Variants of the Fiat-Shamir Identification and Signature Scheme

The present invention relates to a method and apparatus for implementing an identification and signature scheme and, more particularly, relates to an improvement to the method and apparatus disclosed and claimed in, and is a continuation-in-part of, U.S. patent application Ser. No. 883,247 filed July 9, 1986 in the names of Adi Shamir and Amos Fiat, now U.S. Pat. No. 4,748,668.

In the aforementioned parent patent application (t which we refer henceforth as the original Fiat-Shamir scheme), a method and apparatus are disclosed and claimed which enable an entity to generate proofs of identity and signatures of messages that everyone can verify but no one can forge. Compared to other schemes of this type (such as the RSA), the Fiat-Shamir scheme is faster by one to two orders of magnitude. In the present invention disclosed and claimed in this patent application, two improved variants of the Fiat-Shamir scheme are described which are even faster. One variant uses small public values to optimize the verification procedure, and the other variant uses particular secret values to optimize the generation procedure. To keep these variants secure and to fully exploit their improved performance, some of the details of the original method and apparatus disclosed and claimed in the aforementioned parent patent application have been modified. In particular, the public key of an entity is no longer computed as a function of its identity I, the quadratic expressions are generalized to d-th powers, and the execution of the protocol is usually not iterated, but may be.

Other and further objects and advantages of the present invention will be evident from the following detailed description taken in conjunction with the drawings.

FIG. 1 is a schematic showing the novel method and apparatus for proving identity; and

FIG. 2 is a schematic showing the novel method and apparatus for generating and verifying a signature to a message.

Referring now to the drawings, the basic structure of the new and improved variants of the method and apparatus of the aforementioned parent patent application is illustrated in FIG. 1 for a proof of identity. As shown, an entity either selects or is given a public key, see block 10, consisting of a modulus n which is the product of at least two prime numbers, and a sequence of k numbers v.sub.1, . . . , v.sub.k. This public key can be stored in a public key directory, or sent to the verifier along with a trusted center's signature on it to establish its authenticity. The entity computes a corresponding private key, block 12, consisting of numbers s.sub.1, . . . , s.sub.k satisfying s.sub.j.sup.d v.sub.j =1 (mod n) for all 1.ltoreq.j.ltoreq.k, where d is some universally known constant larger than 1.

Accordingly, the identification technique proceeds as follows.

To prove his, hers or its identity, the entity chooses a random r in the range 0<r<n, block 14, and sends x=r.sup.d (mod n) to the verifier, block 16 and line 18, where it is received by the verifier, block 20. Upon receipt of x, line 22, the verifier chooses k random numbers e.sub.1, . . . , e.sub.k in the range 0.ltoreq.e.sub.j <d, block 24, and sends them to the prover, line 26. The prover, in response to receipt of e.sub.1, . . . , e.sub.k, block 28, computes and sends ##EQU1## (mod n) to the verifier, block 32, line 34. Line 36 sends the s.sub.j values from block 12 to block 32. The verifier upon receipt of y value in block 38, via line 40, computes in block 42 the value ##EQU2## This value is sent via line 44 to block 46 where it is compared with the value x, via line 48. The result is passed by line 50 to block 52 where the judgment is made. The verifier accepts the proof of identity in block 54 if ##EQU3## and rejects the proof of identity if this equality does not hold.

The signature technique of the present invention is schematically portrayed in FIG. 2. As shown, the identification scheme of FIG. 1 is turned into a signature scheme by using some publicly known cryptographically strong pseudo random function f which maps its inputs into a sequence of k numbers e=e.sub.1, . . . , e.sub.k in the range 0.ltoreq.e.sub.j <d, block 70. To generate a signature for message m according to the present invention, first, a random r in the range 0<r<n is chosen. Then, e=f(r.sup.d (mod n), m) is computed in block 72. Next ##EQU4## is computed in block 74. The signature of m consists of e and y. These values are either stored or sent to the verifier where they are received in block 76.

To verify the stored or transmitted signature, the verifier computes ##EQU5## block 78. These values are compared with e, block 80. If equal, block 82, the signature is accepted as genuine and, if not, the signature is rejected as a forgery.

The best known attack on these identification and signature techniques has a probability of success of d.sup.-k per attempt. To make the interactive identification protocol secure, it usually suffices to choose d.sup.k larger than or equal to 2.sup.20 since a cheater has only one chance to forge a proof of identity. To make the non-interactive signature technique secure against repeated attempts to forge a signature, d.sup.k should be at least 2.sup.64. This can be achieved either by using sufficiently large values of d and k, or by iterating the executions of these schemes t times and making sure that d.sup.kt .gtoreq.2.sup.64. In most applications, the first option is recommended since it minimizes both the number of modular multiplications and the number of communicated bits.

In the first variant of the present invention, small v.sub.j values are used. In the preferred embodiment of this variant, d is 2 or 3, and all the entities choose the numbers v.sub.j as the first k primes v.sub.1 =2, v.sub.2 =3, v.sub.3 =5, etc. Since the v.sub.j are universal, only the modulus n has to be specified in the public-key directory. In a typical implementation of this variant with k=64 numbers and 512 bit modulus, the size of each entry in the public-key directory is 64 bytes, and the size of each signature is 72 bytes, which are comparable with those of the RSA signature scheme. The size of the private key is about 4 kilobytes, but since each entity has to store only one such file, it can fit into almost any microcomputer based device (with the possible exception of a smart card). With optimized implementations, it is possible to generate proofs of identity in less than 10 modular multiplications and to generate signatures in less than 30 modular multiplications, which is the same as in the original Fiat-Shamir scheme. However, since multiplications by the small v.sub.j values (most of which fit in a single byte) have negligible complexity, the verification of identities and signatures in this variant requires only 1 or 2 full-size modular multiplications. This is an order of magnitude faster than in the original Fiat-Shamir scheme, and is expected to take only tens of milliseconds on a standard microprocessor and less than a millisecond on a mainframe computer. The high efficiency of verification can be crucial when a central computer has to verify many access requests or signed documents generated in thousands of terminals, or when a remotely controlled robot has to verify in real time a stream of signed instructions generated by a more powerful central computer.

The choice of exponent d depends on the relative importance of efficiency and convenience. When d=2, the scheme requires the fewest number of modular multiplications, but the square roots of the v.sub.j do not always exist. It is thus necessary to modify the scheme in one of the following ways.

1. Each entity can choose its own set of small v.sub.j values, making sure that all of them are quadratic residues modulo the entity's n. These v.sub.j 's should be published in full in the public-key directory.

2. Use a longer list of standard v.sub.j values and each entity can choose a subset of k values which are quadratic residues modulo its own n (when n is the product of two primes, the universal list should contain about 4k numbers). The entity's chosen subset should be specified in the public-key directory or sent as part of the signature.

3. Each entity is allowed to modify the standard v.sub.j which are quadratic non-residues. A particularly simple way to achieve this is to pick a modulus n=pq where p=3 (mod 8) and q=7 (mod 8), since then exactly one of v.sub.j, -v.sub.j, 2v.sub.j, -2v.sub.j is a quadratic residue mod n for any v.sub.j. The appropriate variant of each v.sub.j can be specified in the public-key directory, sent as part of the signature, or deduced by the verifier himself during the verification of given signatures.

Alternatively, the entities can use d=3 and avoid all these complications. If the totient function of n is not divisible by 3, then any v.sub.j value has a cubic root mod n. However, this choice of d requires an additional modular multiplication during the generation and the verification of signatures.

The choice of the v.sub.j as the first k primes in the preferred embodiment is based on the fact that large values can make the scheme less efficient, and multiplicatively related values can make the scheme less secure. However, the v.sub.j can be chosen as any other sequence of k numbers.

As described in the parent Fiat-Shamir application, there are many possible tradeoffs between the size of the keys, the number of communicated bits, and the number of modular multiplications. All the optimization ideas described in the parent patent application are equally applicable to this variant.

In a second variant of the present invention, particular s.sub.j values are used. In this variant, the entities can choose their own moduli n, or can use a universal modulus n published by a trusted center (the first option is more secure, while the second option reduces the size of the public key). Each entity chooses a sequence of particular secret numbers s.sub.1, . . . , s.sub.k, and computes the corresponding public numbers v.sub.1, . . . , v.sub.k as ##EQU6## (note that this computation can be carried out even when the factorization of n is unknown). Each s.sub.j should be at least 64 bits long to protect it from exhaustive search attacks, and the exponent d should be large enough to guarantee that s.sub.j.sup.d >n (e.g., when .vertline.n.vertline.=512 and .vertline.s.sub.j .vertline.=64, d should be at least 16 to guarantee sufficient wraparound). The computed values of v.sub.1, . . . , v.sub.k (and the entity's modulus n, if applicable) are placed in the public key directory. The actual generation and verification of proofs of identity and signatures is carried out in the way specified above.

Known components can be used for the apparatus of the present invention as portrayed in FIG. 1 and FIG. 2, the means to carry out the several steps of the process being apparent to those skilled in the art.

Although the invention has been shown and described in terms of a specific preferred embodiment and variants, changes and modifications are possible which do not depart from the spirit, scope or contemplation of the inventive concepts disclosed herein. Such are deemed to fall within the purview of the invention as claimed.