Method and apparatus for compressing rabin signatures

A method and apparatus are disclosed for compressing Rabin signatures. The disclosed compression scheme compresses a Rabin signature, s, for a user having a public key, n, based on a continued fraction expansion of s/n. The continued fraction expansion of s/n can be performed by (i) computing principal convergents, ui/vi, for i equal to 1 to k, of a continued fraction expansion of s/n, where k is a largest integer for which principal convergents are defined; establishing an index l, such that vl<√{square root over (n)}≦vl+1; and generating a compressed Rabin signature (vl, m) for a message, m.

FIELD OF THE INVENTION

The present invention relates to Rabin signature schemes and, more particularly, to a method and apparatus for compressing Rabin signatures.

BACKGROUND OF THE INVENTION

Digital signatures are often employed to ensure the authenticity of transmitted information. A message generator generates a digital signature, s, using a public-key method, such as RSA public key cryptography techniques or the Rabin signature scheme. The message generator sends a message, m, and the signature, s, to a receiver. A Rabin signature, s, typically has a length on the order of 1024 bits. Thus, the Rabin signature scheme adds a significant overhead to a transmitted message. A number of techniques have thus been proposed or suggested for compressing Rabin signatures. Generally, the compression techniques aim to send only a portion of the Rabin signature, such that the transmitted portion is sufficient to reconstruct the full signature.

For example, Coron and Naccache have shown that a Rabin signature can be reconstructed if, for example, more than half of the most significant bits of s are known. See, International Published Patent Application No. WO 03/021864 A2, “Method and Apparatus of Reducing the Size of an RSA or Rabin Signature,” to Jean Sebastien Coron and David Nacacche, Published Mar. 13, 2003. Generally, Coron and Naccache use Coppersmith's LLL-based root finding method, as described in Don Coppersmith, “Finding a Small Root of a Univariate Modular Equation,” Advances in Cryptology, EUROCRYPT '96, Vol. 1070 of Lecture Notes in Computer Science, 155-165 (1996; Springer Verlag). The Coppersmith LLL-based root finding method leads to a slow decompression when the fraction of known bits is close to fifty percent (50%).

It has been suggested that a fast decompression method can be found when at least ⅔ of the bits are given. As used herein, a “fast compression method” means significantly faster than generating a signature (e.g., faster than 1 millisecond on a 1 MHz computer) and a “slow decompression method” means significantly slower than generating a signature (e.g., longer than 1 second on a 1 MHz computer). A need therefore exists for a fast compression method that can compress a Rabin signature by fifty percent.

SUMMARY OF THE INVENTION

Generally, a method and apparatus are disclosed for compressing Rabin signatures. The disclosed compression scheme compresses a Rabin signature, s, for a user having a public key, n, based on a continued fraction expansion of s/n. In one implementation, the continued fraction expansion of s/n is performed by (i) computing principal convergents, ui/vi, for i equal to 1 to k, of a continued fraction expansion of s/n, where k is a largest integer for which principal convergents are defined; establishing an index l, such that vl<√{square root over (n)}≦vl+1; and generating a compressed Rabin signature (vl, m) for a message, m.

DETAILED DESCRIPTION

FIG. 1illustrates a network environment100in which the present invention can operate. As shown inFIG. 1, a message generator120provides a message, m, and digital signature, s, to a compression server200, discussed further below in conjunction withFIG. 2. The compression server200in turn compresses the signature, s, and transmits the message, together with the compressed Rabin signature (vl, m) to a decompression server300, discussed further below in conjunction withFIG. 3. The decompression server300decompresses the message and compressed Rabin signature (vl, m) and provides the message and signature (s, m) to a message receiver180. Thus, the decompression server300receives the message m, and a portion vlof the signature and must solve for the unknown portion of the signature.

FIG. 2is a schematic block diagram of the compression server200ofFIG. 1. As shown inFIG. 2, the compression server200includes a memory210and a processor220. Memory210will configure the processor220to implement the methods, steps, and functions disclosed herein. The memory210could be distributed or local and the processor220could be distributed or singular. The memory210could be implemented as an electrical, magnetic or optical memory, or any combination of these or other types of storage devices. The term “memory” should be construed broadly enough to encompass any information able to be read from or written to an address in the addressable space accessed by processor220. With this definition, information on a network is still within memory210because the processor220can retrieve the information from the network. As shown inFIG. 2, the memory210includes a Rabin compression scheme400, discussed further below in conjunction withFIG. 4, that compresses Rabin signatures according to the present invention.

FIG. 3is a schematic block diagram of the decompression server300ofFIG. 1. As shown inFIG. 3, the decompression server300includes a memory310and a processor320that operate in the same manner asFIG. 2. The memory310includes a Rabin decompression scheme500, discussed further below in conjunction withFIG. 5, that decompresses Rabin signatures that were compressed according to the present invention.

Rabin Signatures

Using the Rabin scheme, the message generator randomly selects two prime numbers, p and q, as the private key of the message generator. The public key is the value n, equal to the product of p and q (n=p*q). For a detailed discussion of the Rabin scheme, see, for example, Michael O. Rabin, “Digitalized Signatures,” Foundation of Secure Computation, 155-69 (1978), incorporated by reference herein.

In order to apply a signature to a message, m, the message generator calculates the signature, s, as follows:
s2≡h(m) (mod n),
where h is a message formatting function. The above computation is often expressed as follows:
s=h(m)1/2mod(p*q).
The message generator sends the message, m, and the signature, s, to a receiver. The receiver can verify the signature based on the following expression:
h(m)=s2mod n.
In other words, the receiver of a Rabin signature can verify the signature by (i) squaring the signature, s, (ii) reducing the result modulo the message generator's public key, n, and (iii) comparing the result with the message digest of the message to be signed. The receiver accepts the message if the two values are equal.

Compression of Rabin Signatures

As previously indicated, compression techniques aim to send only a portion of the Rabin signature, such that the transmitted portion is sufficient to reconstruct the full signature. The compression scheme of the present invention computes a continued fraction expansion, discussed below, of the real number s/n. A signature is reconstructed given the largest integer that is a numerator of a principal convergent of s/n and that is smaller than the square root of n (√{square root over (n)}).

Thus, the compression scheme of the present invention replaces the signature, s, by a positive integer v smaller than √{square root over (n)}, such that v, n and m are sufficient to recover the signature s, without knowledge of the secret key. It is assumed that the message formatting function, h, is deterministic. In other words, the value h(m) can be computed without knowledge of the signature, s. For example, the signature scheme described in PKCS #1 Version 1.5 RSA Encryption Standard from RSA Data Security, Inc. of Redwood City, Calif., uses a deterministic formatting.

Continued Fractions

As previously indicated, the present invention computes a continued fraction expansion of the real number s/n. Let a be a real positive number. Define α0=α, qi=└αi┘ and define recursively αi+1=1/{αi} for all i≧0 until {αi}=0, where “└ ┘” indicates rounding down to the next integer and “{ }” indicates the fractional part of a number. Then, the partial convergents ui=viof s can be computed by u0=q0; v0=1; u1=q0q1; v1=q1+1 and ui+2=qi+2ui+1+ui; vi+2=qi+2vi+1+vi. The theory of continued fractions asserts that the principal convergents ui=viare close rational approximations of α. In particular, the following equation is satisfied:
|viα−ui|≦1/vi+1(1)
See, e.g., Donald E. Knuth, The Art of Computer Programming, Seminumerical Algorithms, Vol. 2, §4.5.3, Eq. (12), Addison Wesley (2nd edition, 1981); or Serge Lang, “Introduction to Diophantine Approximations,” Ch. 1, Theorem 5, Springer Verlag, (1995). If α is rational, then there exists an integer k with {αk}=0 and uk/vk=α.

Compression

FIG. 4is a flow chart describing an exemplary implementation of a Rabin compression scheme400incorporating features of the present invention. As shown inFIG. 4, the Rabin compression scheme400compresses a signature (s; m) as follows: If it is determined during step410that gcd(s, n)≠1 (where “gcd” indicates the greatest common denominator), then output an error during step420and stop. Otherwise, during step430let ui/vi, i=1, . . . , k be the principal convergents of the continued fraction expansion of s/n. During step440, let l be such that vl<√{square root over (n)}≦vi+1. Then, the compressed Rabin signature is (vl, m), where k is the largest integer for which principal convergents are defined.

Verification and Decompression

FIG. 5is a flow chart describing an exemplary implementation of a Rabin decompression scheme500incorporating features of the present invention. As shown inFIG. 5, the Rabin decompression scheme500initially receives (v, m), a compressed signature, during step510. If it is determined during step520that gcd(v, n)≠1, then output an error during step530and stop. Otherwise, during step540, compute 0≦t<n such that:
t≡h(m)v2(mod n).
The compressed signature is valid if and only if t is a square in Z. If the compressed signature is determined to be valid during step550, then set w=√{square root over (t)} and s=w/v (mod n) during step560and output (s, m) during step570.

Analysis

Thus, the Rabin compression scheme400and Rabin decompression scheme500of the present invention do not need to use the secret key. The following theorem shows that any valid Rabin signature can be converted into a valid compressed signature and vice versa. Thus, Rabin signatures and compressed signatures are equally difficult to forge.

Theorem 1. Let n be a Rabin public key that is square free.(I) If (s, m) is a valid Rabin signature, then the compression algorithm400generates a valid compressed signature for m or finds a nontrivial factor of n.(II) If (v, m) is a valid compressed signature, then the decompression algorithm500generates a valid Rabin signature for m.

Time Complexity

The Rabin compression scheme400requires a continued fraction expansion and takes time O(log(n)2). The Rabin decompression scheme500requires two multiplications and an inverse over Z/nZ and a square root in Z and hence also takes time O(log(n)2). It is noted that these bounds are obtained by using known methods. Asymptotically faster algorithms (e.g., FFT based gcd) are not optimal for typical key sizes.

Variant

An alternative compressed signature is (|r|, m), where r∈Z is such that |r|≦n and r≡vls (mod n). It can be shown that such an r exists when vl<√{square root over (n)}<vl+1. A compressed signature is valid if h(m)/r2mod n is a square in Z. Decompression is done using the equality (vl)2≡h(m)/r2(mod n). This variant is more expensive, because the verifier has to compute an additional modular inverse, but the variant has the advantage that the verification accepts both compressed and uncompressed signatures without modification.

Extension to RSA Signatures

The present invention can be extended to RSA signatures with small public exponent (i.e., e=3), but the benefits are smaller. For e equal to 3, the signature can be compressed to ⅔ of its size as follows.

Assume that:
s3≡h(m)(mod n),
is an RSA signature, where h is again a deterministic formatting function. To compress a signature, one computes the continued fraction expansion of s/n and selects the principal convergent ul/vlsatisfying vl<n2/3≦vl+1. The compressed RSA signature is (vl, m).

Given h(m) and vl, this value r can be found by checking whether either of h(m)(vl)3mod n or n-h(m)(vl)3mod n is a cube in Z. Finally, one can reconstruct the signature by setting s≡r/vl(mod n).