Current trends in cloud computing, big data, and Machine Learning (ML) have led to greater needs for distributed computing systems. Such distributed computing systems can include a large number of networked compute servers or worker nodes that perform computations needed for an application executed by a client device or master node. Distributed computation can take advantage of coding schemes, such as Erasure Coding (EC) or Berrut Approximated Coded Computing (BACC), which add redundant data or parity data to allow for a certain number of the worker nodes to fail or return incorrect or late results to the master node and still derive a final result using the correct results received from other worker nodes. Some coding schemes, such as BACC, may provide an approximate result that may require a lower threshold number of correct results to be returned from the worker nodes or a lower amount of computational complexity for the result to be recovered with sufficient accuracy.
In some cases, the data being processed in computing systems can include sensitive data, such as medical data, financial data, or other personal data. With large scale computations, such as computations for ML or big data analysis, it becomes difficult to distribute such computations to worker nodes in a network while maintaining privacy of the data throughout the network. Although the provider of the worker nodes (e.g., a cloud service provider) may be trusted by the master node (e.g., a customer of the cloud service provider), the worker node or its network may become compromised, thereby exposing the data sent by the master node to the worker node.
Various types of homomorphic encryption have been developed to allow encrypted data to be sent to a single worker node and remain encrypted during one or more computations at the worker node without the worker node decrypting the encrypted data before performing the computation or computations. The worker node then sends an encrypted result back to the master node, which decrypts the encrypted result using a secret key. Such homomorphic encryption can safeguard the data provided by the master node to the worker node if the worker node or the communication network between the master node and the worker node becomes vulnerable.
However, such homomorphic encryption schemes have not been used with distributed computing systems including multiple worker nodes since operations performed on data using such homomorphic encryption schemes introduce noise or error into the result, which quickly deteriorates the accuracy of a final result with the addition of more worker nodes and more computations. This has prevented the practical use of such homomorphic encryption schemes in distributed computing systems. The accuracy and recoverability of results further deteriorate when using approximate coding schemes, such as BACC.
The features and advantages of the embodiments of the present disclosure will become more apparent from the detailed description set forth below when taken in conjunction with the drawings. The drawings and the associated descriptions are provided to illustrate embodiments of the disclosure and not to limit the scope of what is claimed.
In the following detailed description, numerous specific details are set forth to provide a full understanding of the present disclosure. It will be apparent, however, to one of ordinary skill in the art that the various embodiments disclosed may be practiced without some of these specific details. In other instances, well-known structures and techniques have not been shown in detail to avoid unnecessarily obscuring the various embodiments.
Network 110 can include, for example, a Storage Area Network (SAN), a Local Area Network (LAN), and/or a Wide Area Network (WAN), such as the Internet. In this regard, one or more of client device 102 and servers 112 may not be physically co-located. Client device 102 and servers 112 may communicate using one or more standards such as, for example, Ethernet or Fibre Channel.
Client device 102 includes one or more processors 104, interface 108, and memory 106. Processor(s) 104 can execute instructions, such as instructions from one or more applications loaded from memory 106, and can include circuitry such as, for example, a Central Processing Unit (CPU) (e.g., one or more Reduced Instruction Set Computer (RISC)-V cores), a Graphics Processing Unit (GPU), a microcontroller, a Digital Signal Processor (DSP), an Application-Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA), hard-wired logic, analog circuitry and/or a combination thereof. In some implementations, processor(s) 104 can include a System on a Chip (SoC), which may be combined with memory 106.
Memory 106 can include, for example, a volatile Random Access Memory (RAM) such as Static RAM (SRAM), Dynamic RAM (DRAM), or a non-volatile RAM, or other solid-state memory that is used by processor(s) 104. Data stored in memory 106 can include, for example, data to be encoded and encrypted before being sent to servers 112 and results received from servers 112 that are decrypted and decoded to derive a final result, in addition to instructions loaded from one or more applications for execution by processor(s) 104, and/or data used in executing such applications, such as keys 18.
While the description herein refers to solid-state memory generally, it is understood that solid-state memory may comprise one or more of various types of memory devices such as flash integrated circuits, NAND memory (e.g., Single-Level Cell (SLC) memory, Multi-Level Cell (MLC) memory (i.e., two or more levels), or any combination thereof), NOR memory, EEPROM, other discrete Non-Volatile Memory (NVM) chips, or any combination thereof. In other implementations, memory 106 may include a Storage Class Memory (SCM), such as, Chalcogenide RAM (C-RAM), Phase Change Memory (PCM), Programmable Metallization Cell RAM (PMC-RAM or PMCm), Ovonic Unified Memory (OUM), Resistive RAM (RRAM), Ferroelectric Memory (FeRAM), Magnetoresistive RAM (MRAM), 3D-XPoint memory, and/or other types of solid-state memory, for example.
As shown in the example of
Each of servers 1121, 1122, and 1123 in the example of
Memory 116 can include, for example, a volatile RAM such as SRAM, DRAM, or a non-volatile RAM, or other solid-state memory that is used by processor(s) 114. Data stored in memory 116 can include, for example, encrypted data to be used as an input for a function 20 or encrypted data resulting from the function 20. In addition, each memory 116 can store instructions loaded from one or more applications for execution by processor(s) 114, such as computing module 19, and/or data used in executing such applications, such as optional key or keys 22. As discussed in more detail below, keys 22 can include one or more evaluation keys used as part of a Fully Homomorphic Encryption (FHE) scheme that allows for function 20 to be performed on encrypted data and return an encrypted result that can be decrypted using a secret key stored in memory 106 of client device 102.
For its part, interface 118 may communicate with client device 102 via network 110 using, for example, Ethernet or Fibre Channel. Each network interface 118 may include, for example, a NIC, a network interface controller, or a network adapter.
Those of ordinary skill in the art will appreciate with reference to the present disclosure that distributed system 100 in
It follows that the multiplication of matrices A and B can be represented as AB=Σi=1KAiBi=C, with the multiplications of the portions of the larger matrices A and B being broken down into smaller matrix multiplication operations for smaller matrices A1 through AK and B1 through BK. In other implementations, each Ai and Bi may instead be a numerical value as opposed to a sub-matrix or sub-dataset.
In a case where client device 102 sends portions of these datasets (i.e., Ai and Bi) to servers 112 for multiplying the matrices A and B (i.e., matrix multiplication), distributed system 100 can include K servers that each receive an encoded and encrypted version of a data portion from A and an encoded and encrypted version of a data portion from B to multiply these inputs together and return an encoded and encrypted product to client device 102. For its part, client device 102 decrypts and approximately decodes (e.g., interpolates) the results received from the K servers 112 or from at least a threshold number of the K servers (i.e., M servers) to decode a final result with sufficient accuracy.
Encoding module 10 may use the same encoding function for each dataset or may use different encoding functions for the datasets. In one example, the encoding functions may be similar to a Berrut Approximated Coded Computing (BACC) as shown in Equations 2 below for encoding datasets A and B.
As used herein, for a positive integer N, [N] denotes the set {1, 2, . . . , N}. Each of the original data portions of A and B have then been transformed from K submatrices or values into N encoded data portions that can each be represented as UA(Zi) and uB(zi) with
In some implementations, client device 102 can determine N or the total number of encoded data portions to be derived from encoding one of A or B at least based in part on a degree of the function to be evaluated by each server 112 and/or the number of sub-datasets or smaller datasets (e.g., submatrices) created from one of the larger, initial datasets (i.e., K). In some examples of such implementations, the relationship of the number encoded data portions for one of the datasets (i.e., N) to the number of smaller datasets (i.e., K) and the degree of the function to be evaluated by each server 112 can be expressed as: N>K·deg ƒ, where deg ƒ is the degree of the function.
As shown in the example of
In more detail, approximate FHE is a public key encryption scheme that can encrypt a vector over complex numbers from set as a message. Servers 112 using an evaluation key, evk, can perform component-wise addition and multiplication of the underlying message vectors without decrypting the message vectors. An approximate FHE scheme can include encryption module 12 of client device 102 generating the secret key (sk), the evaluation key (evk), and the public key (pk) based on parameters for the computations including a level parameter (L), a message dimension (n), and a security parameter (λ).
The messages, or encoded portions of the matrices discussed above (i.e., Ai and Bi), are encrypted using the public key (pk) into ciphertext (ct), which can be expressed as Encpk(m)→ct. As part of the encryption, the messages or encoded portions of the matrices discussed above (i.e., Ai and Bi), are transformed using a complex number set so that message m∈n. Each server 112 receives one or more encrypted data portions as ciphertext cti and evaluates a function 20 using the evaluation key (evk) to calculate an encrypted result as ciphertext ctri, which can be expressed as Evalevk(F,{cti}i∈l)→ctri, where function 20 is represented as F.
In some implementations, each server 112 can receive multiple encrypted data portions and use the encrypted data portions as inputs into a multivariate function. The function F:(n)I→ can be composed of component-wise addition and/or multiplications with input ciphertexts {cti}i∈l and can output the ciphertext result ctri. In keeping with FHE, the multiplicative depth of the function F is less than or equal to the level parameter L. Client device 102 receives the encrypted results from servers 112 and decrypts the encrypted results using decryption module 14 with the secret key sk, which can be expressed as Decsk(ctri)→mri.
The approximate FHE scheme satisfies the following security and correctness properties. As used herein, the notation aA denotes that a is randomly sampled or generated from the set or algorithm A, and Pr[aA:P(a)=True] denotes the probability that an element randomly sampled or generated from A satisfies the property P. For semantic security of the approximate FHE scheme, it holds that for any Probabilistic Poly(n)-Time (PPT) adversary and m0≠m1∈n:
For correctness, the approximate FHE scheme holds that for all (mi)i∈I)∈(n)I and any admissible function F: (n)I→:
While the security is the same as a typical Indistinguishability-Chosen Plaintext Attack (IND-CPA) security of a public-key encryption scheme, the correctness property in Equation 4 above for the disclosed approximate FHE scheme only guarantees that the encrypted result ctri is an approximated result with an error bound by ϵ.
In this regard, the approximate coding schemes disclosed herein (e.g., the encoding functions of Equations 2 above and Equations 8 and 10 below with the corresponding decoding functions of Equations 5, 9, and 11 below) are robust against small numerical errors introduced by the approximate FHE scheme. For bounded inputs, the output of the approximate coding schemes is also bounded, which facilitates use of approximate FHE. More information on implementing an approximate FHE scheme can be found in the paper by J.H Cheon, A. Kim, M. Kim, and Y. Song, “Homomorphic Encryption for Arithmetic of Approximate Numbers”, in International Conference on the Theory and Application of Cryptology and Information Security, Springer, November 2017, pgs. 409-437 and in the paper by R. Datharthri, B. Kostova, O. Saarikivi, W. Dai, K. Laine, and M. Musuvathi, “Eva: An Encrypted Vector Arithmetic Language and Compiler for Efficient Homomorphic Computation”, in Proceedings of the 41st SIGPLAN Conference on Programming Language Design and Implementation, 2020, pgs. 546-561, each of which is hereby incorporated by reference in its entirety.
In experimental results using Equations 2 above for approximate coding without encryption, the approximation error was measured for different numbers of M servers or worker nodes returning the correct value. In this experiment, the column size of the matrix A was set as n=105 and the total number of servers or worker nodes was N=200. As shown in
In another experiment, the encoded portions sent to the worker nodes or servers for distributed computations were first encrypted using approximate FHE as discussed above and compared to the same distributed computations without any encryption.
Returning to the example of
In calculating the encrypted result, each server 112 may use an evaluation key, evk, as part of optional keys 22 in
The encrypted results Ŷi are returned from servers 112i to client device 102, which uses decryption module 14 to decrypt the encrypted results it receives from servers 112i. Decryption module 14 uses a secret key, sk, which may be known only to client device 102, to decrypt the encrypted results. The decrypted results are passed from decryption module 14 to decoding module 16 of client device 102 to decode the decrypted results into an approximated final result, C, representing the multiplication of matrices A and B.
In decoding the decrypted results Y, decoding module 16 can determine when enough decrypted results have been obtained to accurately decode the results into a final result. As discussed above, the encoding enables using only a subset M of all the N results (i.e., from N servers 112) to derive a final result C with sufficient accuracy. Decoding module 16 of client device 102 in some implementations may have a predetermined threshold value for M corresponding to a known or estimated relative error so that client device 102 may wait until the number of received results is equal to or greater than the threshold value for M. In the example where the encoding functions are Equations 2 above, a decoding function is performed by decoding module 16 to interpolate Equation 5 below with the decrypted results Y.
where AB=Σi=1KAiBi≈Σi=1KrBerrut,M(αi)≈C. The accuracy of the final result C improves as the number of decrypted results increases towards the total number of servers 112 (i.e., as M approaches N). With reference to the original division of datasets A and B into K sub-datasets, an exact recovery of the final result is obtained when M≥2K−1. For approximating the final result, a sufficient recovery threshold for most applications can be obtained when M=K, as discussed above with reference to
In another example implementation, the datasets A and B can be represented as matrices of real numbers A∈r×n and B∈n×c as discussed above, but with matrices A and B being divided into K and K′ portions, respectively, as follows so that the matrices A and B are divided into different numbers of submatrices or sub-datasets.
It follows that the multiplication of matrices A and B can be represented as follows.
Encoding module 10 may then instead use encoding functions of Equations 8 below in place of Equations 2 above to encode matrices A and B.
are used as interpolation points.
Each of the original data portions of A and B have then been transformed from K submatrices and K′ submatrices, respectively, into N encoded data portions that can each be represented as uA(zi) and uB(zi) with
Client device 102 may determine N or the total number of encoded data portions to be derived from encoding A or B at least based in part on a degree of the function to be evaluated by each server 112 and/or the number of sub-datasets or smaller datasets (e.g., submatrices) created from one of the larger, initial datasets (i.e., K or K′).
As in the example discussed above, encryption module 12 encrypts the encoded data portions uA(z) and uB(z) using an approximate FHE scheme to generate encrypted data portions represented in
As with the first example above, each server 112i calculates an encrypted result Ŷi by inputting the encrypted data portions Âi and {circumflex over (B)}i it receives from client device 102 into a respective function 20i, which can be represented as Ŷi=Âi{circumflex over (B)}i. The encrypted results Ŷi are returned from servers 112i to client device 102, which uses decryption module 14 to decrypt the encrypted results it receives from servers 112i. The decrypted results are passed from decryption module 14 to decoding module 16 of client device 102 to decode the decrypted results into the approximated final result, C, representing the multiplication of matrices A and B.
In decoding the decrypted results Y in this second example implementation, decoding module 16 can determine when enough decrypted results have been obtained to accurately decode the results into a final result. As discussed above, the encoding enables using only a subset M of all the N results (i.e., from N servers 112) to approximate a final result C with sufficient accuracy. In the example where the encoding functions are Equations 8 above, a decoding function is performed by decoding module 16 to interpolate Equation 5 above with the decrypted results Y.
Unlike the first example implementation above where Equations 2 are used as the encoding functions, decoding module 16 can approximate the final result with a recovery threshold of
for the second example implementation, as opposed to the recovery threshold discussed above for the first example implementation of M=K. Client device 102 approximates the final result C as:
C=AjBl≈rBerrut,M(α(j−1)K′+1) for all j∈[K] and l ∈[K′]. Eq. 9
As a variation, K and K′ can be selected so as to be coprime, i.e., gcd(K, K′)=1, the encoding function for uA(z) in Equations 8 above can then be replaced with Equation 10 below.
When decoding, Equation 9 is replaced with:
C=A
imodK
B
imodK′
≈r
Berrut,M(αi). Eq 11
The second example implementation using the encoding functions of Equations 8 and/or Equation 10 above in place of the encoding functions of the first example implementation with Equations 2 provides for a lower communication complexity with less data needing to be sent to servers 112 and less data being received from servers 112, but the second example implementation has a higher recovery threshold requiring more correct results to be received from servers 112 for a similar relative error in the final result. In more detail, each server 112 in the first example (i.e., using Equations 2 and 5) performs matrix multiplication on two matrices of dimensions
resulting in a matrix having dimensions r×c. In contrast, each server 112 in the second example (i.e., using Equations 8 and/or 10 for encoding and Equations 5 and 10 or 11 for decoding) performs matrix multiplication on two matrices of dimensions
resulting in a matrix having dimensions
The recovery threshold for the first and second examples are K and
respectively, and the communication cost for recovery is Krc and
respectively. The communication complexity of the first example is larger by a factor of K, but its recovery threshold is smaller by a factor of K′.
Example implementations 1 and 2 above can be viewed as extremes of a tradeoff between communication complexity and recovery threshold. In other implementations, client device 102 may divide A and B into submatrices differently than in examples 1 and 2 to provide a communication complexity and recovery threshold between the extremes of examples 1 and 2. In such implementations, client device 102 divides A and B into respective n rows and m columns of submatrices or sub-datasets, as opposed to the nA32 1×mA=K submatrices of A and nB=K×mB=1 submatrices of B in example 1 (i.e., as in Equations 1 above), or the nA=K×mA=1 submatrices of A and nB=1×mB=K submatrices of B in example 2 (i.e., as in Equations 6 above).
By approximating the final result (e.g., C) using the coding disclosed herein, it is possible to distribute the computations among many servers 112 with a lower recovery threshold while using a fixed communication complexity, as compared to computing an exact result. The computational cost of using the approximate distributed computing and approximate FHE is similar to using conventional FHE where only one server 112 performs the encrypted computations, while the computational cost per server is significantly less due to the greater number of servers 112 permitted with the disclosed approximate coding.
For example, using conventional FHE for matrix multiplication could be performed with one server computing AB=Σi=1KAiBi, which means performing K matrix multiplications. As discussed for example 1 above, each server 112 using the systems and methods of the present disclosure means each server 112 performs one matrix multiplication using approximate FHE with K servers 112 to obtain an approximate final result. On the other hand, providing an exact final result would require 2K servers 112. In summary, the distributed systems and methods disclosed herein use an approximate coding with approximate FHE with a similar computational complexity as conventional FHE used with a single server (i.e., K matrix multiplications) if allowing for an approximate result, while the computational cost per server is significantly less.
Although the above examples are discussed in terms of matrix multiplication operations being performed by servers 112, distributed system 100 can be used for other types of bilinear functions. A function B: V×W→Z is considered bilinear if it satisfies Equations 12 below.
a·(v1,w1)+b·(v2,w1)=z,62 (av1+bv2,w1)
a·(v1,w1)+b·(v1,w2)=(v1,aw1+bw2) Eqs. 12
where V and W are vector spaces over F including real or complex numbers (e.g., n, n) with a and b from F (e.g., ). To compute (Σi=1KAi,Σi=1KBi), the same encoding functions from Equations 2 can be used. Each server 112, computes =() and returns the encrypted result to client device 102. Such bilinear functions can include matrix or vector arithmetic operations (e.g., matrix multiplication as discussed above), inner product operations, convolutions, or cross-product operations.
Unlike conventional BACC, which approximates a function ƒ over a dataset X=(X0. . . ,XK−1), the disclosed systems and methods can approximate multivariate functions with more than one input argument (e.g., approximating a function g(X,Y)). Although matrix multiplication is discussed in the above examples, distributed system 100 and the disclosed methods can approximate a multivariate function over multiple datasets (e.g., vectors, matrices).
In addition, the systems and methods disclosed herein can preserve the privacy of the input datasets with ciphertext indistinguishability (e.g., IND-CPA secure), unlike the current use of BACC. The rational functions used for the second example pass through each submatrix multiple times with different interpolation points, whereas in BACC, the rational function only passes through each submatrix once. As noted above, and in contrast to BACC, the above disclosed distributed systems and methods can exploit the tradeoff between communication complexity and threshold recovery for approximate matrix multiplication.
Those of ordinary skill in the art will appreciate with reference to the present disclosure that implementations of the privacy-preserving distributed computing disclosed herein can be practiced with many more servers 112 and many more corresponding encrypted data portions. For example, and as noted above, some results using the privacy-preserving distributed computing disclosed herein have been obtained with 200 servers or worker nodes for matrix multiplication of an A matrix of 105 columns with a B matrix of 105 rows.
In block 602, at least two datasets are divided or partitioned into a plurality of smaller datasets. In some cases, the datasets can include large matrices or vectors that are divided into multiple smaller submatrices or portions of the larger vectors. In other implementations of the distributed computation process of
In implementations where the number of values in the largest dataset exceed the number of worker nodes or servers, the at least two datasets can be divided into equal numbers of smaller datasets, as with the first example discussed above for Equations 1 (e.g., K submatrices from each of larger matrices A and B). In other implementations, the at least two datasets can be divided into different numbers of smaller datasets, as with the second example discussed above for Equations 6 (e.g., K submatrices from larger matrix A and K′ submatrices from larger matrix B).
In block 604, the at least two datasets, whether having been divided into smaller datasets in block 602 or not, are encoded using one or more encoding functions to generate encoded data portions. As discussed above, the encoding functions used for each of the at least two datasets can be the same, as with the example of Equations 2. In other implementations, the encoding functions used for each of the at least two datasets can be different, as with the example of Equations 8 above. The number of encoded data portions from each larger dataset that are generated by the encoding function or functions in most implementations equals the number of worker nodes or servers that will be used to perform the distributed computations (e.g., N encoded data portions generated from matrix A that are sent to N servers 112). The encoded data portions may also be indexed (e.g., i=1, . . . ,N) so that encoded data portions with the same index are sent to the same server as inputs into a multivariate function evaluated by the server.
In block 606, the encoded data portions generated in block 604 are encrypted using a first key according to an approximate FHE encryption scheme to generate encrypted data portions. As discussed above, the first key can be a public key, pk. The indexing of the encrypted data portions may remain the same as for the encoded data portions to ensure the correct encrypted data portions are sent to the same server for computation.
In block 608, the client device sends the encrypted data portions to a plurality of servers (i.e., N servers) for the servers to perform operations on the encrypted data portions. By using an approximate FHE scheme, the servers can perform the operations (e.g., multiplication, addition, subtraction) using the encrypted data portions without having to decrypt the encrypted data portions. The input data or original datasets therefore remain encrypted to protect the privacy of the data while still taking advantage of the processing and fault-tolerance benefits of distributed computing, which may otherwise present data privacy concerns. The client device may also send an evaluation key to each server with the one or more encrypted data portions sent to each server that is used as part of the approximate FHE scheme to enable the computations to be performed with the encrypted data portions. In other implementations, each server may already store an evaluation key for performing the encrypted operations (e.g., keys 22 in
In block 610, the client device receives a plurality of encrypted results from at least a subset of the servers with each server having calculated a respective encrypted result using at least two encrypted data portions. As discussed above, the approximate coding allows for recovery of an approximate result with less than all of the results from the servers. A recovery threshold M can be set so that the client device may only need M out of N results to recover the final result from the distributed computations with sufficient accuracy. The recovery threshold may vary by application and by the chosen coding scheme, as discussed above.
In this regard, the first coding example discussed above using Equations 2 and 5 may have a recovery threshold of M=K, where K would be the number of sub-datasets formed in block 602. In such an example, the client device may begin decoding the received encrypted results after decrypting the results and in response to receiving at least K encrypted results. The client device may also perform error detection, such as with a Cyclic Redundancy Check (CRC) or other algorithm known in the art, by using information included in one or more packets received from the server to determine if a payload of the one or more packets including the encrypted result has been transmitted without errors. In such examples, the client device may only decrypt the encrypted results that pass the error detection and wait until receiving the threshold number (i.e., M) encrypted results that pass the error detection before performing decoding to derive the approximate final result.
If using the second coding example discussed above with Equations 8, 9, and/or 10 and 11, the recovery threshold may instead be
where K and K′ would be the respective numbers of sub-datasets formed in block 602. As with the example above where the recovery threshold is M=K, the client device may decrypt the encrypted results as they are received, assuming the encrypted results pass an error check in some implementations, and then begin decoding the decrypted results after decrypting at least M results.
In block 612, the client device can decrypt the plurality of encrypted results using a secret key, sk, according to the approximate FHE scheme to derive a plurality of decrypted encoded results. As noted above, the decryption may be performed as the encrypted results are received and may also follow an error check to ensure that the data in the encrypted result has not been corrupted in transit from the server to the client device. The approximate coding disclosed herein can advantageously allow for a certain number of results to be discarded, not received, or received late (i.e., stragglers) and still facilitate recovery of an approximate final result with sufficient accuracy. Moreover, the approximate coding disclosed herein also enables the use of approximate FHE, which introduces a bounded error into the calculations performed by the servers.
In block 614, the decrypted encoded results from block 612 are decoded using an approximate decoding function, such as the functions from Equations 5, 9, or 11 above. In some implementations, the decoding may take place as the encrypted results are being decrypted due to the nature of the decoding function, which may allow for decoded results to be added piecemeal with a more accurate final result being updated with each additional decrypted result being used in the decoding function. In other words, the client device in some implementations may not wait for a certain threshold number of encrypted results to be received and may begin decoding before all of the threshold number of M results have been received. In some implementations, the client device may stop decoding after using the recovery threshold number M of decrypted results in the decoding function. In other implementations, the client device may continue updating the final result until reaching a timeout value for receiving additional straggler encrypted results from additional servers.
Those of ordinary skill in the art will appreciate with reference to the present disclosure that the distributed computing process of
In block 702, the server receives at least one encrypted data portion from a client device. In some implementations, each server can receive two or more encrypted data portions that represent encoded and encrypted portions of larger datasets that are to be multiplied together or to have another type of operation performed on them. For example, each encrypted data portion can serve as an input into a bilinear function, such as matrix multiplication, an inner product function, a convolution, or a cross-product function. In other examples, the function performed by the computing module of the server can include multivariate functions of a higher degree, such as the multiplication of three encrypted data portions.
In block 704, the server evaluates a function (e.g., function 20 in
In block 706, the server sends the encrypted result to the client device that had sent the encrypted data portions. As discussed above with reference to the distributed computation process of
Those of ordinary skill in the art will appreciate with reference to the present disclosure that the server-side, distributed computing process of
The foregoing distributed computation systems and processes using the disclosed approximate coding schemes can enable the use of approximate FHE in large scale distributed systems to maintain data privacy, while providing the benefits of distributed computing, such as a lower computational cost per server and improved error and failure tolerance. In addition, the foregoing distributed computation systems and processes can provide sufficiently accurate final results despite the errors or noise introduced by approximate FHE and with lower recovery thresholds as compared to exact coding schemes.
Those of ordinary skill in the art will appreciate that the various illustrative logical blocks, modules, and processes described in connection with the examples disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. Furthermore, the foregoing processes can be embodied on a computer readable medium which causes processor or controller circuitry to perform or execute certain functions.
To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, and modules have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Those of ordinary skill in the art may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present disclosure.
The various illustrative logical blocks, units, modules, processor circuitry, and controller circuitry described in connection with the examples disclosed herein may be implemented or performed with a general purpose processor, a GPU, a DSP, an ASIC, an FPGA or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. Processor or controller circuitry may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, an SoC, one or more microprocessors in conjunction with a DSP core, or any other such configuration.
The activities of a method or process described in connection with the examples disclosed herein may be embodied directly in hardware, in a software module executed by processor or controller circuitry, or in a combination of the two. The steps of the method or algorithm may also be performed in an alternate order from those provided in the examples. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable media, an optical media, or any other form of storage medium known in the art. An exemplary storage medium is coupled to processor or controller circuitry such that the processor or controller circuitry can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to processor or controller circuitry. The processor or controller circuitry and the storage medium may reside in an ASIC or an SoC.
The foregoing description of the disclosed example embodiments is provided to enable any person of ordinary skill in the art to make or use the embodiments in the present disclosure. Various modifications to these examples will be readily apparent to those of ordinary skill in the art, and the principles disclosed herein may be applied to other examples without departing from the spirit or scope of the present disclosure. The described embodiments are to be considered in all respects only as illustrative and not restrictive. In addition, the use of language in the form of “at least one of A and B” in the following claims should be understood to mean “only A, only B, or both A and B.”
This application claims the benefit of U.S. Provisional Application No. 63/416,365 titled “PRIVACY-PRESERVING DISTRIBUTED COMPUTING” (Atty. Docket No. WDA-6487P-US), filed on Oct. 14, 2022, which is hereby incorporated by reference in its entirety.
Number | Date | Country | |
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63416365 | Oct 2022 | US |