Secure user identification based on ring homomorphisms

This application claims priority from U.S. Provisional Patent Application No. 60/132,199, filed May 3, 1999, and said Provisional Patent Application is incorporated herein by reference.

The present invention relates generally to secure communication and document identification over computer networks or other types of communication systems and, more particularly, to secure user identification and digital signature techniques based on ring homomorphisms. The invention also has application to communication between a card, such as a “smart card”, or other media, and a user terminal.

User identification techniques provide data security in a computer network or other communications system by allowing a given user to prove its identity to one or more other system users before communicating with those users. The other system users are thereby assured that they are in fact communicating with the given user. The users may represent individual computers or other types of terminals in the system. A typical user identification process of the challenge-response type is initiated when one system user, referred to as the Prover, sends certain information in the form of a commitment to another system user, referred to as the Verifier. Upon receipt of the commitment, the verifier sends a challenge to the Prover. The Prover uses the commitment, the challenge, and its private key to generate a response, which is sent to the Verifier. The Verifier uses the commitment, the response and a public key to verify that the response was generated by a legitimate prover. The information passed between the Prover and the Verifier is generated in accordance with cryptographic techniques which insure that eavesdroppers or other attackers cannot interfere with or forge the identification process.

It is well known that a challenge-response user identification technique can be converted to a digital signature technique by the Prover utilizing a one-way hash function to simulate a challenge from a Verifier. In such a digital signature technique, a Prover generates a commitment and applies the one-way hash function to it and a message to generate the simulated challenge. The Prover then utilizes the simulated challenge, the commitment and a private key to generate a digital signature, which is sent along with the message to the Verifier. The Verifier applies the same one-way hash function to the commitment and the message to recover the simulated challenge and uses the challenge, the commitment, and a public key to validate the digital signature.

One type of user identification technique relies on the one-way property of the exponentiation function in the multiplicative group of a finite field or in the group of points on an elliptic curve defined over a finite field. This technique is described in U.S. Pat. No. 4,995,082 and in C. P. Schnorr, “Efficient Identification and Signatures for Smart Cards,” in G. Brassard, ed., Advances in Cryptology—Crypto '89, Lecture Notes in Computer Science 435 , Springer-Verlag, 1990, pp. 239-252. This technique involves the Prover exponentiating a fixed base element g of the group to some randomly selected power k and sending it to the verifier. An instance of the Schnorr technique uses two prime numbers p and q chosen at random such that q divides p−1, and a number g of order q modulo p is selected. The numbers p, q, and g are made available to all users. The private key of the Prover is x modulo q and the public key y of the Prover is g −x modulo p. The Prover initiates the identification process by selecting a random non-zero number z modulo q. The Prover computes the quantity gz modulo p and sends it as a commitment to the Verifier. The Verifies selects a random number w from the set of integers {1,2, . . . , 2 t } where t is a security number which depends on the application and in the above-cited article is selected as 72. The Verifier sends w as a challenge to the Prover. The Prover computes a quantity u that is equal to the quantity z+xw modulo q as a response and sends it to the Verifier. The Verifier accepts the Prover as securely identified if gz is found to be congruent modulo p to the quantity g u y z .

Another type of user identification technique relies on the difficulty of factoring a product of two large prime numbers. A user identification technique of this type is described in L. C. Guillou and J. J. Quisquater, “A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing Both Transmission and Memory,” in C. G. Gunther, Ed. Advances in Cryptology—Eurocrypt '88, Lecture Notes in Computer Science 330 , Springer-Verlag, 1988, pp. 123-128. This technique involves a Prover raising a randomly selected argument g to a power b modulo n and sending it to a Verifier. An instance of the Guillou-Quisquater technique uses two prime numbers p and q selected at random, a number n generated as the product of p and q, and a large prime number b also selected at random. The numbers n and b are made available to all users. The private key of the Prover is x modulo n and the public key y of the Prover is x −b modulo n. The Prover initiates the identification process by randomly selecting the number g from the set of non-zero numbers modulo n. The Prover computes the quantity g b modulo n and sends it as a commitment to the Verifier. The Verifier randomly selects a number c from the set of non-zero numbers modulo b and sends c as a challenge to the Prover. The Prover computes the number h that is equal to the quantity gx c modulo n as a response and sends it to the Verifier. The Verifier accepts the Prover as securely identified if g b is found to be congruent modulo n to h b y c .

Although the above-described Schnorr and Guillou-Quisquater techniques can provide acceptable performance in many applications, there is a need for an improved technique which can provide greater computational efficiency than these and other prior art techniques, and which relies for security on features other than discrete logarithms and integer factorization.

The present invention provides a method, system and apparatus for performing user identification, digital signatures and other secure communication functions based on ring homomorphisms. The ring homomorphism in accordance with the invention may utilize two rings R and B, a ring homomorphism ø:R→B, and four subsets R f , R g , R h , and R c , of R. One element f in the set R f serves as a private key for a given user. The result ø(f) of evaluating the homomorphism ø at the element f serves as the public key of the given user.

Copending U.S. patent application Ser. No. 08/954,712, filed Oct. 20, 1997, and assigned, in joint ownership, to the same assignee as the present Application, discloses a user identification technique and digital signature technique based on partial evaluation of constrained polynomials over a finite field, and describes use of a response signal (such as in a commitment/challenge/response type of technique) that is generated by computing a polynomial as the product of a commitment polynomial with the sum of a private key and a challenge polynomial. The techniques hereof provide substantial improvements in computational efficiencies and lowering of processing requirements at equivalent security levels.

In accordance with one aspect of the invention, a secure user identification technique is provided in which one of the system users, referred to as a Prover, randomly selects an element g from the set R g . The Prover evaluates the homomorphism ø at the element g and transmits the result ø(g) to another user referred to as the Verifier. The Verifier randomly selects a challenge element c from the set R c . The Verifier transmits c to the Prover. The Prover generates a response element h using the private key f and the elements c and g. The element h may be generated in the form g*(f+c*g) using addition + and multiplication * in the ring R; or more generally by choosing a set of elements g i , receiving a set of challenge elements c i , creating modified challenge elements d i from the challenge elements c i , transmitting the modified challenge elements d i to the Verifier, and generating the response element h as a polynomial function of the secret key f and the selected elements g i , c i , and d i . The Verifier checks that the element h is in the set R h . The Verifier also evaluates the homomorphism ø at the element h and compares the result ø(h) to a function of ø(g), ø(c), and the public key ø(f) of the Prover. For example, if the element h is generated in the form g*(f+c*g), then the verifier may check if the value ø(h) is equal to the value ø(g)*(ø(f)+ø(c)*ø(g)) using addition + and multiplication * in the ring B. If the element h is in the set R h and if the comparison of ø(h) to the function of ø(g), ø(c), and the public key ø(f) is correct, then the Verifier accepts the identity of the Prover. The Verifier may use the above-noted comparison for secure identification of the Prover, for authentication of data transmitted by the Prover, or for other secure communication functions.

In accordance with another aspect of the invention, a secure user identification technique is provided in which one of the system users, referred to as a Verifier, randomly selects a challenge element c from the set R c . The Verifier transmits c to another user referred to as the Prover. The Prover randomly chooses an element g from the set R g and generates a response element h using the private key f and the elements c and g. The element h may be generated in the form g*(f+c*g) using addition + and multiplication * in the ring R; or more generally by generating the response element h as a polynomial function P(f,c,g) of the secret key f and the selected elements g and c. The Verifier checks that the element h is in the set R h . The Verifier also evaluates the homomorphism φ at the element h and verifies that the polynomial equation P(ø(f),ø(c),X)−ø(h)=0 has a solution X in the ring B. For example, if the element h is generated in the form g*(f+c*g), then the verifier may check if the polynomial ø(c)X 2 +ø(f)X−ø(h)= 0 has a solution in B by checking if the element ø(f) 2 +4ø(c)ø(h) is the square of an element in B. If the element h is in the set R h and if the polynomial equation P(ø(f),ø(c),X)−ø(h)=0 has a solution X in the ring B, then the Verifier accepts the identity of the Prover. The Verifier may use the above-noted comparison for secure identification of the Prover, for authentication of data transmitted by the Prover, or for other secure communication functions.

In accordance with another aspect of the invention, a digital signature technique is provided. A Prover randomly selects an element g from the set R g . The Prover then computes ø(g) and applies a hash function to the element ø(g) and a message m to generate a challenge element c=Hash(ø(g),m) in the set R c . The Prover utilizes g, c, and the private key f to generate an element h. The element h may be generated in the form g*(f+c*g) using addition + and multiplication * in the ring R, or more generally by choosing a set of polynomials g i , generating a corresponding set of elements c i using the hash function, and generating the response element h as a polynomial function h=P(f,c i ,g i ). The Prover than transmits m, ø(g) and h to the Verifier. The Verifier checks that the element h is in the set R h . The Verifier computes c=Hash(ø(g),m), evaluates ø(c) and ø(h), and compares the values of ø(g), ø(c), and ø(h) with the public key ø(f) of the Prover. For example, if the element h is generated in the form g*(f+c*g), then the verifier may check if the value ø(h) is equal to the value ø(g)*(ø(f)+(c)*ø(g)) using addition + and multiplication * in the ring B. If the element h is in the set R h and if the comparison of ø(h) to the function of ø(g), ø(c), and the public key ø(f) is correct, then the Verifier accepts the signature of the Prover on the message m.

In accordance with another aspect of the invention, a digital signature technique is provided. A Prover randomly selects an element g from the set R g . The Prover then applies a hash function to a message m to generate a challenge element c=Hash(m) in the set R c . The Prover utilizes g, c, and the private key f to generate an element h. The element h may be generated in the form g*(f+c*g) using addition + and multiplication * in the ring R; or more generally by generating the response element h as a polynomial function P(f,c,g) of the secret key f and the selected elements g and c. The Prover than transmits m and h to the Verifier. The Verifier checks that the element h is in the set R h . The Verifier computes c=Hash(m), evaluates ø(c) and ø(h), and verifies that the polynomial equation ø(P)(ø(f),ø(c), X)−ø(h)= 0 has a solution X in the ring B, where ø(P) is the polynomial P with the homomorphism øapplied to its coefficients. For example, if the element h is generated in the form g*(f+c*g), then the verifier may check if the polynomial ø(c)X 2 +ø(f)X−ø(h)= 0 has a solution in B by checking if the element ø(f) 2 +4ø(c)ø(h) is the square of an element in B. If the element h is in the set R h and if the polynomial equation ø(P)(ø(f),ø(c),X)−ø(h)= 0 has a solution X in the ring B, then the Verifier accepts the signature of the Prover on the message m.

The present invention provides a method, system and apparatus for performing user identification, digital signatures and other secure communication functions based more particularly on ring homomorphisms given by partial evaluation of constrained polynomials over a finite field. The ring R in accordance with the invention may utilize polynomials of degree less than N with coefficients in the field F q of q elements, where N divides q-1 and q is a power of a prime number. An exemplary predetermined condition on the subsets R f , R g and R c of R may specify that the coefficients are chosen from a predetermined set of values such as, for example, the values 0, 1, and −1 in the field F q , and an exemplary predetermined condition on the subset R h may specify that the coefficients are small, as for example the number q is a prime number, the coefficients of h are chosen between −q/2 and q/2, and the sum of the squares of the coefficients of h is smaller than q 2 . A number of other conditions on the subsets R f , R g and R c may be used in conjunction with or in place of these exemplary conditions. The partial evaluation ring homomorphism in accordance with the invention may consist of a ring B=F q s and a set of elements a 1 , . . . , a s in a public subset S of F q and a homomorphism ø:R→B corresponding to evaluation of a polynomial at the values in S according to the formula ø(p(X))=(p(a 1 ), p(a 2 ), . . . , p(a 3 )). An exemplary condition on the ring R may specify that R is the ring of polynomials modulo the relation X N −1 and an exemplary condition on the set of elements S may specify that each element a i in the set S satisfies the formula a i N =1. A number of other conditions on the ring R and on the set S may be used in conjunction with or in place of these exemplary conditions.

The use of ring homomorphisms, and more particularly ring homomorphisms given by partial evaluation of constrained polynomials over a finite field, in accordance with the invention provides user identification and digital signature techniques which are computationally more efficient than prior art techniques. The security of the techniques of the present invention depend on the fact that recovering an element of a ring from its value by a homomorphism, and more particularly recovering a polynomial from its partial evaluation, can, in certain circumstances, be a particularly difficult task.

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.

FIG. 1 is a block diagram of a type of system that can be used in practicing embodiments of the invention, for example when the processors thereof are suitably programmed in accordance with the flow diagrams hereof.

FIG. 2 is a flow diagram which illustrates a key creation technique in accordance with an exemplary embodiment of the present invention.

FIG. 3 is a flow diagram which illustrates a user identification technique in accordance with an exemplary embodiment of the present invention.

FIG. 4 is a flow diagram which illustrates a further user identification technique in accordance with another exemplary embodiment of the present invention.

FIG. 5 is a flow diagram which illustrates a digital signature technique in accordance with an exemplary embodiment of the present invention.

FIG. 6 is a flow diagram which illustrates a further digital signature technique in accordance with another exemplary embodiment of the present invention.

FIG. 1 is a block diagram of a system that can be used in practicing embodiments of the invention. A number of processor-based subsystems, represented at 105 , 155 , 185 , and 195 , are shown as being in communication over an insecure channel or network 50 , which may be, for example, any wired, optical, and/or wireless communication channel such as a telephone or internet communication channel or network. The subsystem 105 includes processor 110 and the subsystem 155 includes processor 160 . When programmed in the manner to be described, the processors 110 and 160 and their associated circuits can be used to practice embodiments 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 processors may be, for example, Intel Pentium processors. The subsystem 105 may typically include memories 123 , clock and timing circuitry 121 , input/output functions 118 , and monitor 125 , which may all be of conventional types. Inputs can include a keyboard input as represented at 103 and any other suitable input. Communication is via transceiver 135 , which may comprise a modem, high speed coupler, or any suitable device for communicating signals. The subsystem 155 in this illustrative system 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 monitor 176 . Inputs include a keyboard 163 and any other suitable input. Communication of subsystem 155 with the outside world is via transceiver 162 which, again, may comprise a modem, high speed coupler, or any suitable device for communicating signals. As represented in the subsystem 155 , a terminal 181 can be provided for receiving a smart card 182 or other media. A “user” can also be a person's or entity's “smart card”, the card and its owner typically communicating with a terminal in which the card is inserted. The terminal can be an intelligent terminal, or can communicate with an intelligent terminal. It will be understood that the processing and communications media that are described are exemplary, and that the invention can have application in many other settings. The blocks 185 and 195 represent further subsystems on the channel or network.

The present invention will be illustrated below in conjunction with exemplary user identification and digital signature techniques carried out by a Prover and a Verifier in a communication network such as that of FIG. 1 in which, for example, for a particular communication or transaction, any of the subsystems can serve either role. It should be understood, however, that the present invention is not limited to any particular type of application. For example, the invention may be applied to a variety of other user and data authentication applications. The term “user” may refer to both a user terminal as well as an individual using that terminal, and, as indicated above, the terminal maybe any type of computer or other digital data processor suitable for directing data communication operations. The term “Prover” as used herein is intended to include any user which initiates an identification, digital signature or other secure communication process. The term “Verifier” is intended to include any user which makes a determination as to whether a particular communication is legitimate. The term “user identification” is intended to include identification techniques of the challenge-response type as well as other types of identification, authentication and verification techniques.

The user identification and digital signature techniques in accordance with the present invention are based on evaluation of ring homomorphisms. An exemplary embodiment of the present invention is based on the partial evaluation homomorphism of constrained polynomials over a finite field. An exemplary finite field F q =Z/qZ is defined for a prime number q. An exemplary ring R=F q [X]/(X q−1 −1) is a ring of polynomials with coefficients in the finite field F q modulo the ideal generated by the polynomial X q−1 −1. An exemplary homomorphism ø:R→F q s is a homomorphism ø(f(X))=(f(a 1 ), . . . , f(a t )) for an ordered set S={a 1 , . . . , a t } of non-zero integers modulo q. An additional exemplary condition is that if a is in S, then a −1 is also in S. With suitable restrictions on f(X) and a suitable choice of set S, it is infeasible to recover f(X) when given only ø(f(X)). As will be described in greater detail below, this provides a one-way function which is particularly well-suited to use in implementing efficient user identification and digital signatures.

The identification and digital signature techniques make use of the multiplication rule in the ring R. Given a polynomial A(X)=A 0 +A 1 X+ . . . +A q−2 X q−2 in R and a polynomial B(X)=B 0 +B 1 X+ . . . +B q−2 X q−2 in R, an exemplary product may be given by:

C ( X )= A ( X ) B ( X=C 0 +C 1 X+ . . . +C q−2 X q−2

where C 0 , . . . , C q−2 are given by:

C i =A 0 B i +A 1 B 1−1 + . . . A i B 0 A i+1 B q−2 B q−3 + . . . +A q−2 B i+1 (modulo q).

All reference to multiplication of polynomials in the remaining description should be understood to refer to the above-described exemplary multiplication in R. It should also be noted that the above-described multiplication rule is not a requirement of the invention, and alternative embodiments may use other types of multiplication rules.

An exemplary set of constrained polynomials R f is the set of polynomials in R with bounded coefficients. Given the prime number q and the polynomial f(X), it is relatively easy to generate ø(f)=(f(a 1 ), . . . , f(a t )). However, appropriately selected restrictions on the polynomials in R f can make it extremely difficult to invert this function to determine a polynomial F(X) in R f such that ø(F)=ø(f). The difficulty of the inversion is generally dependent on the type of restrictions placed on the polynomials in R f . For example, if easily satisfied restrictions are placed on the polynomials, basic interpolation techniques could be used to find some polynomial F(X) in R f such that ø(F)=ø(f). It will be shown in greater detail below that establishing appropriate restrictions on the polynomials in R f can provide adequate levels of security. An exemplary identification technique in accordance with the invention uses a number of system parameters which are established by a central authority and made public to all users. These system parameters include the above-noted prime number q and set S={a 1 , . . . , a t } of t non-zero elements of the finite field F q and appropriate sets of bounded coefficient polynomials R f ,R g ,R c . FIG. 2 illustrates the creation of a public/private key pair. After establishment of parameters (block 220 ) a Prover randomly chooses a secret polynomial f(X) in R f as its private key (block 230 ). The public key of the Prover is then generated as ø(f)=(f(a 1 ), . . . , f(a t )) which represents the ordered evaluation of the secret polynomial f(X) at the t elements of S, and the public key can be published (block 240 ).

FIG. 3 illustrates an exemplary identification process. The identification process is initiated in the Commitment Phase (block 310 ) by the Prover generating a polynomial g(X) with bounded coefficients. The polynomial g(X) may be selected at random from a set R g that is restricted in a manner to be described below. The Prover uses the polynomial g(X) and the public set of values S={a 1 , . . . , a t } to compute a commitment ø(g)=(g(a 1 ), . . . , g(a t )) and sends the commitment to the Verifier.

The Verifier initiates the Challenge Phase (block 330 ) by generating a challenge polynomial c(X) with bounded coefficients and sending it to the Prover. The polynomial c(X) may be generated by random selection from a set of polynomials R c that is restricted in a manner to be described below. The Prover initiates the Response Phase (block 350 ) by verifying that the challenge polynomial c(X) is in the restricted set of polynomials R c and then using the polynomials c(X),g(X) and the secret polynomial f(X) to generate the response polynomial h(X) given by

h ( X )= g ( X )( f ( X )+ c ( X ) g ( X ))

and sending the response polynomial h(X) to the Verifier. The Verifier initiates the Verification Phase (block 360 ) by using its knowledge of ø(g), c(X), and the public key ø(f) to check that the response polynomial h(X) was generated using the private key f(X) of the Prover by comparing:

h ( a 1 ) to g ( a j )( f ( a 1 )+ c ( a i ) g ( a j )) for i= 1,2 , . . . , t.

This check may be expressed as comparing whether ø(h) is equal to ø(g)(ø(f)+ø(c)ø(g)). The Verifier in the Verification Phase also checks whether or not the coefficients of h(X) are appropriately bounded, given that a legitimate h(X) will have bounded coefficients and will belong to a restricted set R h of polynomials. The restrictions on the set R h depend on the choice of the above noted sets R f ,R g and R c . The Verifier accepts the Prover as legitimate if the response polynomial h(X) transmitted by the Prover passes the checks of steps (A) and (B) of the Verification Phase. The Verifier may perform a number of other checks as part of the identification process. For example, prior to performing steps (A) and (B) of the Verification Phase, the Verifier may check that g(1), provided by the Prover as an element of the commitment ø(g), has a particular expected value.

A first exemplary set of system parameters suitable for use with the above-described identification technique will now be described. It should be emphasized that these and other exemplary parameters described herein are illustrative only and that numerous alternative sets of parameters could also be used. In the first exemplary set of parameters, the prime number q is selected as 769 , and the set S includes t=384 non-zero integers modulo q. The set S is constructed such that if a is an element of S, then a −1 is also an element of S. It should be noted that a given implementation may utilize only a subset of the t elements of S. The set R f is the set of all polynomials f(X) of degree less than 768 constructed with 51 coefficients of value 1, with 51 coefficients of value −1, and all other coefficients set to zero. The set R g is the set of all polynomials g(X) of degree less than 768 constructed with 51 coefficients of value 1, with 51 coefficients of value −1, and all other coefficients set to zero. The set R c is the set of all polynomials c(X) of degree less than 768 constructed with 5 coefficients of value 1, with 5 coefficients of value −1, and all other coefficients set to zero. Finally, the set R h is the set of polynomials h(X)=h 0 +h 1 X+ . . . +h 767 X 767 of degree less than 768 whose coefficients are between −384 and 384 and which satisfy the inequality h 0 2 +h 1 2 + . . . +h 767 2 <769 2 =591361. The user identification technique described in conjunction with FIG. 3 above is then implemented using polynomials selected from the sets R f ,R g ,R c and R h .

Alternative embodiments of the invention may utilize several private key polynomials f 1 , . . . f n , several commitment polynomials g 1 , . . . g r and several challenge polynomials c 1 , . . . , c s and may further utilize other functions of the key polynomials, commitment polynomials, and challenge polynomials to generate several response polynomials h 1 , . . . h u . For example, h i , could be generated as the value

h i =P i ( f 1 , . . . , f n , g 1 , . . . , g r , c 1 , . . . , c g )

for polynomials P i (U 1 , . . . , f n ,g 1 , . . . , g r , c 1 , . . . , c s ) with coefficients in R. The Verification Phase then consists of the two verification steps: (A) verify that h is in the set R h ; and (B) verify that the value ø(h i ) is equal to the value

ø( P i )(ø( f 1 ), . . . , ø( f n )ø( g 1 ), . . . ø( g r ), ø( c 1 ), . . . , ø( c s )) for i=1,2 , . . . , u,

where ø(P i ) is the polynomial P i with the homomorphism ø applied to its coefficients.

A second exemplary identification technique in accordance with the invention uses the same systems parameters and public/private key pairs as described above. FIG. 4 illustrates the second exemplary identification process. The identification process is initiated in the Challenge Phase (block 430 ) by the Verifier generating a challenge polynomial c(X) with bounded coefficients and sending it to the Prover. The polynomial c(X) may be generated by random selection from a set of polynomials R c as described above. The Prover initiates the Response Phase (block 450 ) by verifying that the challenge polynomial c(X) is in the restricted set of polynomials R c and then generating a polynomial g(X) with bounded coefficients, where the polynomial g(X) may be selected at random from a set R g as described above. The Prover uses the polynomials c(X),g(X) and the secret polynomial f(X) to generate the response polynomial h(X) given by

h ( X )= g ( X )( f ( X )+ c ( X ) g ( X ))

and sending the response polynomial h(X) to the Verifier. The Verifier initiates the Verification Phase (block 460 ) by using its knowledge of c(X), and the public key (f) to check that the response polynomial h(X) was generated using the private key f(X) of the Prover by verifying that:

f ( a i ) 2 +4 c ( a i ) h ( a i ) equals a square modulo q for i=1,2 , . . . , t.

This check my be expressed as verifying that ø(f) 2 +4ø(c)ø(h) is equal to a square in the ring B. The Verifier in the Verification Phase also checks whether or not the coefficients of h(X) are appropriately bounded, given that a legitimate h(X) will have bounded coefficients and will belong to a restricted set R h of polynomials. The restrictions on the set R h depend on the choice of the above noted sets R f ,R g and R c . The Verifier accepts the Prover as legitimate if the response polynomial h(X) transmitted by the Prover passes the checks of steps (A) and (B) of the Verification Phase.

A second exemplary set of system parameters suitable for use with the above-described identification technique will now be described. In the second exemplary set of parameters, the prime number q is selected as 641 , and the set S includes t=320 non-zero integers modulo q. The set S is constructed such that if a is an element of S, then a −1 is also an element of S. It should be noted that a given implementation may utilize only a subset of the t elements of S. The set R f is the set of all polynomials f(X) of degree less than 640 constructed with 214 coefficients of value 1, with 214 coefficients of value −1, and all other coefficients set to zero. The set R g is the set of all polynomials g(X) of degree less than 640 constructed with 43 coefficients of value I, with 43 coefficients of value −1, and all other coefficients set to zero. The set R c is the set of all polynomials c(X) of degree less than 640 constructed with 5 coefficients of value 1, with 5 coefficients of value −1, and all other coefficients set to zero. Finally, the set R h is the set of polynomials h(X)=h 0 +h 1 X+ . . . +h 767 X 767 of degree less than 640 whose coefficients are between −320 and 320 and which satisfy the inequality h 0 2 +h 1 2 + . . . +h 767 2 <641 2 =410881. The user identification technique described in conjunction with FIG. 4 above is then implemented using polynomials selected from the sets R f ,R g ,R c and R h .

FIG. 5 illustrates the operation of an exemplary digital signature technique implemented using the above-described ring homomorphism method. In a digital signature technique, the Prover generates a simulated challenge polynomial by applying a one-way hash function to a message m and a commitment ø(g). The one-way hash function is also available to the Verifier and will be used to validate the digital signature. As shown in FIG. 5 , in the Message and Commitment Phase (block 505 ), the Prover generates a polynomial g(X) in the set R g as previously described and uses g(X) to generate the commitment ø(g). The Prover also selects a message m to be signed. In the Challenge Phase (block 530 ) the Prover computes a challenge polynomial c(X) by applying a hash function Hash(o,o) such that c(X) is generated as Hash(m,ø(g)). The message m and commitment ø(g) are suitably formatted as an input to the function Hash(o,o) and the output c(X) of Hash(o,o) maps uniformly onto the set R c . In the Digital Signature Phase (block 545 ) the Prover computes a response polynomial as in the above-described user identification embodiments. For example, h(X) may be computed as g(X)(f(X)+c(X)g(X)). The Prover then sends the message m to the Verifier, along with the pair (ø(g),h(X)) as a digital signature on the message m. In the Verification Phase of (block 560 ), the Verifier uses the one-way hash function to compute c(X)=Hash(m,ø(g)). The Verifier accepts the signature as valid if h(X) is within in the set R h and if ø(h) is equal to ø(g)(ø(f)+ø(c)ø(g)). As in the identification embodiments, alternative embodiments may use several private keys, several commitments, several challenges, and different functions to generate the response.

FIG. 6 illustrates the operation of a second exemplary digital signature technique implemented using the above-described ring homomorphism method. In a digital signature technique, the Prover generates a simulated challenge polynomial by applying a one-way hash function to a message m. The one-way hash function is also available to the Verifier and will be used to validate the digital signature. As shown in the Message Phase (block 610 ), the Prover selects a message m to be signed. In the Challenge Phase (block 630 ), the Prover computes a challenge polynomial c(X) by applying a hash function Hash(o) such that c(X) is generated as Hash(m). The message m is suitably formatted as an input to the function Hash(o) and the output c(X) of Hash(o) maps uniformly onto the set R c . In the Digital Signature Phase (block 654 ), the Prover randomly selects a polynomial g(X) from the set R g and computes a response polynomial as in the above-described user identification embodiments. For example, h(X) may be computed as g(X)(f(X)+c(X)g(X)). The Prover then sends the message m to the Verifier, along with the polynomial h(X) as a digital signature on the message m. In the Verification r Phase (block 660 ), the Verifier uses the one-way hash function to compute c(X))=Hash(m). The Verifier accepts the signature as valid if h(X) is within in the set R h and if the quantity (f) 2 +4ø(c)ø(h) is a square in B. As in the identification embodiments, alternative embodiments may use several private keys, several commitments, several challenges, and different functions to generate the response.

Examples of operation of embodiments hereof will be provided below using very small numbers. These examples are not cryptographically secure and are meant only to illustrate the process. For further detail, see Appendix I (published as J. Hofffstein, D. Lieman, J. H. Silverman, Polynomial Rings and Effect Public Key Authentication, in Proceeding of the International Workshop on Cryptographic Techniques and E-Commerce (CrypTEC '99),Hong Kong, (M. Blum and C. H. Lee, eds.), City University of Hong Kong Press) and Appendix II (J. Hoffstein, J. H. Slverman, Polynomial Rings and Efficient Public Key Authentication II, CCNT '99 Proceedings, to appear.) The technique is called “PASS” (for Polynomial Authentication And Signature Scheme), and has a variation called PASS2.

The numbers used by PASS are integers modulo q. This means that each integer is divided by q and replaced by its remainder. For example, if q=7, then the number 39 would be replaced by 4, since

39 divided by 7 equals 5 with a remainder of 4.

The objects used by PASS are polynomials of degree N−1

a 0 +a 1 x+a 2 x 2 + . . . +a N−1 x N−1 ,

where the coefficients a 0 , . . . ,a N−1 are integers modulo q. (It is sometimes more convenient to represent a polynomial by an N-tuple of numbers [a 0 ,a 1 , . . . ,a N−1 ]. In this situation the star product becomes a convolution product Convolution products can be computed very efficiently using Fast Fourier Transforms.) PASS uses a special kind of multiplication where x N is replaced by 1, and x N+1 is replaced by x, and x N+2 is replaced by x 2 , and so on (In mathematical terms, this version of PASS uses the ring of polynomials with mod q coefficients modulo the ideal consisting of all multiples of the polynomial x N−1 . More generally, one could use polynomials modulo a different ideal; and even more generally, one could use some other ring. The basic definitions and properties of rings and ideals can be found, for example, in Topics in Algebra , I. N. Herstein, Xerox College Publishing, Lexington, Mass., 2 nd edition, 1975.) A * will be used to indicate this special polynomial multiplication.

Here is a sample multiplication using N=6:

( 5 + x + 2 ⁢ x 3 + 3 ⁢ x 5 ) * ⁢ ( 3 + x 2 + 2 ⁢ x 3 + 4 ⁢ x 4 + x 5 ) = ⁢ 15 + 3 ⁢ x + 5 ⁢ x 2 + 17 ⁢ x 3 + 25 ⁢ x 4 + 20 ⁢ x 5 + 6 ⁢ x 6 + ⁢ 13 ⁢ x 7 + 12 ⁢ x 8 + 13 ⁢ x 9 + 3 ⁢ x 10 ⁢   ⁢ ( use ⁢   ⁢ the ⁢   ⁢ rule ⁢   ⁢ x 6 = 1 , x 7 = x , x 8 = x 2 , x 9 = x 3 , x 10 = x 4 ) = ⁢ 21 + 16 ⁢ x + 17 ⁢ x 2 + 30 ⁢ x 3 + 28 ⁢ x 4 + 20 ⁢ x 5 ⁢ ( reduce ⁢   ⁢ the ⁢   ⁢ coefficients ⁢   ⁢ modulo ⁢   ⁢ 7 ) = ⁢ 2 ⁢ x + 3 ⁢ x 2 + 2 ⁢ x 3 + 6 ⁢ x 5

Polynomials whose coefficients consist entirely of 0's and 1's play a special role in PASS. (In some versions, one also allows coefficients to equal −1.) These polynomials with only 0's and 1's as coefficients are called binary polynomials. For example,

1 +x 2 +x 3 +x 5

is a binary polynomial. In practice one may also want to specify how many 1's are allowed.

The PASS2 authentication scheme is next described, using a small numerical example.

PASS2 Parameters

The first step is to choose a prime number q and to take N=q−1. For this example, take

q=7 and N=6.

One also needs to choose a set S consisting of half of the numbers between 1 and q−1, so for our example, half of the numbers between 1 and 6. Take the set

S={ 2, 4, 6}.

(There is one other condition on the set S. This condition says that if b is in S, then S must also contain the number c that satisfies the equation bc=1 (modulo q). In our example, 2*4=1 (modulo 7) and 6*6=1 (modulo 7), so the set S={2,4,6} has the required property.) Finally, one needs to specify two numbers A h and B h that will be used in the verification process. For this example, take

A h =5 and B h =22.

PASS2 Key Creation

The key creator Bob chooses a binary polynomial f(x) of degree less than N. This means that f(x) has only 0's and 1's as its coefficients. For example, Bob might choose the polynomial

f ( x )=1 +x 2 +x 3 +x 5 .

The polynomial f(x) is his private key, so he must keep it secret.

Next Bob computes the values of f(x) modulo q for the numbers in S. In this example the set S is S { 2 , 4 , 6 }, so Bob computes

f (2)=1+4+8+32=45=3 (modulo 7)

f (4)=1+16+64+1024=1105=6 (modulo 7)

f (6)=1+36+216+7776=8029=0 (modulo 7).

This set of values

f ( S )={3, 6, 0}

is Bob's public key. He publishes it so that people can use it to verify his identity.

PASS2 Commitment Step

The first step in the PASS2 authentication process is for Bob to make a Commitment and send it to Alice. He does this by choosing a binary polynomial g 1 (x) and computing the set of values g 1 (S), in much the same way that he chose f(x) and computed the values of f(x). He keeps the polynomial g 1 (x) secret, but he sends the set of values g 1 (S) to Alice as his Commitment.

For our example we will suppose that Bob chooses the polynomial

g 1 ( x ) x+x 3 +x 4 +x 5 .

He computes the values

g 1 (2)=58=2 (modulo 7)

g 1 (4)=1348=4 (modulo 7)

g 1 (6)=9294=5 (modulo 7)

and sends the set of values

g 1 ( S )={ g 1 (2), g 1 (4), g 1 (6)}={ 2 , 4 , 5 }

to Alice as his Commitment.

PASS2 Challenge Step

The second step in the PASS2 authentication process is for Alice to send a Challenge to Bob. Alice's challenge consists of two binary polynomials c 1 (x) and c 2 (x), possibly satisfying some additional conditions. (The principal extra condition is that the polynomials c 1 (x) should not vanish modulo q for all nonzero values of x not in the set S. In this example, we have c 1 (x)=X 5 +x 3 , and the values of c 1 (x) at nonzero numbers not in S are c 1 (1)=2 (modulo 7), c 1 (3)=4 (modulo 7), and c 1 (S)=2 (modulo 7).) For our example we suppose that Alice chooses the polynomials

c 1 ( x )= x 3 +x 5 and c 2 ( x )= x+x 2 .

Alice sends the two challenge polynomials c, and c 2 to Bob.

PASS2 Response Step

The third step in the PASS2 authentication process is for Bob to use his private key f(x), his commitment polynomial g 1 (x), and Alice's challenge polynomials c 1 (x) and c 2 (x) to create his Response. He does this by choosing another binary polynomial g 2 (x) and computing the polynomial

h ( x )=( f ( x )+ c 1 ( x )* g 1 ( x )+ c 2 ( x )* g 2 ( x )* g 2 ( x ).

Note that this computation is done using star multiplication (i.e., with X N =1) and that the coefficients are always computed modulo q. Bob sends the polynomial h(x) to Alice as his Response. He does not reveal the polynomial g 2 (x), and indeed he may discard it as soon as he has computed h(x).

Suppose that in our example Bob chooses the polynomial

g 2 ( x )=1 +x+x 5 .

Then

h ⁡ ( x ) = ⁢ ( ( 1 + x 2 + x 3 + x 5 ) + ( x 3 + x 5 ) * ( x + x 3 + x 4 + x 5 ) + ⁢ ( x + x 2 ) * ( 1 + x + x 5 ) ) * ( 1 + x + x 5 ) = ⁢ 1 + 5 ⁢ x + 4 ⁢ x 2 + 3 ⁢ x 3 + 6 ⁢ x 4 ⁢   ⁢ ( modulo ⁢   ⁢ 7 , with ⁢   ⁢ the ⁢   ⁢ rule ⁢   ⁢ x 6 = 1 ) .

PASS2 Verification Step

The fourth and final step in the PASS2 authentication process is for Alice to use Bob's public key f(S), Bob's commitment g(S), and her challenge polynomials c 1 (x) and c 2 (x) to verify that Bob's response is a valid response. This Verification consists of two parts.

[A] Recall that the PASS2 parameters included two numbers A h and B h . Alice writes the polynomial h(x) as h 0 +h 1 x+h 2 x 2 + . . . +h N−1 x N−1 with coefficients h 0 ,h 1 , . . . ,h N−1 taken modulo q and lying as close as possible to the number Ah. She then computes the quantity

C =( h 0 −A h ) 2 +( h 1 −A h ) 2 +( h 2 −Ah ) 2 + . . . +( h N−1 −A H ) 2 .

She compares the number C to the number B h . If C is smaller than B h , then Bob's response passes the first test. If C is larger than B h , then Bob's response fails the first test.

[B] For each number b in the set S, Alice computes the number

( f ( b )+ c 1 ( b ) g 1 ( b )) 2 +4 c ( b ) h ( b ) modulo q.

(Note that Alice possesses enough information to compute this number, since she knows the polynomials c 1 (x), c 2 (x), and h(x) and she knows the values of f(b) and g 1 (b) for every number b in the set S.) Alice checks if this number is equal to the square of a number modulo q. If it is equal to a square modulo q for every number b in the set S, then Bob's response passes the second test. If it fails to be a square for even a single number in the set S, then Bob's response fails the second test.

In the present example, this works as follows. The example quantities are A h =5 and B h =22, and the response polynomial is h(x)=1+5x+4x 2 +3x 3 +6x 4 . For the first verification test, which is test [A], Alice writes h(x) using coefficients modulo 7 that are as close as possible to 5; in other words, she uses the numbers 2,3,4,5,6,7,8 as coefficients of h(x), which means she writes h(x) as

h ( x )=8+5 x+ 4 x 2 +3 x 3 +6 x 4 +7 x 5 .

Alice then computes

(8−5) 2 +(5−5) 2 +(3−5) 2 +(3−5) 2 +(6−5) 2 +(7−5) 2 =19.

This value is smaller than 22 (i.e., it is smaller than B h ), so Bob's response passes the first verification test.

For the second verification test, which is test [B], Alice uses the known quantities

{ f (2), f (4), f (6)}={4, 3, 0}

{ g 1 (2), g 1 (4), g 1 (6)}={4, 2, 4}

c 1 ( x )= x 3 +x 5 , so { c (2), c 1 (4), c 1 (6)}={2, 5, 4}

c 2 ( x )= x+x 2 , so { c 2 (2), c 2 (4), c 2 (6)}={2, 6, 5}

h ( x )=1+5 x+ 4 x 2 +3 x 3 +6 x 4 , so { h (2), h (4), h (6)}={5, 0, 3}

These values let her compute

( f (2)+ c 1 (2) g 1 (2)) 2 +4 c 2 (2) h (2)=2 (modulo 7)

( f (4)+ c 1 (4) g 1 (4)) 2 +4 c 2 (4) h (4)=1 (modulo 7)

( f (6)+ c 1 (6) g 1 (6)) 2 +4 c 2 (6) h (6)=1 (modulo 7)

Each of these numbers is a square modulo 7, since

1=1 2 and 2=3 2 (modulo 7).

(The numbers 0, 1, 2, and 4 are squares modulo 7, and the numbers 3, 5, and 6 are not squares modulo 7.) Bob's response passes the second verification test. Since it has now passed both tests [A] and [B], Alice accepts that Bob has proven his identity.

Any authentication scheme involving the steps of

Commitment/Challenge/Response/Verification

can be turned into a digital signature scheme. The basic idea is to use a hash function (see below) to create the challenge from the commitment and the digital document to be signed. The steps that go into a PASS2 Digital Signature are as follows.

PASS2 Key Creation (Digital Signature)

Same as for PASS2 Authentication: Bob creates his private key f(x) and his public key consisting of the partial set of values f(S).

PASS2 Commitment Step (Digital Signature)

Same as for PASS2 Authentication: Bob chooses a polynomial g 1 (x) and computes the partial set of values g 1 (S) to serve as his commitment.

PASS2 Challenge Step (Digital Signature)

Bob takes his commitment g 1 (S) and the digital document D that he wants to sign and runs them through a hash function H (see below) to produce challenge polynomials c 1 (x) and c 2 (x).

PASS2 Response Step (Digital Signature)

Same as for PASS2 Authentication: Bob uses his private key f(x), the polynomial g 1 (x),another polynomial g 2 (x), and the challenge polynomials c 1 (x) and c 2 (x) to compute the response polynomial h(x)=f(x)+c 1 (x)*g 1 (x)+c 2 (x)*g 2 (x))*g 2 (x). Bob publishes the D, g 1 (S), and h(x). The quantities g 1 (S) and h(x) are his digital signature for the digital document D.

PASS2 Verification Step (Digital Signature)

When Alice wants to check Bob's digital signature on the digital document D, she begins by running g 1 (S) and D through the hash function H to reproduce the challenge polynomials c 1 (x) and c 2 (x). She now has all of the information needed to verify that h(x) is a valid response for the public key f(S), the commitment g 1 (S), and the challenge c 1 (x) and c 2 (x). If h(x) is a valid response, she accepts Bob's signature on the document D.

Notice how Bob's signature is inextricably tied to the digital document D. If even one bit of D is changed or if one bit of the commitment g 1 (S) is changed, then the hash function will produce different challenge polynomials c 1 (x) and c 2 (x), so the verification step will fail and the signature will be rejected.

Hash functions, which are well known in the art, are used herein. The purpose of a hash function is to take an arbitrary amount of data as input and produce as output a small amount of data (typically between 80 and 160 bits) in such a way that it is very hard to predict from the input exactly what the output will be. For example, it should be extremely difficult to find two different sets of inputs that produce the exact same output. Hash functions are used for a variety of purposes in cryptography and other areas of computer science.

It is a nontrivial problem to construct good hash functions. Typical hash function such as SHA1 and RD5 proceed by taking a chunk of the input, breaking it into pieces, and doing various simple logical operations (e.g., and, or, shift) with the pieces. This is generally done many times. For example, SHA1 takes as input 512 bits of data, it does 80 rounds of breaking apart and recombining, and it returns 160 bits to the user. This process can be repeated for longer messages.

The PASS2 scheme described above is a variation of an earlier version of PASS. Both schemes have the same level of security, but the operating characteristics (key sizes, communication requirements, etc.) of PASS are not as good as those of PASS2. Next, PASS is demonstrated with a small numerical example, to illustrate the similarities and differences between the two systems. The fundamental similarity is that the security depends on the difficulty of reproducing a binary polynomial from a partial set of its values.

PASS Parameters

PASS and PASS2 use the same parameters q, N (with N=q−1), a set of numbers S, and two quantities A h and B h , although the actual values of these parameters may differ.

q= 7 , N= 6 , S ={2,4,6 }, A h =5 , B h =9.

PASS Key Creation

Bob chooses two binary polynomials f 1 (x) and f 2 (x) as his private key. The partial sets of values f 1 (S) and f 2 (S) form his public key.

f 1 ⁡ ( x ) = x 4 + 1 f 1 ⁡ ( S ) = { f 1 ⁡ ( 2 ) , f 1 ⁡ ( 4 ) , f 1 ⁡ ( 6 ) } = { 3 , 5 , 2 } f 2 ⁡ ( x ) = x 5 + x f 2 ⁡ ( S ) = { f 2 ⁡ ( 2 ) , f 2 ⁡ ( 4 ) , f 2 ⁡ ( 6 ) } = { 6 , 6 , 5 }

PASS Commitment Step

Bob chooses two binary polynomials g 1 (x) and g 2 (x). He computes and sends to Alice the partial sets of values g 1 (S) and g 2 (S) as his commitment.

g 1 ⁡ ( x ) = x 5 + x 4 g 1 ⁡ ( S ) = { g 1 ⁡ ( 2 ) , g 1 ⁡ ( 4 ) , g 1 ⁡ ( 6 ) } = { 6 , 6 , 0 } g 2 ⁡ ( x ) = x + 1 g 2 ⁡ ( S ) = { g 2 ⁡ ( 2 ) , g 2 ⁡ ( 4 ) , g 2 ⁡ ( 6 ) } = { 3 , 5 , 0 }

PASS Challenge Step

Alice choose four binary polynomials c 1 (x), c 2 (x), C 3 (x), and c 4 (x) (possibly satisfying some other constraints) and sends them to Bob as her challenge.

c 1 ⁡ ( x ) = x 3 + x c 1 ⁡ ( S ) = { c 1 ⁡ ( 2 ) , c 1 ⁡ ( 4 ) , c 1 ⁡ ( 6 ) } = { 3 , 5 , 5 } c 2 ⁡ ( x ) = x 5 + x 4 c 2 ⁡ ( S ) = { c 2 ⁡ ( 2 ) , c 2 ⁡ ( 4 ) , c 2 ⁡ ( 6 ) } = { 6 , 6 , 0 } c 3 ⁡ ( x ) = x 5 + x 2 c 3 ⁡ ( S ) = { c 3 ⁡ ( 2 ) , c 3 ⁡ ( 4 ) , c 3 ⁡ ( 6 ) } = { 1 , 4 , 0 } c 4 ⁡ ( x ) = x 5 + x c 4 ⁡ ( S ) = { c 4 ⁡ ( 2 ) , c 4 ⁡ ( 4 ) , c 4 ⁡ ( 6 ) } = { 6 , 6 , 5 }

PASS Response Step

Bob computes the polynomial

h ( x )= f 1 ( x ) g 1 ( x ) c 1 ( x )+ f 1 ( x )g 2 ( x ) c 2 ( x )+ f 2 ( x ) g 1 ( x ) c 3 ( x )+ f 2 ( x ) g 2 ( x ) c 4 ( x ).

and sends h(x) to Alice as his response. (Remember that h(x) is computed using the rule x N=1 and that the coefficients are computed modulo q.)

h ⁡ ( x ) = ⁢ ( x 4 + 1 ) ⁢ ( x 5 + x 4 ) ⁢ ( x 3 + x ) + ( x 4 + 1 ) ⁢ ( x + 1 ) ⁢ ( x 5 + x 4 ) + ⁢ ( x 5 + x ) ⁢ ( x 5 + x 4 ) ⁢ ( x 5 + x 2 ) + ( x 5 + x ) ⁢ ( x + 1 ) ⁢ ( x 5 + x ) = ⁢ 5 ⁢ x 4 + 5 ⁢ x 3 + 5 ⁢ x 2 + 4 ⁢ x + 6

PASS Verification Step

Verification consists of two steps. First Alice writes the polynomial h(x) as h 0 +h 1 +h 2 x 2 + . . . +h N−1 x N−1 with coefficients h 0 h 1 . . . h N−1 modulo q taken as close as possible to A h and she computes the quantity

C =( h 0 −Ah ) 2 +( h 1 −A h ) 2 +( h 2 −A h ) 2 + . . . +( h N−1 −A H ) 2 .

She compares the number C to the number B h . If C is smaller than B h , then Bob's response passes the first test. If C is larger than B h , then Bob's response fails the first test.

Second, for each number b in the set S, Alice computes the two numbers

h ( b ) (modulo q )

and

f ( b ) g 1 ( b )c 1 ( b ) c 1 ( b ) c 2 ( b )+f 2 ( b ) g 1 ( b ) c 3 ( b )+ f 2 ( b ) g 2 ( b ) c 4 ( b ) (modulo q ).

If they are the same for every number b in the set S, then Bob's response passes the second test; otherwise his response fails the second test.

Note that Alice has enough information to compute these quantities, because she knows the polynomials h(x), c 1 (x), c 2 (x), C 3 (x) and C 4 (x) and she knows the values of f 1 (b), f 2 (b), g 1 (b), and g 2 ,b) for every number b in the set S.

For the example, the polynomial h(x) is 5x 4 +5x 3 +5x 2 +4x+6 and the number A h is equal to 5.

This means that Alice should write h(x) as

h ( x )=7 x 5 +5x 4 +5x 3 +5x 2 +4x+6

since she wants the coefficients, which are numbers modulo 7, to be as close to 5 as possible.

Then she computes

C =(7−5) 2 +(5−5) 2 +(5−5) 2 +(5−5) 2 +( 4−5) 2 +(6−5) 2 =6.

This is smaller than the bound B h =9, so Bob's response passes the first test.

Next Alice computes the values

h (2)=0 (modulo 7),

h (4)=1 (modulo 7),

h (6)=0 (modulo 7).

and

f 1 (2) g 1 (2) c 1 (2)+ f 1 (2) g 2 (2) c 2 (2)+ f 2 (2) g 1 (2) c 3 (2)+f 2 (2) g 2 (2) c 4 (2)=0 (modulo 7),

f 1 (4) g 1 (4) c 2 (4)+ f 2 (4) g 2 (4) c 2 (4)+ f 2 (4) g 1 (4) c 3 (4)+ f 2 (4) g 2 (4) c 4 (4)=1 (modulo 7),

f 1 (6) g 1 (6) c 2 (6)+ g 1 (6) g 2 (6) c 2 (6)+ f 2 (6) g 1 (6) c 3 (6)+ f 2 (6) g 2 (6) c 4 (6)=0 (modulo 7).

Since these values match the values of h, Bob's response passes the second test, so Alice accepts that Bob is really who he says he is.

The user identification and digital signature techniques of the present invention provide significantly improved computational efficiency relative to prior art techniques at equivalent security levels, while also reducing the amount of information which must be stored by the Prover and Verifier and communicated between the Prover and Verifier. It should be emphasized that the techniques described above are exemplary and should not be construed as limiting the present invention to a particular group of illustrative embodiments. Alternative embodiments within the scope of the appended claims will be readily apparent to those skilled in the art.