CA1295706C - Method and system for authentication of accreditations and of messages with zero-knowledge proof and for the signing of messages, and a station for use in such system, in particular executed as a smart card station - Google Patents

Abstract

PHQ 87.030 02.08.1988 Abstract A method and system for authentication of accreditations and of messages with zero-knowledge proof and for the signing of messages, and a station for use in such system, in particular executed as a smart card station.

Instead of using multiple accreditations and an iterative process of verification, use is made of a comprehensive accreditation (high exponent p) and a number D is drawn at random, which number is within the range between 0 and p-1. The operations of verification proceed by the computation of the D-th power of the inverse accreditation B.
Application in particular, to smart cards and, more specifically, to bank cards.

Figure 7.

Description

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PHQ 87.030 1 02.08.19~8 A method and system for authentication of accreditations and of messages with zero-knowled~e proof and for the signing of messages, and a station for use in such system, in particular executed as a smart card station.

BACKGROUND TO THE INVENTION
The invention relates to a method for authentication of accreditations or of messages with zero-knowledge proof and to a method for the signing of messages. The invention also relates to a system for executing such authentication method and signing method and to a station for use in such a system.
The invention finds application in the verification of the authenticity of bank cards referred to as "smart" or, more generaly, of the authenticity of any medium permitting its holder to control an access (to a set of premises, to a safe, to a database, to a computerized system, to a telephone line, etc.). It is also applicable to the verification of the authenticity of messages of any type, that are capable of controlling an opening or a closing, activating or deactivating a system, controlling the starting of an engine, controlling a satellite, triggerring an alarm etc.. Finally, the invention permits the signing of a message in such a manner that its addressee is assured of its origin, and can in turn convince a third party as to such origin.

DESCRIPTION OF SELECTED PRIOR ART
The invention is based on two branches of cryptography, which are public-key cryptography and zero-knowledge-proof verification procedures respectively. For reasons of reference, these two techniques are recalled briefly. Encoding and decoding techniques have developed considerably, inter alia through the availability of data processors and ; telecommunciation facilities.
In a cryptographic system, a message in clear M is transformed with the aid of a key E, to give an encoded message ElM). To the key E corresponds an lnverse key D, which permits retrieval of the message in clear by an inverse transformation: D(E(M))=M.
Traditionally, the keys E and D are kept secret and are known only to the interlocutors.

31.2~5~6 PHQ 87.030 2 02.0~.1988 A novel cryptographic system has been developed, in which the encoding key is no longer kept secret but is published.
Paradoxically, such revelation does not weaken the security of the system. The reason is that the knowleclge of the key E does not, in practice, permit retrieval of the decoding key D. Such encoding function is called a "trap" function which is particularly difficult to invert, except for a person who knows the value of the trap.
The general principles of these systems have been described in the article by W. DIFFIE and M. HEI,LMANN entitled "New Directions in Cryptography" in IEEE Trans. on Information Theory, vol.
IT-22, pp. 644-654, Nov. 1976. See also, the article by M. HEL~MANN "The Mathematics of Public-Key Cryptography", Scientific American of August 1979, vol. 241, No. 2, pp. 130-139.
Public-key cryptography may be applied in a particularly effective manner in a system referred to as RSA. This system is described by Martin GARDNER in "A new kind of cipher that would take miilions of years to break", Scientific American, August 1977, pp. 120-121. In the RSA system, the trap function is the factorization of a number into prime components. Factorization is a difficult operation.
For example, several minutes are required to find, manually, the prime factors of a modest number of 5 digits, such as 29,083. These numbers are 127 and 223. However, the obtaining of the product of 127 and 229 takes only a few seconds. The asymmetry of such an operation is therefore obvious. Recourse to a computer accelerates the factorization, but the fact remains that in order to factorize a number having two hundred digits the most powerful computers would be required. Thus, in practice, it is not possible to factorize a very large number.
These properties are exploited in the RSA system in the following manner. Two distinct prime numbers are chosen, namely a and b, and the product is formed: N = a.b, wherein N may have, for example, 500 bits. Further, an integer p is selected, which is prime with the smallest common multiple of (a-1) and (b-1). In order to encode a message, previously put into digital- form M, M being in the range between 0 and N-l, the p-th power of M is calculated in the ring of integers modulo N, viz. C=MP mod N. The function "raise to the power p modulo N" then defines a permutation of the integers from 0 to N-1.
In order to decode a message C, it is necessary to 9~570~i PHQ 87.030 3 02.08.1988 extract the p-th root of the encoded message C in the ring of integers modulo N. This operation amounts to raising the number C to the power d, d being the reciprocal of the exponent p modulo the smallest common multiple of the numbers (a-1) and (b-1). If the prime factors a and b are not known, the determination of d is impossible and, kherefore, the deciphering operation.
For example, selection of a=47 and b=59 gives N=47.59=2773. It is possible to take p=17. The coding key is therefore defined by the two numbers 2773 and 11. In practice, the numbers used are much larger.
The encoding of a word M wh:ich is presented in the form of the number 920 is as follows:
C = 92017 mod 2773 = 948 mod 2773.
Conversely, in order to decipher the number 948, use will be made of an exponent d which is the inverse of 17 modulo 1334, which is the LCM of 46 and 58. This exponent d is 157, since 157 . 17 = 2669, i.e. 1 modulo 133~. Thus, deciphering of 948 amounts to calculation of 948157 mod 2773, i.e. 920, which is indeed the initial message.
Thus, in the RSA system, the numbers N and p can themselves be public treference is then made to the "public power p"), but the numbers a and b must remain secret. Naturally, it is possible to use more than two prime factors in order to form the number N.
The RSA system is described in Vnited States Patent No. 4,405,829 issued on 20th September 1983. Such a system may serve not only to encipher but also to sign a message.
In the context of the si~ning of messages, an entity considered to emit messages M, being an entity referred to as a signatory, is considered to operate with a public key (N, p). This entity, in order to sign its messages before transmission, adds a redundancy thereto in order to obtain a component of the ring of integers modulo N, and then raises this component to the power d (inverse of p) modulo N. The entity in question, holding the secret parameters of the public key, that is to say the two prime factors a and b, has knowledge of d. The signed message is therefore S=[Red(M)]d mod N .
In order to verify the signature, the addressee of the message uses the public key (N, p) associated with the emitting entity 70~i PHQ 87.030 4 02.08.1988 and calculates SP mod N, which, by hypothesis, gives once again the component encoding the message M with its redundancy. By retrieving the redundancy, the addressee thus concludes therefrom that the message could have been sent only by the entity which claims to have done this, since only that entity was capable of processing the message in this manner.
The operations of enciphering and of signature can be combined. In this case, the transmitter commences by signing its message while makinq use of its secret key; then, it enciphers it by making use of the public key of its correspondent. On receipt, the correspondent deciphers the message with the aid of its secret key, and then authenticates the message by using the public key of the emitter.
These techniques of cryptography thus lead to various method of authentication. In order to explain in greater detail this aspect, there will he taken by way of example the authentication of bank "smart" cards without, this constituting in any sense whatsoever a limitation of the scope of the invention.
A bank "smart" card possesses an identity, which is constituted by a string of information items such as the serial number of the chip, the number of the bank account, a period of validity and an application code. The card can, on request, present these identity information items, in the form of a sequence of bits forming a word I.
With the aid of redundancy rules, it is possible to form a number J which is twice as long as I, which will be hereinafter designated Red(I)=J. For example, if the number I is written in the form of quartets, each quartet can be supplemented by a redundancy quartet in such a manner as to form as many octets of the HAMMING encoding type.
The number J is frequently referred to as the "shaded" identity, the shadow being constituted by the redundancy which accompanies the identity.
The International Standardization Organization ~ISO) has specifically stated these solutions in the note ISO/TC97/SC20/N207 entitled "~igital Si~nature with Shadow" which became a preliminary draft standard DP9796.
The authority empowered to issue such cards, in the present case the bank, chooses a public-key system (N, p). It publishes the number N and p, but keeps the factorization of N secret. The shaded o~

PHQ 87.030 5 02.08.19B8 identity J of each card is then considered as a component of the ring of inteqers modulo N. The bank can extract therefrom the p~th root in this ring, which, as stated above, requires the knowledge of the prime factors of N, which is the case. This number, hereinafter ~esignated ~h is, to some extent, the identity of the card signed by the bank. This is referred to as "accreditation". The result of this is, by definition, A=J1/P mod N.
Authentication of an accreditation now amounts to reading the identity of the card, either in the simple form I or in the shaded form J, and then to reading the accreditation ~ from the card, raisiny the latter to the power p in the ring of integers modulo N, which is possible because the parameters N and p are known, and, finally, comparing the result, namely AP mod N, with J. If AP mod N is equal to J, then the accreditation A is authentic.
Althouqh this method permits detection of false cards, it nevertheless presents a disadvantage which is that of revealing the accreditation of the authentic cards. A verifier lacking scrupulousness might therefore reproduce cards identical to that which he has just verified (cards which might be referred to as "clones") by reproducing the accreditation which he has read from the authentic card.
Now, the authentication of an accreditation does not, strictly speaking, require the communication of the latter to the verifier, but only the establishment of a conviction that the card has an authentic accreditation. The problem is, therefoxe, finally that of demonstrating that the card has an authentic accreditation, without revealing the latter.
This problem can be resoved by a procedure referred to as "zero-knowledge proof". In such a procedure, the entity which attempts to adduce the proof, the "verified" entity, and the entity which awaits this proof, the "verifier", adopt an interactive and probabilistic behaviour.
By itself, this technique has been described by Shafi GOLDWASSER, Silvio MICALI and Charles RACKOFF in their paper at the "17th ACM Symposium on Theory of Computing, May 1985, this paper being entitled "The Knowledge Complexity of Interactive Proof Systems" and published in the Reports, pp. 291-304. The prime examples were found in graph theory.

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PHQ 87.030 6 02.08.1988 Adi SHAMIR was the first to think of using this process in theory of numbers, and it might be applied to smart ca~ds in t:he following manner. This so-called S process is as follows:
At the start of the authentication operation, the card proclaims its identity I. The redundancy rules, which are publicly known, permit the deduction of J, twice as long as I, which gives the shaded identity. The card and the ver:ifier both know the numbers N and p published by the card issuer, but only the latter has the factorization of the number N available, which is the trap information used to calculate the accreditations.
The authentication operation is continued by repeating the following processing:
- the card draws, at random, a component r from the ring of integers modulo N, computes therefrom the p-th power (rP mod N) in the ring, and transmits this power to the verifier in the guise of title T for iteration;
- the verifier draws, at random, a bit d (O or 1) to interrogate the card in the guise of marker t : for a=O the component r, and for d-1 the product of the component r and the accreditation in the ring (r.A mod N); in other words, if the draw is uncertain, the verified must have available r and r.A mod N, which implies the knowledge of A;
- the verifie~ raises the marker t to the power p modulo N, to retrieve : for d=O, the title T, and for d=1 the product, in the ring, of the title T and the shaded identity J.
Thus, on the one hand, it is necessary to have available the accreditation A in order to possess simultaneously the two possible values of the marker t, viz. r and rA. On the other hand, the verified cannot deduce from this operation the value A of the accreditation since, even if he requests the verified to supply rA to him, he does not know r, which was drawn at random by the verified. The verifier does indeed know rP, supplied by way of title by the verified, but is incapable of extracting therefrom the p-th root modulo N, because he does not know the factorization of N.
A veriEied who does not hold an authentic accreditation might bluff by attempting to guess the draw by the veriEier. If he bets on "O" ("tails"), he estimates that the verifier will raise the title to the power p modulo N and that the verifier will compare the result :~2~

P~Q 87.030 7 02.08.1988 obtained with the title T. In order to convince the verifier, the verified will have to supply by way of title T the marker raised to the power p. If the bluffer bets, on the other hand, on "1" (Cheads~ he estimates that the verifier will raise the title to the power p and will then multiply the result obtained by J. In order to be convincing, he therefore must transmit, by way of title T, the marker raised to the power p multiplied by J.
In other words, the verifiecl has one chance in two of giving a correct response if he reverses the chronology of the events, that is to say if he does not first of all determine the title T and then the marker t, but if he bets on the draw by the verifier and if he forms the title a posteriori with the aid of a marker drawn at random.
In this probabilistic process, the chances of the verified guessing the correct response are one in two in each processing, so that, in repeating this processing k times, the chances of the bluffer fall to 1/2k. The safety factor of this authentication process is therefore 2k. In practice, k is of the order of 16 to 24.
In such a process the number p is small, for example 3.
It is also possible to use 2, but in this case certain precautions must be taken int he choice of the prime factors of the number N in order that the function "raise to the square modulo N" should be a permutation on the quadratic residues of the ring of integers modulo N. The numbers a and b must be integers of the form 4x+3; it is recalled that quadratic residues are components which are squares in the ring and the shaded identity J must be capable of being modified in a representative quadratic residue before computing the accreditation. This solution is described in the document already mentioned ISO/TC97/SC20/N207.
The integer N, as in the bank cards of today, may be of the form N=K+2320 where K is an integer of 240 bits which is published and known to all terminals. Only the issuer of cards has the factorization of N available. It is nevertheless recommended to use larger numbers.
The identity I, as in the bank cards of toda~, may be a symbol of 160 bits, which is obtained by the chaining of a serial number of the chip of 4~1 bits, of a bank account number of 76 bits, of an application code of 8 bits and of a period of validity of 32 bits. In these circumstances, the shaded identity has 320 bits. The accreditation ~Z~5~
- ~ - 2010~-8~72 is then the cube root of this word, modulo N. This is a number of 320 bits.
An improvemen-t -to this technique consists in using not the accreditation itselE A (AP mod N=J) but its inverse, designa-ted B. The result of this is BPJ mod N=l, which permits simplification of the comparison of the title and the marker. It is then sufficient to transmit a mar~er equal to r(dB-d+l), which is equal either to r if d=0 or to rB if d=l, and to compute tP(dJ-d+l) mod N in order to find the title T. It is then possible to -transmit only a part thereof, for example about one hundred o:E its bits, or, even better, after a compression by a one-way function.
It is recalled that a compression function causes correspondence between a set of n components and a set of m other components, m being less than n, and such that it is virtually impossible to locate two components having the same image.
The invention may be summarized, according to one aspect as a method for authentication of an object element viz à viz verifier element, said method comprising the following initialization steps: a. generating and disclosing a first integer p of at least ten bits and a product ~ of two secret pr-ime factors; b. generating a first personalized digital quantity I
with added redundancy to form a second personalized digital quantity J; c. generating and storing in the objact element an accreditation number B as the inverse of A = Jl/Pmod ~, so that - 8a - 20104-8472 BPJmod N = l; said method furthermore comprising the following authentication steps: d. in the object element drawing a first random integer r which is a member of the ring of integers modulo N and calculating a title number T according to T = rPmod N and forwarding at least a prede-termined multibit field o:E the title number to the veriEier element as a first comparison data; e. in the verifier element drawing a second random number D within the closed interval [o,p-l] and forwarding this second random number to the object element; f. in the object element generating a marker number t according to t = r.BDmod N and forwarding the marker number to the verifie:r element; g. in the verifier element calculating a second comparison data equal to tPJDmod N and comparing said first comparison data to a corresponding multibit field of the second comparison data, and upon correspondence directly generating an authentification approbation.
BRIEF DESCRIPTION OF THE DRAWIN~S
The state of the art and the invention are furthermore explained with respect to the following Figures~
Figure 1 represents the abovementioned probabilistic process;
Figure 2 likewise illustrates the FS process;
Figures 3, 4 likewise illustrate the processes for the technique S and FS;
Figures 5, 6 likewise illustrate a signing process for the techniques S and FS;
Figure 7 illustrates the process for -the authentication o~ an accreditation according to the invention;

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- 8b - 20104-8472 Figure 8 illustrates ~he proces.s for the authentication of a message according to the invention;
Figure 9 illustrates the process ~or ~signing a message according to the invention;
Figure 10 shows diagramMatically an assembly permitting the implementation of the processes of the invention.
EXTENSIVE EXPLANATION OF PARTICULAR TECHNIQUES
Figure 1 il].ustrates thi.s process. The arrows extending from left to right represent a transmission from the verified to the verifier (identit~ I, title T, marker t) and the arrows extending from right -to left a transmission in the opposite direction (bit d drawn at random). A draw at random is represented by a circle associated with a question mark. The symbol E signifies "is a member of" and -the numbers between brackets designate the set of integers in -the range between the two indicated limits, including the limits. The final comparison ~f~5~

PHQ 87.03~ 9 02.08.1988 deciding on the authenticity of the accreditatlon is schematically represented by an equality sign surmounted by a question mark. The dashed block indicates a set of operations which are executed k times (iteration).
A process which is even further improved has recently been proposed, which makes use of multiple accreditations. This process has been described in the paper by Amos FIAT and Adi SHAMIR, published in the Compte Rendu de CRYPT0 86, Santa ~arbara, CA, ~SA, ~ugust 1986, "How to Prove Yourself : Practical Solutions to Identification and Signature Problems", Springer Verlag, Lecture Notes in Computer Science, No. 263, pp. 186-194.
Noting several accreditations in the card leads to an increase in the efficiency of the processing, and to a reduction in the number of iterations required in order to achieve a given level of security in relation to the luc~ left to the bluffer. In this method, n diversified identities I1 ... In, are produced, which, supplemented by their shadows, give n shaded diver.~ified identities J1 ... Jn. The card contains the n inverse accreditations B1 ... Bn, which verify the relations Ji.~iP mod N=1.
In this process, which will be designated as FS, each processing or iteration then becomes the following (taking 2 for the public exponent):
- the card draws at random a component r in the ring of integers modulo N, and then transmits to the verifier 128 bits of the square of this component in the guise of title T;
- the verifier draws at random a word of ln bits, i.e. b1 ... bn, which he transmits to the card;
- the card then computes the product of the component r and the inverse accreditations designated by the "1" bits in the words of n bits b1 ... bn. Furthermore, the card transmits in the guise marker, the value t thus obtained:
t=r.(b1.B1-b1+1). ... .(bn.~n-bn+1) mod N
- the verifier tests thi.s Inarker t by raising it to the square in the ring, and then by multiplying this square by the diversified shaded identities designated by the "1" bits of the word of n bits, i.e.: tP.(b1.J1-b1+1). ... .(bn.Jn-bn+1) mod N.
The authenticity is proven if the published bits of the C)6 PHQ 87.030 10 02.08.1988 title T are retrieved.
Any person would be able to draw at random a marker t, and then, in the ring, to raise this to the square and to multiply by a selection of diversified identities in order to form a title T. In fact, if this title is given at the start of the processing, if the question asked is indeed the expected selection, then the marker t is an acceptable response which authenticates the card.
Thus a winning strategy exists for a person who guesses or knows in advance the draw by the verifier.
In order to pass successfully through an iteration, the bluffer must, this time, guess a word of n bits, and no longer just a single bit, as in the GMR process. If the 2n values are equally probable, the product of the multiplication of the accreditations ~n) by the number of iterations (k) reduces exponentially the chances left to the bluffer. The security factor of the authentication operation is then 2k.n ~
At each iteration, the verified transmits, for example, 128 bits (one quarter of the 512 bits) and a component of the ring, and the verifier transmits n bits. At each iteration, the verifier and the card compute a square and execute a number of multiplications which is equal to the number of "1" bits in the word of n bits (HAMMING
weighting).
As another compromise between efficiency of the iteration and the maximum number of multiplications to be executed during the ~æ~ e~
iteration, it is possible to limit the number of ~ bits in the word of n bits, to a certain fraction.
In connection with this technique, reference is made to the paper by Amos FIAT and Adi SHAMIR at the "5th World Congress on Computing and Communications Protection and Security", Paris on 4th -6th March 1987, under the title "Unforgeable Proofs of Identity".
Figure 2 attached illustrates diagrammatically this FSprocess, with the same conventions as for Figure 1.
These processes for the authentication of accreditation may readily be adapted for the authentication of a message emitted by an entity considered to be accredited. In this case, the title T
transmitted by the verified is no longer formed exc]usively by rP mod N, as in the preceding case, but also by.the message m to be ~lZ~;70~

PHQ 87.030 11 02.08.19~8 authenticated. More specifically, .it is the result obtained by a A - ~ function, designated f, the arguments of which are m and rP mod N.
Figures 3 and 4 show schematically these processes in the 5 case of the techniques S and FS.
Finally, these techniques may likewise serve to sign a message. The compression function then plays the part assigned to the draw by the verififer. More specifically, in the process of type S, the signatory draws k components r in the ring of integers modulo N, namely r1, r2, ..., rk, which will play the part of the various values of r drawn in the course of the k iterations. The signatory raises these integers to the square mod N and computes the compression function f(m, r2 mod N, ... rk2 mod N), which provides a number D having as bits d1, d2, ..., dk. Each bit di of this number plays the part which was played by the bit drawn at random in the process for the authentication of accreditation described hereinabove. The signatory then forms k mal~ 4~^S
-titlcs ti=riBdl mod N, with i=1,2 ... k. 'rhe signed message is then a multiplet formed by m, I, d1, ... dk, t1, ..., tk.
In order to verify such a signed message, the titles t1 modulo N are raised to the square and each square is multiplied by Jdi modulo N. A computation is then made of the compression function f(m, t12Jd1 mod N, ..., tk2Jdk mod N), and the result obtained is compared with the number D, i.e. with the bits d1, d2, .,.., dk.
When applied to the signing of messages, the number k is larger than in authentication. It often has of the order of 60 to 80 bits and, more precisely, at least 30 bits. In fact, the verification is no longer undertaken in real time, and the fraudulent person therefore has plenty of time to formulate a false signature.
Figures S and 6 schematically represent this process in the case of the techniques S and FS, respectively. These techniques of the prior art present certain disadvantages. In particular, the method FS, with the multiplicity of its accreditations, takes up a large space in memory. Moreover, the need to undertake a repetition of the processings extends the duration of the exchanges. Finally, the multiplicity of markers extends the information items to be added to a message in order to sign it.

3L2~i7~j PHQ 87.030 12 02.08.1988 SUMMARY OF THE INVENTION
Among other things, it is an object of the invention to remedy this disadvantage. To this end, it utilizes a single accreditation (and no longer multiple accreditations) and a single processing ~and no longer a repetition of processings).
More specifically, the subjects of the present invention are a method for the authentication o:E an accreditation and a method for the authentication of a message, these methods both utilizing on the one hand, the formulation of an accreditation based on the public-key ~ystem 0 and, on the other, zero-knowledge proof;
e~ e c ~ t~'on a) so far as concerns the operation of the ~rm.}~ of the accreditation, it comprises the following operat.ions:
- an authority to issue accreditations chooses two prime numbers, forms the product (N) of these two numbers, keeps secret these numbers, which then constitute the prime factors of N, chooses an integer p and publishes N and p;
- for each holder of an accreditation, a digital identity I is formed, and is then supplemented by redundancy in order to form a shaded identity word ~, - an accreditation A is formulated by the authority by taking the p-th root of the shaded identity in the ring of integers modulo N, (AP
mod N=J ), - into an appropriate medium containing a memory, the authority loads the inverse modulo N of the accreditation A, i.e. a number B
referred to as the inverse accreditation (BPJ mod N=1), this number B
constituting the accreditation vhich is to be authenticated;
b) so far as concerns the authentication of the accreditation thus formulated, this operation comprises an interactive and probabilistic digital process of the zero-knowledge proof type and taking place between a medium containing an accreditation, this medium being referred to as "the verified" and an authentication element referred to as "the verifier", this process comprising at least one digital processing comprising the following known operations-- the verified draws first a random integer r which is a member of the ring of integers modulo N, - the verified raises this integer r to the power p modulo N, the result being a title T, 70~

PHQ 87.030 13 02.08.1988 - the verified then issues at least a portion of the bits of the title T, - the verifier then draws, at random, a number D and re~uests the verified to undertake certain oiperations on r and on the inverse accreditation ~, these operations being associated with the number D
drawn at random by the verifier and being executed in the ring of integers modulo N, - the verified issues to the verifier a number t, referred to as the marker, being the result of these operations, - the verifier undertakes, in his turn, operations relating to the marker t issued by the verified and to the shaded identity J of the verified, these operations being themselves also associated with the number D drawn at random by the verifier and undertaken in the ring of integers modulo N, - the verifier compares the result thus obtained with the bits of the title T which the verified had issued at the start of the verification process, and acknowledges the authenticity of the accreditation if he retrieves these bits.
The process for the authentication of accreditation according to the invention is characterized in that:
a) in the course of the formulation of the accreditation (B) the number p serving to extract the p-th root of the shaded identity (J) chosen to comprise at least ten bits, b) for the authentication of the accreditation the process comprises only a single interactive and probabilistic processing (and not a repetition of such a processing), this single processing consisting of the following operations:
- the number which the verifier draws at random is an integer D
within the range between 0 and p-1 (including the limits), - the operation which the verified executes in order to issue a marker t is the product, in the ring of integers modulo N, of the component r which it has itself drawn at random, and the D-th power of the inverse accreditation B, the title being then t=r.BD mod ~, - the operations which the verifier executes are the product, in the ring of integers modulo N, of the p-th power of the marker t and the D-th power of the shaded identity J, i.e. tPJD mod N, - the comparison operation executed by the verifier then relates to PHQ 87.030 14 02.08.1~88 the bits of the title T which are issued by the ver.ified and to the bits obtained by the preceding operation, the authenticity of the accreditation being acquired in a single processiny since all the bits of the title issued by the verified are retrieved b~ the verifier in tpJD mod N.
On the other hand, so far as concerns the process for the authentication of a message, the process according tot he invention is characterized in that:
a) in the course of the formulation of the accreditation of the principal, the number p serving to extract the p-th root of the shaded identity comprises at least ten bits, b) for the authentication of the message, the process comprises only a single interactive and probabilistic processing (and not a repetition of such a processing), this single processing consisting of the following operations:
- the number which the verifier draws at random is an integer D
within the range between O an p-1 (including the limits), - the operation which the verified executes in order to issue the marker t is the product, in the ring of integers modulo N, of the component r which it has itself drawn, and the D-th power of the inverse accreditation B, the marker then being r.BD mod N, - the operations which the verifier executes are generating the product, in the ring of integers modulo N, of the p-th power of the marker t, and the D-th power of the shaded identity J, - the verifier forms a compression function of the message and of the result of the preceding operations, i.e. f(m, tPJDmod N), - the comparison which the verifier executes then relates to the compression function which he has obtained and to the title T which the verified has issued to him at the start of the verification process, the authenticity of the message being acquired in a single processing since, at the end of this processing, there is equality between all the bits of the compression function obtained by the verifier and the corresponding bits of the title which are issued by t he verified.
Finally, the subject of the invention is a process for the signing of a message. In this case, the accreditation of the signatory is formulated according to the known public-key process described hereinabove and the signature consists of a known 7~6 PHQ 87.030 15 02.08.1988 probabilistic digital processing comprising the following operations:
- the signatory draws, at random, at least one integer r which is a member of the ring of integers modulo N, - the signatory raises this integer r to the power p modulo N, - the signatory computes a compression function f by adopting as arguments the message m to be signed and the power rDmodN obtained, - the signatory forms at least one marker t by executing certain operations on r and on the inverse acc:reditation B, these operations being associated with the number D drawn at random and being executed in the ring of integers modulo N, - the signatory transmits the message m, its identity I, the word D, the marker or markers t, the total forming a signed message.
The process for the signing of a message according to the invention is characterized in that:
a) in the course of the formulation of lthe accreditation of the signatory the number p serving to extract the p-th root of the shaded identi.ty is chosen to be large and comprises a plurality of tens of bits, b) for the operation of signing the message:
- the signatory draws, a random single integer r which is a member of the ring of integers modulo N, - the compression function has, as arguments, the message m and the p-th power of r, which provides a number D, ~7 c~7~e~
- the sole titlc produced by the signatory is the product, in the ring of integers modulo N, of the integer r and the D-th power of the inverse accreditation B, - the signato~ provides, with the message m, its identity I, the r word D and the ~ t.
In the first two processes, the number p comprises at least 10 bits and preferably between 16 and 24 bits.
In the process of signing, the number p is larger and comprises a plurality of tens of bits, for example from 60 to 80 bits, or, anyway, at least thirty.
The value of p is, in fact, the security factor of an elementary processing. If p is appropriate for the sought object, although only a single comprehensive accreditation is available, it is possible to be content with a single processing.

~L2~
- 16 - 20104-8~72 DESCRIPTION OF A PREFERRED EMBODIMENT
The conventions used in Figures 7 to 9 are the same as -those of Figures l to 6. In all cases, the accreditation is obtained by the use of a number p which is Large. This accredita-tion is called "comprehensive", as opposed ~o -the customary accreditations for whic-h p was the order oE 2 or 3 and which are relatively "superficial".
Thus, in the case oE smart cards, the -following processing is done:
- the card draws, at random, a component r (l r<N<l) in the ring of integers modulo N, and gives, in -the guise of title T, 1~8 bits of the public power of this component (rP mod N);
- the verifier draws, at random, an e~ponent D from O to p-l and transmits it -to the card;
- the card computes the marker t which is the product (in the ring) of the componen~ r and the D-th power of the accreditation B (t=r.BD mod N);
- the verifier computes, in the ring, -the product of the p-th power of the marker t and the D-th power of the shaded iden-tity J, i.e. tP.JD mod N.
The proof is accepted if all the published bits of the title T are thus retrieved.
Any one that has guessed the question of the verifier PHQ 87.030 17 02.08.1988 (the exponent D) can draw, at random, a marker t, and then undertake in advance the computations of the verifier, that is to say form the product, in the ring, of the marker t to the exponent p and the shaded identity J to the exponent D. If the title T is given at th~ start of S the iteration, and if the question posed is indeed D, then the marker t is an acceptable response.
This reasoning, indicating a winning strategy for the person guessing, shows that it is not possible to distinguish data originating from the recording of a successful operation and data originating from a masquerade constructed by inverting the chronology of the iteration, that is to say by choosing the exponents before the titles. The verifier collects information items which are impossible to distinguish from those which he could have produced alone, without interaction with the verified r which shows that the accreditation does indeed remain secret in the card.
In order to complete successfully an authentication processing, the bluffer must guess an exponent D. If the p-values of D
are equally probahle, the chance left to the bluffer is 1/p. The security factory is therefore p.
In such a processing, the verified transmits approximately one hundred bits and a component of the ring; the verifier transmits an exponent (D). In order to complete a processing, the verified computes, first of all, the public power of the component r drawn at random, and then the product of this component r and the D-th power of the accreditation B. The verifier undertakes slightly less computation for the purpose of retrieving the title T, since he can intelligently combine the power computations- the D-th power of the shaded identity J and the p-th power of the marker t.
If the exponent p has the value of 216, then the exponent D is a number having 16 bits. Thus, in a single processing, with a security factor of 216, i.e. 65536, the verified computes in the ring 16 squares, and then 16 squares and a mean of 8 multiplications. The verifier computes only 16 squares and proceeds, on average, with 8 multiplications.
The process for the authentication of a message is illustrated in Figure 8 with the same conventions, the diference in relation to Figure 7 consisting solely in the formation of the ~29S70~

PHQ 87.030 18 02.08.1988 compression function f.
In order to sign a message, the holder of the accreditation B of comprehensiveness p commences by drawing, at random, a component r in the ring, and then computes the public power of the component rtrP mod N). He then produces an exponent D by virtue of the compression ~unction f applied to the concatenation of the message m and of the public power of the component r. The marker t is the product of the component r and the D-th power of the accreditation B. The signed message is the concatenation of the identity I, of the message m, of the exponent d and of the marker t.
In order to verify a signature, the verifier computes in the ring the product of the marker t to the power p and the shaded identity J (reconstructed from the proclaimed identity I) to the power D
in order to reconstitute what must be the public power of the component r. Finally, the verifier must retrieve the exponent D by applying the function f to the concatenation of the message and of the reconstituted public power. This is illustrated in Figure 9.
If p is written in 64 arbitrary bits, the signatory undertakes approximately 192 multiplications in the ring (he must compute successively two powers with exponents of 64 bits without being able to combine them) in order to complete the operation. This complexity is already markedly less than that of the RSA process: 768 multiplications, on average, for an exponential modulo a composite number in 512 bits, and 1536 for a composite number in 1024 bits.
If p is written in 64 arbitrary bits, the verifier undertakes only approximately 112 multiplications, since he combines his operations into 64 squares and three times 16 multiplications, on average, in order to compute a single step tP.JD mod N. It is fortunate that the authentication is simpler than the formulation of the signature, since each signature is called upon to be verified several times.
However, it is possible to choose 264 (that is to say a power of 2) as exponent p in order to simplify the computations, without modifying the security of the system. The raising to the power p is then undertaken by 64 squares. The exponent D is a number of 64 bits. The signature is then undertaken in 160 multiplications. The verification of a signature is ref]ected, on average, in 96 multiplications in the ring, '7~

PHQ 87.030 19 02.08.1988 i.e. 12.5~ of the RSA with 512 bits and 6.2% of the RSA with 1024 bits.
In the earlier process FS described hereinabove with 64 multiple accreditations in the same card, one squaLe and an average of 32 multipliations are required. Thus, the method of the invention involving comprehensive accreditation is utilized at the cost of a surplus of computations by a multiplicative factor of the order of 3.
When, in the prior art, there is a limitation to 8 accreditations in the card, with 8 markers in the signature (8 iterations at 5 multiplications, i.e. 40 multiplications), the ratio dimishes slightly further, in favour of the invention.
It is certainly also possible to choose 264~1 as public exponent (that is to say an odd number). At ~he cost of only one supplementary multiplication to compute a public power, certain restrictions are lifted in this way with regard to the quadratic residues in the ring (when the exponent p is even, several elements of the ring may correspond to the p-th root, but only one is appropriate).
Figure 10 shows diagrammatically a computer which permits the invention to be implemented, among other things for ex~cuting the authentication process. For simplicity, the other station communicating with this computer, has not been shown. Either or both stations may be physically realized in a so-called smart card, that, for other purposes, has been published extensively.
The computer shown comprises an input-output interface IO, a central unit CU, a programmable memory of the read-only type (PROM), a read-only memory (ROM) and a random access memory (RAM). The computer further comprises an element G of the noise-generator or random-generator type.
The accreditation and the identity information items recorded in the memory PROM are inaccessible from outside. The programs are recorded in the memory ROM. The memory RAM serves to store computation results. The generator G is used for drawing of the various numbers participating in the process (r, D).
The central unit and the memories may be structured as the monolithic self-programmable microcomputer described in Vnited States Patent 4,382,279.
The compression function may rely upon the DES (Data Encryption Standard) algorithm. There is in existence a smart card, which executes this DES algorithm.

Claims (9)

1. A method for authentication of an object element viz à viz verifier element, said method comprising the following initialization steps:
a. generating and disclosing a first integer p of at least ten bits and a product N of two secret prime factors;
b. generating a first personalized digital quantity I with added redundancy to form a second personalized digital quantity J;
c. generating and storing in the object element an accreditation number B as the inverse of A = J1/Pmod N, so that BPJmod N = 1;
said method furthermore comprising the following authentication steps:
d. in the object element drawing a first random integer r which is a member of the ring of integers modulo N and calculating a title number T according to T = rPmod N and forwarding at least a predetermined multibit field of the title number to the verifier element as a first comparison data;
e. in the verifier element drawing a second random number D within the closed interval [o, p-1] and forwarding this second random number to the object element;
f. in the object element generating a marker number t according to t = r.BDmod N and forwarding the marker number to the verifier element;
g. in the verifier element calculating a second comparison data equal to tPJDmod N and comparing said first comparison data to a corresponding multibit field of the second comparison data, and upon correspondence directly generating an authentificationapprobation.

2. A method for authentication of a message m of a principal viz à viz a verifier element, said method comprising the following initialization steps:
a. generating and disclosing a first integer p of at least ten bits and a product N of two secret prime factors;
b. generating a first personalized digital quantity I with added redundancy to form a second personalized digital quantity J;
c. generating and storing in a medium element associated to the principal an accreditation number B as the inverse of: A=J1/Pmod N so that BPJmod N=1; said method furthermore comprising the following authentication steps:
d. in the medium element drawing a first random integer r which is a member of the ring of integers modulo N, calculating a first intermediate value rPmod N and therefrom a title number T by a hash function having as argument m and the firstintermediate value, and forwarding at least a predetermined multibit field of the title number to the verifier element as a first comparison data;
e. in the verifier element drawing a second random number D within the closed interval [0, p-1] and forwarding this second random number to the medium element;
f. in the medium element generating a marker number t according to t=r.BDmod N
and forwarding the marker number to the verifier element;
g. in the verifier element calculating a second intermediate value according to tPJDmod N and therefrom a second comparison data by a hash function having as argument m and the second intermediate value, and comparing said first comparison data to a corresponding second multibit field of the second comparison data, and upon correspondence directly generating an authentification signal.

3. A method for signing a message m by an accredited entity, said method comprising the following accreditation steps by an accrediting authority:
a. generating and disclosing a first integer p of at least third bits and a product N of two secret prime factors;
b. generating a first personalized digital quantity I with added redundancy to form a second personalized digital quantity J;
c. generating and storing into a medium held by the accredited entity an accreditation number B as the inverse of A=J1/P mod N so that BPJ mod N=1;
d. drawing a second first random integer r which is a member of the ring of integers modulo N, calculating a first intermediate value D=rP mod N; and a hash function f by taking as arguments the message m to be signed and the first intermediate value D;
e. generating a sole marker t as the product trod mod N
f. transmitting by way of signed message, the message m, the identity I, the first intermediate value D and the marker t.

4. A system for the authentication of an accreditation information A with zero-knowlegde proof, this information having been formulated by a process of the public-key type comprising the following operations:
- an authority issuing the accreditation chooses two prime factors, forms the product N of these two factors, keeps secret these factors, chooses an integer p that comprises at least ten bits positions and publishes N and p, - for the holder of the accreditation, a digital identity I is formed, and supplemented by redundancy in order to form a shaded identity word J, - the accreditation information A is formulated by the authority by taking the p-th root of the shaded identity J in the ring of integers modulo N, (A= J1/P mod N=J), said system comprising - a memory for storing an inverse information modulo N of the accreditation information A, i.e. the inverse accreditation information B (BP j mod N=1), which is to be authentificated, - processing a means for executing the authentification operation by means of a single-layer interactive and probabilistic digital process of the zero-knowlegde proof type and comprising communication means for communicating between a medium containing the memory called "the verified" and an authentification element called "the verifier", said processing means comprising:
- in the verified first random number generating means for generating a first random integer r that is a member of the ring of integers modulo N, - power rising means fed by the first random number generating means for raising r to the power p modulo N to produce a title T, - first transmission means fed by the power raising means for transmitting at least a predetermined bit portion of the title T to the verifier, - in the verifier second random number generating means for generating a second random number (D) within the interval 0 and (p-1), including the limits thereof, - request means cum second transmission means fed by the second PHQ 87.030 02.08.1988 random number generating means for generating and transmitting a processing request to the verified, - in the verified first calculating means fed by the second transmission means to calculate the product in the ring of integers modulo N of the first random integer r, and the D-th power of the inverse accreditation information B to feed the result there of as a marker t=r.BD mod N to the first transmission means, - in the verifier second calculating means fed by the first transmission means for calculating the product of the marker t, within the ring of integers modulo N, and the D-th power of the shaded identity J, i.e. tPjD mod N, - in the verifier comparing means fed by the second calculating means and by the first transmission means for comparing said predetermined bit portion to a corresponding bit portion of tPjD
mod N for in a single comparison step upon a detected equality issuing an authentified accreditation signal.

5. A system for the authentification of a message m originating from a presumably accredited principal, by means of a digital word B obtained by a public-key process comprising the following operations:
- an authority issuing the accreditation chooses two prime numbers, forms the product N of these two numbers, chooses an integer p, and publishes N and p, - for the principal a digital identity is formed and supplemented by redundancy to form a shaded identity word J, - an accreditation information A is formulated by taking the p-th root of the shaded identity J in the ring of integers modulo N, (A= J1/P mod N=J), said system comprising - a memory for storing an inverse information modulo N of the accreditation information A, i.e. the inverse accreditation information B (BP j mod N=1), - processing means for executing the authentification operation by means of a single-layer interactive and probabilistic process of the zero-knowledge proof type and comprising communication means for communicating between a medium containing the memory called "the verified" and an authentification element called "the verifier", said PHQ 87.030 02.08.1988 processing means comprising:
- in the verified first random number generating means for generating a first random integer r that is an element of the ring of integers modulo N, - power raising means cum first compression means fed by the first random number generating means for raising r to the power p modulo N and computing a result by means of a compression function that has as arguments the message m and rP mod N, said result constituting a title T, - first transmission means fed by the first compression means for transmiting at least a predetermined bit portion of the title T to the verifier, - in the verifier second random number generating means for generating a second random number (D) within the interval between 0 and (p-1), including the limits thereof, - request means cum second transmission means fed by the second random number generating means for generating and transmitting a processing request to the verified, - in the verified first calculating means fed by the second transmission means to calculate the product in the ring of integers modulo N of the first random integer r, and the D-th power of the inverse accreditation information B to feed said product as a marker t=r.BD mod N to the first transmission means, - in the verifier second calculating means fed by the first transmission means for calculating the product of the marker t, within the ring of integers modulo N, and the D-th power of the shaded identity J, i.e. tPjD mod N, - in the verifier second compression means fed by the second calculating means for computing a result by taking as arguments the message to be authentificated and said product, - in the verifier comparing means fed by said second compression means and by the first transmission means for in a single comparision step comparing said predetermined bit portion to a corresponding bit portion of said result and upon a detected equality issuing an "authentic message signal".

6. A system for signing a message m by a presumably accredited entity, this accreditation having been formulated by a public-key process comprising the following operations:
- an authority issuing the accreditation chooses two prime factors, forms the product of these two factors, keeps secret these factors, chooses an integer p that comprises at least thirty bit positions and publishes N and p, - for any entity that is a signatory a digital identity I is formed and supplemented by redundancy in order -to form a shaded identity word J, - the accreditation information A is formulated by the authority by taking the p-th root of the shaded identity J in the ring of integers modulo N (A=j1/P mod N), said system comprising:
- a memory medium held by the signatory for starting an inverse information modulo N of the accreditation information A, i.e. the inverse accreditation information B (i.e. BPJ mod N=1), - signature generating means for generating a signature according to a probabilistic digital process, and comprising:
- random number generating means for generating a random integer I that is a member of the ring of integers modulo N, - power raising means fed by the random number generating means for raising r to the power p modulo N, - compression means fed by the power raising means for calculating a compression function that has as arguments the message m and rP mod N to yield a result number D, - product forming means fed by the random number generator and by the compression means to form the product of r and D-th power of the inverse accreditation information B to yield a sole marker t, - transmission means fed by the product forming means to transmit a signal message consisting of the message m, the identity I, the result number D, and the sole marker t,

7. A station for use as the "verified" in a system as claimed in either Claim 4 or Claim 5 and manufactured in the shape of a smart card.

8. A station for use as the "verifier" in a system as claimed in either Claim 4 or Claim 5 and manufactured in the shape of a smart card.

9. A signature generating station for use in a system as claimed in Claim 6 and manufactured in the shape of a smart card.