Patent Publication Number: US-7593526-B2

Title: Method and apparatus for compressing rabin signatures

Description:
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&#39;s LLL-based root finding method, as described in Don Coppersmith, “Finding a Small Root of a Univariate Modular Equation,” Advances in Cryptology, EUROCRYPT &#39;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, u i /v i , 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 v l &lt;√{square root over (n)}≦v l+1 ; and generating a compressed Rabin signature (v l , m) for a message, m. 
     A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a network environment in which the present invention can operate; 
         FIG. 2  is a schematic block diagram of the compression server of  FIG. 1 ; 
         FIG. 3  is a schematic block diagram of the decompression server of  FIG. 1 ; 
         FIG. 4  is a flow chart describing an exemplary implementation of a Rabin compression scheme incorporating features of the present invention; and 
         FIG. 5  is a flow chart describing an exemplary implementation of a Rabin decompression scheme incorporating features of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a network environment  100  in which the present invention can operate. As shown in  FIG. 1 , a message generator  120  provides a message, m, and digital signature, s, to a compression server  200 , discussed further below in conjunction with  FIG. 2 . The compression server  200  in turn compresses the signature, s, and transmits the message, together with the compressed Rabin signature (v l , m) to a decompression server  300 , discussed further below in conjunction with  FIG. 3 . The decompression server  300  decompresses the message and compressed Rabin signature (v l , m) and provides the message and signature (s, m) to a message receiver  180 . Thus, the decompression server  300  receives the message m, and a portion v l  of the signature and must solve for the unknown portion of the signature. 
       FIG. 2  is a schematic block diagram of the compression server  200  of  FIG. 1 . As shown in  FIG. 2 , the compression server  200  includes a memory  210  and a processor  220 . Memory  210  will configure the processor  220  to implement the methods, steps, and functions disclosed herein. The memory  210  could be distributed or local and the processor  220  could be distributed or singular. The memory  210  could 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 processor  220 . With this definition, information on a network is still within memory  210  because the processor  220  can retrieve the information from the network. As shown in  FIG. 2 , the memory  210  includes a Rabin compression scheme  400 , discussed further below in conjunction with  FIG. 4 , that compresses Rabin signatures according to the present invention. 
       FIG. 3  is a schematic block diagram of the decompression server  300  of  FIG. 1 . As shown in  FIG. 3 , the decompression server  300  includes a memory  310  and a processor  320  that operate in the same manner as  FIG. 2 . The memory  310  includes a Rabin decompression scheme  500 , discussed further below in conjunction with  FIG. 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:
 
 s   2   ≡h ( m ) ( mod n ),
 
where h is a message formatting function. The above computation is often expressed as follows:
 
 s=h ( m ) 1/2    mod  ( 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 )= s   2    mod 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&#39;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 α 0 =α, q i =└α i ┘ and define recursively α i+1 =1/{α i } for all i≧0 until {α i }=0, where “└ ┘” indicates rounding down to the next integer and “{ }” indicates the fractional part of a number. Then, the partial convergents u i =v i  of s can be computed by u 0 =q 0 ; v 0 =1; u 1 =q 0  q 1 ; v 1 =q 1 +1 and u i+2 =q i+2  u i+1 +u i ; v i+2 =q i+2 v i+1 +v i . The theory of continued fractions asserts that the principal convergents u i =v i  are close rational approximations of α. In particular, the following equation is satisfied:
 
|v i   α−u   i |≦1/ v   i+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 {α k }=0 and u k /v k =α.
 
     Compression 
       FIG. 4  is a flow chart describing an exemplary implementation of a Rabin compression scheme  400  incorporating features of the present invention. As shown in  FIG. 4 , the Rabin compression scheme  400  compresses a signature (s; m) as follows: If it is determined during step  410  that gcd(s, n)≠1 (where “gcd” indicates the greatest common denominator), then output an error during step  420  and stop. Otherwise, during step  430  let u i /v i , i=1, . . . , k be the principal convergents of the continued fraction expansion of s/n. During step  440 , let l be such that v l &lt;√{square root over (n)}≦v i+1 . Then, the compressed Rabin signature is (v l , m), where k is the largest integer for which principal convergents are defined. 
     Verification and Decompression 
       FIG. 5  is a flow chart describing an exemplary implementation of a Rabin decompression scheme  500  incorporating features of the present invention. As shown in  FIG. 5 , the Rabin decompression scheme  500  initially receives (v, m), a compressed signature, during step  510 . If it is determined during step  520  that gcd(v, n)≠1, then output an error during step  530  and stop. Otherwise, during step  540 , compute 0≦t&lt;n such that:
   t≡h ( m ) v   2 ( 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 step  550 , then set w=√{square root over (t)} and s=w/v (mod n) during step  560  and output (s, m) during step  570 .
 
     Analysis 
     Thus, the Rabin compression scheme  400  and Rabin decompression scheme  500  of 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 algorithm  400  generates 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 algorithm  500  generates a valid Rabin signature for m.       

     Time Complexity 
     The Rabin compression scheme  400  requires a continued fraction expansion and takes time O(log(n) 2 ). The Rabin decompression scheme  500  requires 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≡v l s (mod n). It can be shown that such an r exists when v l &lt;√{square root over (n)}&lt;v l+1 . A compressed signature is valid if h(m)/r 2  mod n is a square in Z. Decompression is done using the equality (v l ) 2 ≡h(m)/r 2  (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:
 
 s   3   ≡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 u l /v l  satisfying v l &lt;n 2/3 ≦v l+1 . The compressed RSA signature is (v l , m).
 
     Equation (1) implies
 
| v   1   s−u   l   n|≦n   1/3 ,
 
and thus there exists r∈Z with |r|≦n 1/3  and r 3 ≡h(m)(v l ) 3  (mod n).
 
     Given h(m) and v l , this value r can be found by checking whether either of h(m)(v l ) 3  mod n or n-h(m)(v l ) 3  mod n is a cube in Z. Finally, one can reconstruct the signature by setting s≡r/v l  (mod n). 
     As is known in the art, the methods and apparatus discussed herein may be distributed as an article of manufacture that itself comprises a computer readable medium having computer readable code means embodied thereon. The computer readable program code means is operable, in conjunction with a computer system, to carry out all or some of the steps to perform the methods or create the apparatuses discussed herein. The computer readable medium may be a recordable medium (e.g., floppy disks, hard drives, compact disks such as DVD, or memory cards) or may be a transmission medium (e.g., a network comprising fiber-optics, the world-wide web, cables, or a wireless channel using time-division multiple access, code-division multiple access, or other radio-frequency channel). Any medium known or developed that can store information suitable for use with a computer system may be used. The computer readable code means is any mechanism for allowing a computer to read instructions and data, such as magnetic variations on a magnetic media or height variations on the surface of a compact disk, such as a DVD. 
     It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.