Patent Grant
8150030
This is a U.S. National Phase Application under 35 USC 371 of International Application PCT/FR2007/050836, filed on Feb. 22, 2007.
This application claims the priority of French application No. 06/50619 filed on Feb. 23, 2006, and the content of which is hereby incorporated by reference.
The invention relates to cryptography. To be more precise, the invention concerns the use of a cryptographic condensation function, also known as a “compression” function or a “hashing” function.
Hashing functions (or hash functions) are widely used in cryptography because they can reduce the length of a digital data word, for example to reproduce an electronic signature or certificate to guarantee the integrity and authenticity of a message. Hashing functions can also form essential parts of a wide range of protocols, such as cryptographic entity authentication protocols or the SSL/TLS protocol.
As a general rule, a hashing function H is a compression function that converts a large set of characters (referred to as the message or input) of any length into a smaller set of characters of fixed length (called the output, message digest or hash). Because the hashing function is a “one way” function it is impossible to retrieve the original set from the message digest.
One example of this kind of hashing function is described by R. C. Merkle in the paper “One-way Hash Functions and DES” (CRYPTO, Springer-Verlag 1989), where the message digest is calculated by a chaining principle.
To be more precise, a function is a cryptographic hashing function if it satisfies the following three conditions: resistance to attacks “on the first pre-image” (or antecedent), resistance to attacks “on the second pre-image”, and resistance to “collisions”.
Resistance to an attack on the first pre-image makes it very difficult (i.e. technically virtually impossible) to recover the content of a message or input x from a given message digest or output y. In other words, it is technically virtually impossible (either at the algorithm level or the hardware level) to find an input x such that H(x)=y.
Resistance to an attack on the second pre-image makes it very difficult to produce from a message x and its message digest y another message x′ that gives the same message digest y. In other words, given an input-output pair (x, y) where H(x)=y, it is very difficult to find an input x′ where x′≠x such that H(x′)=y.
Resistance to collisions makes it very difficult to find two random messages that give the same message digest. In other words, it is very difficult to find any two inputs x and x′ such that H(x)=H(x′).
For example, for a message digest having a length of n bits, there are 2n such message digests. Moreover, the “birthdays” theorem indicates that 2n/2 attempts are sufficient to find a collision by chance.
Thus a cryptographic hashing function is of good quality if the minimum number of attempts necessary to solve the above three problems is of the order of 2n, 2n and 2n/2, respectively.
At present, the hashing functions in widespread use are constructed on a one-off basis or are based on using DES (data encryption standard) block encryption. However, their performance and their message digest lengths are not well suited to current uses in cryptography.
The hashing functions most widely used at present are of the MD5 (Message Digest Algorithm) type and the SHA-1 (Secure Hash Standard) type (see http://www.ietf.org/rfc/rfc3174.txt).
However, X. Wang and H. Yu have shown, in their papers “How to Break MD5 and Other Hash Functions” (May 2005) and “Finding Collisions in the Full SHA-1” (August 2005), that these functions have weaknesses enabling faster calculation of collisions than pure chance. Similarly, other types of hashing functions that are also part of the MD5 family are also vulnerable.
Attempts have been made to construct other hashing functions relying, through certain proofs, on the difficulty of solving certain classes of problems in the arithmetic of large numbers. The major drawback of these functions is the lack of efficiency inherent to the arithmetic of large numbers. Moreover, they guarantee resistance only to collisions and therefore do not satisfy the conditions regarding resistance to attacks on the first pre-image and the second pre-image.
One aspect of the present invention relates to a method of cryptographically hashing a message M, including the following steps:
Evaluating systems of polynomials over a finite field is extremely fast, especially for polynomials of low degree (for example of degree 2, 3, or 4), whereas it is very difficult to invert a polynomial evaluation in reasonable time. This is because finding an antecedent of a value taken by a set of polynomials is known to be a very difficult mathematical problem.
Thus new security properties are easily accessed, which is difficult in the prior art, which uses hashing functions that are all constructed on the same principle and suffer many attacks.
The iterative calculation of the successive output p-tuples V1, . . . Vi, . . . , Vc includes the following steps:
each new current iteration i includes the following steps:
Thus the method is simple to use and provides a high level of security.
The current input p-tuple Ei=(ei1, . . . , eip) advantageously corresponds to a chaining p-tuple defined as being equal to the preceding output p-tuple Vi-1 calculated in the preceding iteration.
Thus, for each data m-tuple Mi, the corresponding input p-tuple has already been calculated in the preceding iteration as a function of the preceding chaining p-tuple Vi-1 and the preceding data m-tuple, which further facilitates using the method and at the same time provides high security.
At least one set of multivariate polynomials can advantageously be modified in each iteration i as a function of said current input p-tuple Ei=(ei1, . . . , eip).
Thus inversion of a polynomial evaluation becomes even more complex, which increases security without unduly decreasing the efficiency and speed of calculation of the hashing value.
In one particular embodiment of the invention, the calculation of said output p-tuple Vi corresponding to said data m-tuple Mi includes simultaneous application of said data m-tuple and the corresponding input p-tuple Ei to only one set of multivariate polynomials, said set of multivariate polynomials being of compressive form.
A set of multivariate polynomials is said to be of compressive form when it includes a number of polynomials smaller than the number of variables associated with each polynomial.
Thus the hashing value is calculated with a high security level in an optimum time.
In another embodiment of the invention, the calculation of said output p-tuple Vi corresponding to said data m-tuple Mi includes the following steps:
This increases the level of security because it is even more difficult to recover the message from a hashing value determined by two chained sets of multivariate polynomials, even if those polynomials are of low degree, for example of degree 2.
In another embodiment of the invention, the calculation of said output p-tuple corresponding to said data m-tuple Mi includes the following steps:
This other embodiment is of high security because it is very difficult to find the message using a hashing value determined by two chained sets of multivariate polynomials even if those polynomials are of low degree.
Said first set of multivariate polynomials is advantageously of expansive form and said second set of multivariate polynomials is advantageously of compressive form.
A set of multivariate polynomials is said to be of expansive form when that set includes a number of polynomials greater than the number of variables associated with each polynomial.
The expansive form of the first set of multivariate polynomials guarantees no collision during the first step and increases the resistance to collisions during the next compression step.
Another aspect of the present invention is directed to a device for cryptographically hashing a message M, and including:
Said calculation means advantageously include:
Another aspect of the invention is directed to a computer program downloadable from a communications network and/or stored on a computer-readable medium and/or executable by a microprocessor, comprising instructions for executing the steps of a cryptographic hashing method with one or more of the following features when it is executed in a computer.
Other features and advantages of the invention emerge on reading the description given below by way of non-limiting illustration and with reference to the appended drawings, in which:
The formation means 3 are for forming a sequence M1, . . . , Mi, . . . , Mc of blocks or m-tuples of data from a message M. Thus the message M, which can be of any length, can be represented by a sequence of m-tuples M1=(a1,1, . . . , a1,m), . . . , Mi=(ai,1, . . . , ai,m), . . . , Mc=ac,1, . . . , ac,m), where m is a strictly positive integer. The m-tuples are defined in a finite field F, for example the binary field GF(2).
This can typically be effected by securely adding or concatenating additional messages ΔM to/with the original message M to form a completed message M+ΔM the length of which is a multiple of that of a block Mi. In this example, the completed message M+ΔM has a length equal to c multiplied by m(|M+ΔM|=|cm|). Of course, it is sufficient but not necessary for the length of ΔM to be less than m. The additional elements ΔM can advantageously include information relating to the original message M, such as the initial length of that message M.
Thus the formation means 3 can include padding means 9 and subdivision means 11. The padding means 9 are for padding out the message M with the additional elements ΔM to complete its length so as to be able to subdivide it. The subdivision means 11 are adapted to subdivide the completed message M+ΔM to form the sequence of m-tuples M1, . . . , Mi, . . . , Mc.
The calculation means 5 calculate iteratively successive output p-tuples V1, . . . , Vi, . . . , Vc, where p is a strictly positive integer, corresponding to the sequence (M1, . . . , Mi, . . . , Mc) of data m-tuples as a function of at least one set of multivariate polynomials Π of any degree defined over a finite field F.
Finally, the determination means 7 determine a hashing value H(M) of the message M as a function of the last output p-tuple Vc. Thus the hashing value H(M)=φ(Vc) where φ is any function that can simply be the identity function such that H(M)=Vc.
The calculation means can include definition means 13 and application means 15.
During an initialization phase, the definition means 13 define an initialization input p-tuple E0=(e01, . . . , e0p) in the finite field F. The application means 15 then apply the initialization input p-tuple E0 and the first data m-tuple M1 to said at least one set of multivariate polynomials Π to calculate a first output p-tuple V1 corresponding to the first data m-tuple M1.
Then on each new current iteration number i the definition means 13 define in the finite field F a current input p-tuple Ei=(ei1, . . . , eip).
The initial or current input p-tuple Ei=(ei1, . . . , eip) can be a subset of a data m-tuple Mi. Also, the current input p-tuple Ei=(ei1, . . . , eip) can be defined as a function of one or more of the output p-tuples Vi, . . . , Vi-1 calculated in preceding iterations by the application means 15, i.e. Ei=f(V1, . . . , Vi-1).
The current input p-tuple Ei=(ei1, . . . , eip) is advantageously a function only of the output p-tuple Vi-1 calculated in the preceding iteration, i.e. Ei=f(V1-1) or explicitly (ei1, . . . , eip)=f(Vi-1,1, . . . , Vi-1,p).
The application means 15 apply the current input p-tuple Ei and a current data m-tuple Mi to said at least one set of multivariate polynomials Π to calculate a current output p-tuple Vi corresponding to the current data m-tuple Mi.
Thus the general cryptographic hashing process can be modeled as follows:
V0=E0, Vi=Π(f(Vi-1), Mi) for 1≦i≦c and H(M)=φ(Vc).
To facilitate the calculation, the current input p-tuple Ei=(ei1, . . . , eip) can advantageously correspond to a chaining p-tuple defined as being equal to the preceding output p-tuple Vi-1 calculated in the preceding iteration. Thus the calculation means 15 can calculate successive values Vi of the chaining p-tuple by iteratively applying the m-tuple Mi with the preceding value Vi-1 of the chaining p-tuple to said at least one set of multivariate polynomials Π to determine the hashing value H(M) of the message M as follows:
V0=E0, Vi=Π(Vi-1, Mi) for 1≦i≦c and H(M)=φ(Vc).
To increase security, the definition means 13 can modify said at least one set of multivariate polynomials Π in each iteration i as a function of the current input p-tuple Ei=(ei1, . . . , eip). Thus the set of multivariate polynomials Π can be generated dynamically using a mechanism specified by f(Vi-1).
Thus the application means 15 calculate an output p-tuple Vi corresponding to a data m-tuple Mi by simultaneously applying the data m-tuple and the corresponding input p-tuple Ei to the set of multivariate polynomials Π(p,k).
In other words, the application means 15 form a k-tuple (ai1, . . . , aim, ei1, . . . , eip) where k=m+p by concatenating an input p-tuple with said data m-tuple Mi so that the output p-tuple Vi is calculated by a polynomial evaluation of the set of multivariate polynomials Π(p,k) in the following manner:
Π(p,k)(ai1, . . . , aim, ei1, . . . , eip)=(vi1, . . . , vip).
For example, a multivariate polynomial in the four variables x1, x2, x3, x4 over the finite field GF(2) with exclusive-OR addition and Boolean multiplication between two bits can take the following form:
P(x1, x2, x3, x4)=x1x2+x1x3x4+x4+1.
Π1(t,m)=P11(x1, . . . , xm), . . . , P1t(x1, . . . , xm) and
Π2(p,k)=P21(x1, . . . , xk), . . . , P2p(x1, . . . , xk).
In this example, the first set of multivariate polynomials Π1(t,m) includes t polynomials with m variables and the second set of multivariate polynomials Π2(p,k) includes p polynomials with k variables, and here k=t+p.
Thus to calculate the output p-tuple Vi corresponding to the data m-tuple Mi, the application means 15 include first application means 17 and second application means 19.
The first application means 17 calculate an intermediate data t-tuple (bi1, . . . , bit) by applying the data m-tuple Mi to the first set of multivariate polynomials Π1(t,m) in the following manner:
Π1(t,m)(ai1, . . . , aim)=(bi1, . . . , bit).
The second application means 19 simultaneously apply the intermediate data t-tuple (bi1, . . . , bit) and the input p-tuple Ei corresponding to the data m-tuple Mi to the second set of multivariate polynomials Π2(p,k) to determine the output p-tuple Vi as follows:
Π2(p,k)(bi1, . . . , bit, ei1, . . . , eip)=(vi1, . . . , vip).
Of course, the second set of multivariate polynomials Π(p,k) is of compressive form, having a number p of polynomials smaller than the number k=t+p of variables associated with each polynomial.
The first set of multivariate polynomials Π1(t,m) is advantageously of expansive form, i.e. includes a number t of polynomials greater than the number m of variables associated with each polynomial.
This is because the expansive form of the first set of multivariate polynomials Π1(t,m) guarantees no collision during the first step and increases the resistance to collisions during the next compression step effected by the second set of multivariate polynomials Π2(p,k).
Π1(t,m+p)=P11(x1, . . . , xm+p), . . . , P1t(x1, . . . , xm+p) and
Π2(p,t)=P21(x1, . . . , xt), . . . , P2p(x1, . . . , xt).
According to this example, the first set of multivariate polynomials Π1(t,m+p) includes t polynomials with m+p variables and the second set of multivariate polynomials Π2(p,t) includes p polynomials and t variables.
Thus, to calculate the output p-tuple Vi corresponding to the data m-tuple Mi, the application means 15 include first application means 21 and second application means 23.
The first application means 21 calculate an intermediate data t-tuple (bi1, . . . , bit) by simultaneously applying the data m-tuple and said corresponding input p-tuple Ei to the first set of multivariate polynomials Π1(t,m+p) as follows:
Π1(t,m+p)(ai1, . . . , aim, ei1, . . . , eip)=(bi1, . . . , bit).
The second application means 21 apply the intermediate data t-tuple to the second set of multivariate polynomials Π2(p,t) to determine said output p-tuple Vi as follows:
Π2(p,t)(bi1, . . . , bit)=(vi1, . . . , vip).
The first set of multivariate polynomials Π1(t,m+p)is advantageously of expansive form, including a number t of polynomials greater than the number m+p of variables associated with each polynomial. So, the second set of multivariate polynomials Π2(p,t) is of compressive form, including a number p of polynomials smaller than the number t of variables associated with each polynomial.
Note that for the
To facilitate the calculation, but still with a high security level, the polynomials of the first and second sets of multivariate polynomials can advantageously be of degree 2.
Moreover, this data processing system can be used to execute a computer program including instructions for implementing the cryptographic hashing method of the invention.
The invention is also directed to a computer program downloadable from a communications network and containing instructions for executing the steps of the cryptographic hashing method of the invention when it is executed in a computer. The computer program can be stored in a computer-readable medium and can be executable by a microprocessor.
This program can use any programming language, and take the form of source code, object code or an intermediate code between source code and object code, such as a partially compiled form, or in any other desirable form.
The invention is also directed to a computer-readable information medium containing instructions of the above computer program.
The information medium can be any entity or device capable of storing the program. For example, the medium can include storage means, such as a ROM, for example a CD ROM or a microelectronic circuit ROM, or magnetic storage means, for example a diskette (floppy disk) or a hard disk.
The information medium can be a transmissible medium such as an electrical or optical signal, which can be routed via an electrical or optical cable, by radio or by other means. The program of the invention can in particular be downloaded over an Internet-type network.
Alternatively, the information medium can be an integrated circuit into which the program is incorporated, the circuit being adapted to execute the method in question or to be used in its execution.
| Number | Date | Country | Kind |
|---|---|---|---|
| 06 50619 | Feb 2006 | FR | national |
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/FR2007/050836 | 2/22/2007 | WO | 00 | 8/25/2008 |
| Publishing Document | Publishing Date | Country | Kind |
|---|---|---|---|
| WO2007/096566 | 8/30/2007 | WO | A |
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|---|---|---|---|
| 5263085 | Shamir | Nov 1993 | A |
| 6031464 | Matsumoto et al. | Feb 2000 | A |
| 6269098 | Crayford | Jul 2001 | B1 |
| 20020095581 | Imai et al. | Jul 2002 | A1 |
| 20050271203 | Akiyama et al. | Dec 2005 | A1 |
| Number | Date | Country | |
|---|---|---|---|
| 20090067620 A1 | Mar 2009 | US |
