 | Nov 2016 | US | |
62422883 | Nov 2016 | US | |
62455444 | Feb 2017 | US | |
62458746 | Feb 2017 | US | |
62459652 | Feb 2017 | US |