Abstract:
A method for signing and subsequently verifying a digital message, including the following steps implemented using at least one processor-based subsystem: selecting parameters including an integer q and a relatively smaller integer p that is coprime with q; generating random polynomial f relating to p and random polynomial g relating to q; producing a public key that includes h, where h is equal to a product that can be derived using g and the inverse of f mod q; producing a private key from which f and g can be derived; storing the private key and publishing the public key; producing a message digest by applying a hash function to the digital message; producing a digital signature using the message digest and the private key; and performing a verification procedure utilizing the digital signature and the public key to determine whether the signature is valid.

Description:
RELATED APPLICATION 
     This application claims priority from U.S. Provisional Patent Application No. 61/965,912 filed Feb. 10, 2014, and said Provisional Patent Application is incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to the field of cryptography and, more particularly, to a public key digital signature technique. 
     BACKGROUND OF THE INVENTION 
     Public key digital signatures are important for secure exchange of information between plural parties, for example between computers or mobile devices, or between a smart card and a terminal. 
     A digital signature and authentication method and apparatus, which has in the past demonstrated advantageous operation, is described in U.S. Pat. No. 7,308,097, assigned to the same assignee as the present Application, incorporated by reference, and sometimes referred to as “NTRUSign” (mark of NTRU Cryptosystems, Inc.). Reference can also be made to “NTRUSign: Digital Signatures Using the NTRU Lattice”, J. Hoffstein, N. Howgrave Graham, J. Pipher, J. Silverman, and W. Whyte, Topics In Cryptology-CT-RSA 2003, Lecture Notes in Computer Science, Vol. 2612, Springer, Berlin, 2003, also incorporated by reference. 
     The signing technique in the &#39;097 Patent uses a mixing system based on multiplication in a ring and reduction modulo an ideal q in that ring; while the verification technique uses special properties of products of elements whose validity depends on elementary probability theory. The security of the identification/digital signature scheme comes from the interaction of reduction modulo q and the difficulty of forming products with special properties. In an embodiment of the digital signature scheme of the &#39;097 Patent, the security also relies on the experimentally observed fact that for most lattices, it is very difficult to find a vector whose length is only a little bit longer than the shortest vector, and it is also difficult to find a lattice vector that is quite close to a randomly chosen nonlattice vector. 
     Although the technique of the &#39;092 Patent has provided acceptable performance, and has exhibited good security, there is a need for an improved digital signature technique that is more efficient to use and has even better security. It is among the objectives of the present invention to provide improvement over the technique of the &#39;092 Patent and over other prior art techniques relating to digital signatures. 
     SUMMARY OF THE INVENTION 
     One drawback of the prior art, which is addressed by features of the present invention, is the relative complexity and computational requirements for key generation and signing. Another drawback is that every signature leaked some information about the private signing key, a fact that was eventually exploited to break the vanilla version of “NTRUSign” with no perturbations (see “Learning A Parallelepiped: Crypanalysis of GGH and NTRU Signatures”, P. Q. Nguyen and O. Regev, Advances in Cryptography—Eurocrypt 2006, Lecture Notes, in Computer Science, Vol. 4004, Springer, Berlin, 2006). 
     Applicant has discovered that through the use of two coprime integers, it is possible to create signatures using only a short half-basis. A further feature hereof involves the introduction of a rejection sampling technique in the context of an “NTRUSign” type of signature scheme, which assures that transcript distributions are completely decoupled from the keys that generate them. (Background rejection sampling is described, for example, in Lyubashevsky, V., Fiat-Shamir With Aborts, Applications to Lattice and Factoring-Based Signatures, In: ASIACRYPT 2009, pp. 598-616. Springer (2009). Reference can also be made to copending U.S. patent application Ser. No. 14/121,041, assigned to the same assignee as the present Application.) 
     In accordance with a form of the invention, a method is set forth for signing and subsequently verifying a digital message, comprising the following steps implemented using at least one processor-based subsystem: selecting parameters including an integer q and a relatively smaller integer p that is coprime with q; generating random polynomial f relating to p and random polynomial g relating to q; producing a public key that includes h, where h is equal to a product that can be derived using g and the inverse off mod q; producing a private key from which f and g can be devived; storing the private key and publishing the public key; producing a message digest by applying a hash function to the digital message; producing a digital signature using the message digest and the private key; and performing a verification procedure utilizing the digital signature and the public key to determine whether the signature is valid. In an embodiment of this form of the invention, the step of producing a digital signature comprises the following steps: (A) generating a noise polynomial; (B) deriving a candidate signature using the private key, the message digest, and the noise polynomial; (C) determining whether the coefficients of the candidate signature are within a predetermined range; and (D) repeating steps (A) through (C) until the criterion of step (C) is satisfied, and outputting the resultant candidate signature as the produced digital signature. 
     An embodiment of the invention further comprises transmitting the digital signature and, in this embodiment, the step of performing a verification procedure includes receiving the transmitted digital signature and performing the verification procedure on the received digital signature. In a variation of this form of the invention, the digital message can comprise a challenge communication from a verifier entity, and the digital signature can be transmitted to said verifier entity. 
     The prior art “NTRUSign” technique of U.S. Pat. No. 7,308,097 is based directly on the close vector problem. In other words, given a point in lattice space, the signer demonstrates that they can find a point in the lattice near to it. This requires the signer to know a full basis for the lattice, so during key generation “NTRUSign” has to generate a complete basis. It does this by starting with a half-basis (f, g) and completing the basis by finding (F, G). (See, again, the &#39;097 Patent and the above-referenced “NTRUSign” paper). In contrast, in the present invention, after key generation, the signer demonstrates a different ability: that given one lattice point, they can find another lattice point close by with a particular property. (Call the second lattice point the signature; in this case, the property is that the signature is equal to the message representative when taken mod p). Here, signing doesn&#39;t require the signer to know a full basis. The signer just needs to know enough short lattice vectors to find a vector that has the desired property. An advantage hereof is that this can be done with only a half-basis. This allows key generation to stop after generating half the basis, without requiring the computationally intensive step of completing the basis. It also makes signing more efficient as only the relatively smaller half-basis need be used. 
     Further features and advantages of the invention will become more readily apparent from the following detailed description when taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a system that can be used in practicing embodiments of the invention. 
         FIG. 2  is a flow diagram of a public key digital signature technique which, when taken with the subsidiary flow diagrams referred to therein, can be used in implementing embodiments of the invention. 
         FIG. 3  is a flow diagram, in accordance with an embodiment hereof, of a routine for key generation. 
         FIG. 4  is a flow diagram, in accordance with an embodiment hereof, of a routine for signing a digital message. 
         FIG. 5  is a flow diagram, in accordance with an embodiment hereof, of a routine for verification of a digital signature. 
         FIG. 6  is a flow diagram, in accordance with another embodiment hereof, of a routine for signing a digital message. 
         FIG. 7  s a flow diagram, in accordance with another embodiment hereof, of a routine for verification of a digital signature. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a block diagram of a system that can be used in practicing embodiments of the invention. Two processor-based subsystems  105  and  155  are shown as being in communication over an insecure channel  50 , which may be, for example, any wired or wireless communication channel such as a telephone or internet communication channel. The subsystem  105  includes processor  110  and the subsystem  155  includes processor  160 . The subsystems can typically comprise mobile devices, computers, or terminals. When programmed in the manner to be described, the processors  110  and  160  and their associated circuits can be used to implement an embodiment of the invention and to practice an embodiment of the method of the invention. The processors  110  and  160  may each be any suitable processor, for example an electronic digital processor or microprocessor. It will be understood that any general purpose or special purpose processor, or other machine or circuitry that can perform the functions described herein, electronically, optically, or by other means, can be utilized. The subsystem  105  will typically include memories  123 , clock and timing circuitry  121 , input/output functions  118  and display  125 , which may all be of conventional types. Inputs can include a touchscreen/keyboard input as represented at  103 . Communication is via transceiver  135 , which may comprise a modem or any suitable device for communicating signals. 
     The subsystem  155  in this illustrative embodiment can have a similar configuration to that of subsystem  105 . The processor  160  has associated input/output circuitry  164 , memories  168 , clock and timing circuitry  173 , and a display  176 . Inputs include a touchscreen/keyboard  155 . Communication of subsystem  155  with the outside world is via transceiver  162  which, again, may comprise a modem or any suitable device for communicating signals. 
       FIG. 2  illustrates a basic procedure that can be utilized with a public key digital signature technique, and refers to routines illustrated by other referenced flow diagrams which describe features in accordance with an embodiment of the invention. Reference can also be made to Appendix I for further details of the invention. The block  210  represents the generating of the public key and private key signals and data, and the publishing of the public key. The routine of an embodiment thereof is described in conjunction with the flow diagram of  FIG. 3 . In the present example, this operation can be performed, for example, at the processor-based subsystem  105  of  FIG. 1 . The public key information can be published; that is, made available to any member of the public or to any desired group to whom the private key holder desires to send the digital signatures. Typically, although not necessarily, the public key may be made available at a central public key library facility or website where a directory of public key holders and their public keys are maintained. 
     The block  250  represents a routine that can be employed (that is, in this example, by the user of processor-based subsystem  155  of  FIG. 1 ) for signing the digital message. This routine, in accordance with an embodiment of the invention, is described in conjunction with the flow diagram of  FIG. 4 . In this example, the digital signature is then transmitted over the channel  50  ( FIG. 1 ). 
     The block  270  represents a routine that can be employed (that is, in this example, by the user of processor-based subsystem  155  of  FIG. 1 ) for using, inter alia, the public key to implement a verification procedure to either accept or reject the digital signature. This routine, in accordance with an embodiment of the invention, is described in conjunction with the flow diagram of  FIG. 5 . 
       FIG. 3  is a flow diagram of a routine, represented by the block  210  of  FIG. 2 , in accordance with an embodiment of the invention, for implementing key generation. Reference can also be made to Appendix I. The block  310  represents the defining and/or inputting of parameters used in key generation, which include: R, a polyhnomial quotient ring in which products of small elements are also small; q, an integer; p, a small integer or polynomial coprime with q (as ideals of R); Rq, the ring with coefficients drawn from Zq (where Zq is the integers taken mod q); and (R f , R g ), the space of private keys, two subsets of the ring Rq whose members are “small” relative to arbitrary members of Rq. The block  320  represents the step of randomly selecting f in p*R f  and g in R g . [As described in Appendix I, the random polynomials can be chosen such that f is p times a trinary polynomial and such that ∥g∥≦p/2. Writing f=pF, so F is trinary, it is assumed that F is invertible modulo q and modulo p. If not, this f can be discarded and a new one chosen.] Then, as represented by the block  330 , the inverse of f (that is, f −1 ) in the ring Rq, called f inv , is computed, and h is computed as h=f inv *g in the ring q. The private key f, g and the public key h can then be output, as represented by the block  340 . 
       FIG. 4  is a flow diagram of a routine, represented by the block  240  of  FIG. 2 , in accordance with an embodiment of the invention, for implementing the signing of a digital message using, inter alia, the private key. Reference can also be made to Appendix I. 
     Referring to  FIG. 4 , the block  410  represents the inputting of the following: R, a polynomial quotient ring in which products of small elements are also small; q, an integer; p, a small integer or polynomial coprime with q (as ideals of R); R q , the ring R with coefficients drawn from Z q ; R h , the hash output space, a subset of (R q ×R q ) where every element is equal to itself mod p; B R , the L ∞  norm of the noise=floor ((q−p)/2p); B s , the L ∞  norm of the s component of the signature; B t , the L ∞  norm of the t component of the signature; H, a hash function taking as input a message and a public key; (f, g), the private key; h, the public key; and M, the message to be signed. (M corresponds to μ in Appendix I). 
     As represented by the block  420 , a document hash, mod p, designated (s p , t p ), is calculated as H(M, h); that is the hash of the message and the public key. Next, the loop of blocks  430 ,  440 , and  450  implements the rejection sampling of candidate signatures, and selection of a candidate signature that meets a size criterion (see also Appendix I). The block  430  represents randomly generating noise r with L ∞  norm less than or equal to B R . The block  440  represents the successive calculations of s 0 , t 0 , a, and (s, t) as follows:
 
 s   0   =s   p   +pr  
 
 t   0   =h*s   0  mod  q  
 
 a=g   −1 *( t   p   −t   0 ) mod  q  
 
( s, t )=( s   0   , t   0 )+( a*f, a*g )
 
     Next, the decision block  450  represents the step of determining whether the coefficients of the candidate signature and its components are in a predetermined range, dependent on range-defining integers. In this embodiment, a determination is made of whether all of the following are true:
 
 L   ∞  norm of ( a*f )≦ q/ 2 −B? 
 
 L   ∞  norm of ( a*g )≦ q/ 2 −B? 
 
 L   ∞  norm of  s≦B   s ?
 
L ∞  norm of ≦B t ?
 
If not, the block  430  is re-entered, and the process steps of blocks  430 ,  440  and  450  are repeated until a candidate digital signature which meets the criteria of block  450  is obtained. The block  460  is then entered, this block representing the outputting of the qualifying candidate signature, that is, the encoded signed message s, or (s, t) (see Appendix 1).
 
       FIG. 5  is a flow diagram of a routine, represented by the block  270  of  FIG. 2 , in accordance with an embodiment of the invention, for implementing verification of whether the received digital signature is valid. Reference can also be made to Appendix I. 
     The block  510  represents the inputting of the following: R, a polynomial quotient ring in which products of small elements are also small; q, an integer; p, a small integer or polynomial coprime with q (as ideals of R); R q , the ring R with coefficients drawn from Z q ; R h , the hash output space, a subset of (R q ×R q ) where every element is equal to itself mod p; B s , the L ∞  norm of the s component of the signature; B t , the L ∞  norm of the t component of the signature; H, a hash function taking as input a message and a public key; h, the public key; M, the message; A, the additional data; and s, the signature. (The additional data is typically added to the hash of the message for enhanced security.) 
     Next, as represented by the block  520 , the following calculations are made:
 
( s   p   , t   p )= H ( M, A )
 
 t=s*h  mod  q  
 
A determination is then made (decision block  530 ) as to whether both of the following hold:
 
The L ∞  norm of s≦B s  
 
The L ∞  norm of t≦B t  
 
If not, the signature is rejected (block  550 ). If, however, the inquiry of block  530  is answered affirmatively, the decision block  540  is entered, this block representing the inquiry of whether (s p , t p ) equals (s, t) mod p. If not, the signature is rejected (block  550 ) (s, t) mod p or, if so, the signature is accepted (block  560 ).
 
       FIGS. 6 and 7  respectively illustrate a further embodiment of the signing routine of  FIG. 4  and a further embodiment of the verification routine of  FIG. 5 . The routines are similar to those of their counterparts but, in some respects, are generalized to show that variations can be implemented within the intended scope hereof. 
     In the signing routine of  FIG. 6 , the block  610  corresponds to block  410  of  FIG. 4 , except that in this case, an input is provided for DistR, the distribution function for the random noise, which outputs noise in Rq, and an input is provided for SpaceS, the permitted space for the signatures to lie in. Also, in this case, as represented by block  620 , (s p , t p ) is calculated as H (M, A), where A is the input additional data added to the hash function. The block  630  represents the step of randomly generating noise r from the distribution DistR. Then, the calculations of block  640  correspond to the previously described calculations of block  440 . Inquiry is then made (decision block  650 ) as to whether (s, t) is in SpaceS. If not, block  630  is re-entered, and the steps of blocks  630 ,  640 , and  650  are repeated until a candidate digital signature which meets the criterion of block  650  is obtained. The block  650  is then entered, this block representing the outputting of the qualifying candidate signature; that is, the signed message (s, t). In this manner, rejection sampling is achieved. Regarding the more generalized verification routine of  FIG. 7 , the block  710  corresponds to block  510  of  FIG. 5 , except that in this case, inputs are provided for SpaceS and L h , the lattice defined by the public key. Inquiry is made (block  730 ) as to whether (s, t) is in SpaceS and (s, t) is in L h . If not, the signature is rejected (block  770 ). If so, (s p ,t p ) is calculated as H(M, A) (block  740 ), and inquiry is made (decision block  750 ) as to whether (s p , t p ) equals (s, t) mod p. If not, the signature is rejected (block  770 ). If so, however, the signature is accepted. 
     The invention has been described with reference to particular preferred embodiments, but variations within the spirit and scope of the invention will occur to those skilled in the art. For example, while a digital signature technique has been described, it will be understood that an authentication producer of the challenge-response-verification type can alternatively be implemented, using the technique hereof and employing the challenge as the message to be signed. Also, it will be understood that coefficients of polynomials can alternatively be represented in other forms including, but not limited to, matrices.