GOST R 34.10-2001: Digital Signature Algorithm
draft-dolmatov-cryptocom-gost34102001-08

Internet-Draft                                          V. Dolmatov, Ed.
Intended status: Informational                            Cryptocom Ltd.
Expires:    June  21, 2010                            December 21, 2009

                            GOST R 34.10-2001 
                       digital signature algorithm
               draft-dolmatov-cryptocom-gost34102001-08

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Abstract

   This document is intended to be a source of information about the
   Russian Federal standard for digital signatures (GOST R 34.10-2001),
   which is one of the Russian cryptographic standard algorithms (called
   GOST algorithms). Recently, Russian cryptography is being used in
   Internet applications, and this document has been created as
   as information for developers and users of GOST R 34.10-2001 for
   digital signature generation and verification. 

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Table of Contents

   1. Introduction.....................................................2
      1.1. General information.........................................2
      1.2. The purpose of GOST R 34.10-2001............................3
   2. Applicability....................................................3
   3. Definitions and notations........................................3 
      3.1. Definitions.................................................3 
      3.2. Notations...................................................5 
   4. General statements...............................................6 
   5. Mathematical conventions.........................................7 
      5.1. Mathematical definitions....................................7 
      5.2. Digital signature parameters................................9
      5.3. Binary vectors.............................................10 
   6. Main processes..................................................10 
      6.1. Digital signature generation process.......................11 
      6.2. Digital signature verification.............................11
   7. Test examples (Appendix B to GOST R 34.10-2001).................13 
      7.1. The digital signature scheme parameters....................13 
      7.2. Digital signature process (Algorithm I)....................15 
      7.3. Verification process of digital signature (Algorithm II)...16
   8. Security considerations.........................................17
   9. IANA considerations.............................................17
   10. Normative references...........................................18 
   11. Informative references.........................................18

1. Introduction

1.1. General information

   1. GOST R 34.10-2001 was developed by the Federal Agency for 
   Government Communication and Information under the President of 
   Russian Federation with participation of the All-Russia Scientific 
   and Research Institute of Standardization.

   GOST R 34.10-2001 was submitted by Federal Agency for Government
   Communication and Information at President of Russian Federation.

   2. GOST R 34.10-2001 was accepted and activated by the Act 380-st of
   12.09.2001 issued by the Russian federal committee for standards.

   3. GOST R 34.10-2001 was developed in accordance with terminology and
   concepts of international standards ISO 2382-2-76. "Data processing.
   Dictionary. Part 2. Arithmetic and logic operations", ISO/IEC 9796-91
   "Information technology. Secure methods. Digital signature scheme
   with message recovering", series ISO/ IEC 14888 "Information
   technology.  Secure methods. Digital signatures and application" and
   series ISO/ IEC 10118 "Information technology. Secure methods. Hash
   functions"

   4. GOST R 34.10-2001 replaces GOST R 34.10-94.

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1.2. The purpose of GOST R 34.10-2001

   GOST R 34.10-2001 describes generation and verification processes for
   digital signature, based on operations with elliptic curve points 
   group, defined over prime finite field. 

   GOST R 34.10-2001 is developed to replace GOST R 34.10-94. Necessity 
   for this development is caused by the need to increase the digital
   signature security against unauthorized modification. Digital 
   signature security is based on complexity of discrete logarithm
   calculation in elliptic curve points group and also on the security
   of the hash function used (according to [GOST3411]).

   Terminologically and conceptually GOST R 34.10-2001 is in accord with
   international standards ISO 2382-2 [1], ISO/ IEC 9796 [2], series
   ISO/ IEC 14888 [3]-[5] and series ISO/ IEC 10118 [6]-[9]. 

   Note: the main part of GOST R 34.10-2001 is supplemented with two
   appendixes: 

   extra terms in digital signature area (Appendix A to this memo); 

   test examples (section 7 of this memo);

   a bibliography in digital signature area (section 12 of this memo).

2. Applicability 

   GOST R 34.10-2001 defines an electronic digital signature (or simply
   digital signature) scheme, digital signature generation and
   verification processes for a given message (document), meant for
   transmission via insecure public telecommunication channels in data
   processing systems of different purposes. 

   Use of digital signature based on GOST R 34.10-2001 makes transmitted
   messages more resistant to forgery and loss of integrity, in
   comparison with digital signature scheme prescribed by the previous
   standard. 

   GOST R 34.10-2001 is obligatory to use in Russian Federation in all
   data processing systems providing public services.

3. Definitions and notations 

 3.1. Definitions

   The following terms are used in the standard: 

   3.1.1 Appendix: Bit string, formed by digital signature and by
   arbitrary text field (ISO/IEC 148881-1 [3]). 

   3.1.2 Signature key: Element of secret data, specific to the subject

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   and used only by this subject during the signature generation process
   (ISO/IEC 14888-1 [3]).

   3.1.3 Verification key: Element of data mathematically linked to the
   signature key data element, used by the verifier during the digital
   signature verification process (ISO/IEC 14888-1 [3]). 

   3.1.4 Domain parameter: Element of data which is common for all the
   subjects of the digital signature scheme, known or accessible to all
   the subjects (ISO/IEC 14888-1 [3]).

   3.1.5 Signed message: A set of data elements, that consists of the
   message and the appendix, which is a part of the message.

   3.1.6 Pseudo-random number sequence: A sequence of numbers, which is
   obtained during some arithmetic (calculation) process, used in
   specific case instead of a true random number sequence (ISO 2382-2
   [1]). 

   3.1.7 Random number sequence: A sequence of numbers none of which can
   be predicted (calculated) using only the preceding numbers of the
   same sequence (ISO 2382-2 [1]).

   3.1.8 Verification process: A process using the signed message, the
   verification key and digital signature scheme parameters as initial
   data and giving the conclusion about digital signature validity or
   invalidity as a result. (ISO/IEC 14888-1 [3]).

   3.1.9 Signature generation process: A process using the message, the
   signature key and digital signature scheme parameters as initial data
   and generating the digital signature as the result (ISO/IEC 14888-1
   [3]). 

   3.1.10 Witness: Element of data (resulting from the verification
   process) which states to the verifier whether digital signature is
   valid or invalid (ISO/IEC 148881-1 [3]).

   3.1.11 Random number: A number chosen from the definite number set in
   such a way that every number from the set can be chosen with
   equal probability (ISO 2382-2 [1]). 

   3.1.12 Message: String of bits of a limited length (ISO/IEC 9796
   [2]).

   3.1.13 Hash code: String of bits that is a result of the hash
   function (ISO/IEC 148881-1 [3]). 

   3.1.14 Hash function: The function, mapping bit strings onto bit
   strings of fixed length observing the following properties: 

        1) it is difficult to calculate the input data, that is the 
        pre-image of the given function value; 

        2) it is difficult to find another input data that is the 
        pre-image of the same function value as is the given input data; 

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        3) it is difficult to find a pair of different input data, 
        producing the same hash function value.

   Note: The property 1 in the context of the digital signature area
   means that it is impossible to recover the initial message using the
   digital signature; property 2 means that it is difficult to find
   another (falsificated) message that produces the same digital
   signature as a given message; property 3 means that it is difficult
   to find some pair of different messages, that both produce the same
   signature. 

   3.1.15 [Electronic] Digital signature: String of bits obtained as a
   result of signature generation process. This string has an internal
   structure, depending on the specific signature generation mechanism.

   Note: In GOST R 34.10-2001 terms "Digital signature" and "Electronic
   sigital signature" are synonymous to save terminological succession
   to native legal documents currently in force and scientific
   publications. 

3.2 Notations 

   In GOST R 34.10-2001 the following notations are used: 

   V256 - set of all binary vectors of length 256 bit;

   V_all - set of all binary vectors of an arbitrary finite
   length; 
   
   Z - set of all integers; 
   
   p - prime number, p > 3; 
   
   GF(p) - finite prime field represented by a set of integers 
           {0, 1, ..., p - 1};

   b (mod p) - minimal nonnegative number, congruent to b modulo p; 
   
   M - user's message, M belongs to V_all;
 
   (H1 || H2 ) - concatenation of two binary vectors;

   a,b - elliptic curve coefficients;

   m - points of the elliptic curve group order;
   
   q - subgroup order of group of points of the elliptic curve;
   
   O - zero point of the elliptic curve; 
   
   P - elliptic curve point of order q; 
   
   d - integer - a signature key;
      
   Q - elliptic curve point - a verification key;

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   ^ - the power operator;

   /= - non-equality;

   sqrt - square root;

   zeta - digital signature for the message M.

4. GENERAL STATEMENTS 

   A commonly accepted digital signature scheme (model) (see 6 ISO/IEC
   14888-1 [3]) consists of three processes:

        - generation of a pair of keys (for signature generation and for
          signature verification);

        - signature generation;  

        - signature verification.

      In GOST R 34.10-2001 a process for generating a pair of keys (for
   signature and verification) is not defined. Characteristics and ways
   of the process realization are defined by involved subjects, who
   determine corresponding parameters by their agreement. 
   
   The digital signature mechanism is defined by realization of two main
   processes (see part 7):

        - signature generation (see. 6.1);  
   
        - signature verification (see. 6.2).

   The digital signature is meant for authentication of the signatory of
   the electronic message. Besides, the digital signature usage gives an
   opportunity to provide the following properties during signed message 
   transmission:

        - realization of control of the transmitted signed message 
          integrity,  
   
        - proof of the authorship of the signatory of the message,  
   
        - protection of the message against possible forgery.

   A schematic representation of the signed message is shown in the
   figure 1.
                                  appendix
                                     |
                     +-------------------------------+
                     |                               |
     +-----------+   +------------------------+- - - +
     | message M |---| digital signature zeta | text |
     +-----------+   +------------------------+- - - +

                     Figure 1 - Signed message scheme

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   The field "digital signature" is supplemented by the field "text" 
   (see figure 1), that can contain for example identifiers of the 
   signatory of the message, and/or time label.
   
   The digital signature scheme determined in GOST R 34.10-2001 must be
   implemented using operations of elliptic curve points group, defined
   over a finite prime field, and also with the use of hash function.

   The cryptographic security of the digital signature scheme is based
   on complexity of solving the problem of calculation the discrete 
   logarithm in elliptic curve points group, and also on the security of 
   the hash function used. The hash function calculation algorithm is
   determined in [GOST3411]. 
    
   The digital signature scheme parameters needed for signature
   generation and verification are determined in 5.2. 

   GOST R 34.10-2001 does not determine the process of generating
   parameters needed for digital signature scheme. Possible sets of 
   these parameters are defined for example in [RFC4357].

   The digital signature represented as a binary vector of length 512
   bit, must be calculated using definite set of rules stated in 6.1. 
    
   The digital signature of the received message is accepted or denied
   in accordance with the set of rules, stated in 6.2.

5. Mathematical conventions 

   To define a digital signature scheme it is necessary to describe
   basic mathematical objects, used in the signature generation and
   verification processes. This section lays out basic mathematical
   definitions and requirements for the parameters of the digital
   signature scheme.

5.1 Mathematical definitions

   Suppose a prime number p > 3 is given. Then an elliptic curve E,
   defined over a finite prime field GF(p), is the set of number pairs
   (x,y), x, y belong to Fp , satisfying the identity

   y^2 = x^3 + a*x + b (mod p),                                      (1)

   where a, b belong to GF(p) and 4*a^3 + 27*b^2 is not congruent to 
   zero modulo p.
    
   An invariant of the elliptic curve is the value J(E) satisfying the
   equality

                   4*a^3
   J(E) = 1728 * ------------ (mod p)                                (2)
                 4*a^3+27*b^2

   

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   Elliptic curve E coefficients a,b are defined in the following way
   using the invariant J(E):

   | a=3*k (mod p)
   |                              J(E)
   | b=2*k (mod p), where k = ----------- (mod p), J(E) /= 0 or 1728 (3)
                              1728 - J(E)

   The pairs (x,y) satisfying the identity (1) are called the elliptic
   curve E points, x and y are called x- and y-coordinates of the point
   correspondingly.

   We will denote elliptic curve points as Q(x,y) or just Q. Two 
   elliptic curve points are equal if their x- and y-coordinates are
   equal. 
    
   On the set of all elliptic curve E points we will define the addition
   operation, denoted by "+". For two arbitrary elliptic curve E points
   Q1 (x1, y1) and Q2 (x2, y2) we will consider several variants. 
    
   Suppose coordinates of points Q1 and Q2 satisfy the condition x1 /=
   x2. In this case their sum is defined as a point Q3 (x3,y3) with
   coordinates defined by congruences

   | x3=lambda^2-x1-x2 (mod p),                  y1-y2
   |                              where lambda= ------- (mod p).     (4)
   | y3=lambda*(x1-x3)-y1 (mod p),               x1-x2

   If x1 = x2 and y1 = y2 /= 0, then we will define point Q3
   coordinates in a following way

   | x3=lambda^2-x1*2 (mod p),                    3*x1^2+a
   |                               where lambda= --------- (mod p)   (5)
   | y3=lambda*(x1-x3)-y1 (mod p),                 y1*2

   If x1 = x2 and y1 = - y2 (mod p), then the sum of points Q1 and Q2
   is called a zero point O, without determination of its x- and
   y-coordinates. In this case point Q2 is called a negative of point 
   Q1. For the zero point the equalities hold

   O+Q=Q+O=Q,                                                        (6)

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   where Q is an arbitrary point of elliptic curve E.

   A set of all points of elliptic curve E including zero point forms a
   finite abelian (commutative) group of order m regarding introduced
   addition operation. For m the following unequalities hold: 
   
   p + 1 - 2*sqrt(p) =< m =< p + 1 + 2*sqrt(p).                      (7)

   The point Q is called a point of multiplicity k, or just a multiple
   point of the elliptic curve E, if for some point P the following
   equality holds:

   Q = P + ... + P = k*P.                                            (8)
       -----+-----
            k

5.2 Digital signature parameters 

   The digital signature parameters are:

        - prime number p is an elliptic curve modulus, satisfying the
          inequality p > 2^255. The upper bound for this number must be
          determined for specific realization of digital signature 
          scheme;  

        - elliptic curve E, defined by its invariant J(E) or by 
          coefficients a, b belonging to GF(p).  
     
        - integer m is an elliptic curve E points group order;  
   
        - prime number q is an order of a cyclic subgroup of the 
          elliptic curve E points group, which satisfies the following
          conditions:

   | m = nq, n belongs to Z , n>=1    
   |                              ;                                  (9)
   | 2^254 < q < 2^256

        - point P /= O of an elliptic curve E, with coordinates 
          (x_p, y_p), satisfying the equality q*P=O.

        - hash function h(.):V_all -> V256, which maps the messages
          represented as binary vectors of arbitrary finite length onto
          binary vectors of length 256 bit. The hash function is 
          determined in [GOST3411].

   Every user of the digital signature scheme must have its personal
   keys:  
   
        - signature key, which is an integer d, satisfying the 
          inequality 0 < d < q;

        - verification key, which is an elliptic curve point Q with 
          coordinates (x_q, y_q), satisfying the equality d*P=Q.

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   The previously introduced digital signature parameters must satisfy
   the following requirements:
   
        - it is necessary that the condition p^t/= 1 (mod q ) holds for
          all integers t = 1, 2, ... B where B satisfies the inequality 
          B >= 31;

        - it is necessary that the inequality m /= p holds;  
   
        - the curve invariant must satisfy the condition 
          J(E) /= 0, 1728.

5.3 Binary vectors 

   To determine the digital signature generation and verification
   processes it is necessary to map the set of integers onto the set of
   binary vectors of length 256 bit.

   Consider the following binary vector of length 256 bit where
   low-order bits are placed on the right, and high-order ones are
   placed on the left
                                 
   H = (alpha[255], ... , alpha[0]), H belongs to V256              (10)

   where alpha[i], i = 0, ... , 255 are equal to 1 or to 0. We will say
   that the number alpha belonging to Z is mapped onto the binary vector
   h, if the equality holds

   alpha = alpha[0]*2^0 + alpha[1]*2^1 + ... + alpha[255]*2^255.    (11)
                       
   For two binary vectors H1 and H2 , which correspond to integers alpha
   and beta, we define a concatenation (union) operation in the
   following way. Let 

        H1 = (alpha[255], ... , alpha[0]), 
                                                                    (12) 
        H2 = (beta[255], ..., beta[0]),
   
   then their union is 
 
        H1||H2 = (alpha[255], ... , alpha[0], beta[255], ..., beta[0])
                                                                    (13)
   that is a binary vector of length 512 bit, consisting of coefficients              
   of the vectors H1 and H2.

   On the other hand, the introduced formulae define a way to divide a         
   binary vector H of length 512 bit into two binary vectors of length
   256 bit, where H is the concatenation of the two.

6. Main processes 

   In this section the digital signature generation and verification
   processes of user's message are defined. 
      
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   For the realization of the processes it is necessary, that all users
   know the digital signature scheme parameters, which satisfy the
   requirements of section 5.2. 
   
   Besides, every user must have the signature key d and the
   verification key Q(x[q], y[q]) , which also must satisfy the
   requirements of section 5.2.

6.1 Digital signature generation process 

   It is necessary to perform the following actions (steps) according to
   Algorithm I to obtain the digital signature for the message M
   belonging to V_all:
                                              
   Step 1 - calculate the message hash code M: H = h(M).            (14)

   Step 2 - calculate an integer alpha, binary representation of which 
               
   is the vector H , and determine e = alpha (mod q ) .             (15)
   
   If e = 0, then assign e = 1. 
   
   Step 3 - generate a random (pseudorandom) integer k, satisfying the 
   inequality 
   
   0 < k < q.                                                       (16) 
   
   Step 4 - calculate the elliptic curve point C = k*P and determine 
      
   r = x_C (mod q),                                                 (17) 
   
   where x_C is x-coordinate of the point C. If r = 0, return to step 3.
   
   Step 5 - calculate the value
   
   s = (r*d + k*e) (mod q).                                         (18) 
   
   If s = 0, return to step 3. 
                                     
   Step 6 - calculate the binary vectors R and S , corresponding to r 
                                                   
   and s, and determine the digital signature zeta = (R || S) as 
   concatenation of these two binary vectors.

   The initial data of this process are the signature key d and the
   message M to be signed. The output result is the digital signature
   zeta. 

6.2 Digital signature verification

   To verify digital signature for the received message M belonging to
   V_all it is necessary to perform the following actions (steps)
   according to Algorithm II:

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   Step 1 - calculate the integers r and s using the received signature
   zeta. If the inequalities 0 < r < q, 0 < s < q hold, go to the next 
   step. In the other case the signature is invalid.
 
   Step 2 - calculate the hash code of the received message M

   H = h(M).                                                        (19)

   Step 3 - calculate the integer alpha, the binary representation of 
                   
   which is the vector H , and determine 
   
   e = alpha (mod q).                                               (20)
   
   If e = 0, then assign e = 1. 
   
   Step 4 - calculate the value v = e^(-1) (mod q) .                (21)

   Step 5 - calculate the values 
   
   z1 =  s*v (mod q), z2 = -r*v (mod q).                            (22)
   
   Step 6 - calculate the elliptic curve point C = z1*P + z2*Q and 
   determine
   
   R = x_C (mod q),                                                 (23)
   
   where x_C is x-coordinate of the point .

   Step 7 - if the equality R = r holds, than the signature is accepted,
   in the other case the signature is invalid. 

   The input data of the process are the signed message M, the digital
   signature zeta and the verification key Q. The output result is the
   witness of the signature validity or invalidity.

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7. Test examples (Appendix to GOST R 34.10-2001)

   This sectin is included in GOST R 34.10-2001 as an appendix but is 
   officially mentioned as not a part of the standard.

   This is a reference appendix and is not a part of the standard. The
   values given here for the parameters p, a, b, m, q, P, the signature
   key d and the verification key Q are recommended only for testing 
   correctness of actual realizations of the algorithms described in
   GOST R 34.10-2001. 
   
   All numerical values are introduced in decimal and hexadecimal
   notations. The numbers beginning with 0x are in hexadecimal notation.
   The symbol "\\" denotes a hyphenation of a number to the next line.
   For example, the notation 
   
   12345\\ 
   67890 
   
   0x499602D2
   
   represents 1234567890 in decimal and hexadecimal number systems
   correspondingly. 
   
7.1 The digital signature scheme parameters 

   The following parameters must be used for the digital signature
   generation and verification (see section 5.2). 

7.1.1 Elliptic curve modulus 

   The following value is assigned to parameter p in this example
   
   p= 57896044618658097711785492504343953926\\
   634992332820282019728792003956564821041 
   
   p = 0x8000000000000000000000000000\\
   000000000000000000000000000000000431

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7.1.2 Elliptic curve coefficients 

   Parameters a and b take the following values in this example: 
   
   a = 7
   a = 0x7

   b = 43308876546767276905765904595650931995\\
   942111794451039583252968842033849580414 

   b = 0x5FBFF498AA938CE739B8E022FBAFEF40563\\
   F6E6A3472FC2A514C0CE9DAE23B7E

7.1.3 Elliptic curve points group order 

   Parameter m takes the following value in this example: 
   
   m = 5789604461865809771178549250434395392\\
   7082934583725450622380973592137631069619
   
   m = 0x80000000000000000000000000000\\
   00150FE8A1892976154C59CFC193ACCF5B3
   
7.1.4 Order of cyclic subgroup of elliptic curve points group 

   Parameter q takes the following value in this example: 
   
   q = 5789604461865809771178549250434395392\\
   7082934583725450622380973592137631069619
   
   q = 0x80000000000000000000000000000001\\
   50FE8A1892976154C59CFC193ACCF5B3
   
7.1.5 Elliptic curve point coordinates 

   Point P coordinates take the following values in this example: 
   
   x_p = 2
   x_p = 0x2 
   
   y_p = 40189740565390375033354494229370597\\
   75635739389905545080690979365213431566280
   
   y_p = 0x8E2A8A0E65147D4BD6316030E16D19\\
   C85C97F0A9CA267122B96ABBCEA7E8FC8
   
7.1.6 Signature key 

   It is supposed in this example that the user has the following
   signature key d: 

   d = 554411960653632461263556241303241831\\
   96576709222340016572108097750006097525544

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   d = 0x7A929ADE789BB9BE10ED359DD39A72C\\
   11B60961F49397EEE1D19CE9891EC3B28 

7.1.7 Verification key 

   It is supposed in this example that the user has the verification key
   Q with the following coordinate values: 
   
   x_q = 57520216126176808443631405023338071\\
   176630104906313632182896741342206604859403
   
   x_q = 0x7F2B49E270DB6D90D8595BEC458B5\\
   0C58585BA1D4E9B788F6689DBD8E56FD80B

   y_q = 17614944419213781543809391949654080\\
   031942662045363639260709847859438286763994
   
   y_q = 0x26F1B489D6701DD185C8413A977B3\\
   CBBAF64D1C593D26627DFFB101A87FF77DA 
   
7.2 Digital signature process (Algorithm I) 

   Suppose that after 1-3 steps according to the Algorithm I (6.1) are
   performed the following numerical values are obtained: 
   
   e = 2079889367447645201713406156150827013\\
   0637142515379653289952617252661468872421
   
   e = 0x2DFBC1B372D89A1188C09C52E0EE\\
   C61FCE52032AB1022E8E67ECE6672B043EE5
   
   k = 538541376773484637314038411479966192\\
   41504003434302020712960838528893196233395
   
   k = 0x77105C9B20BCD3122823C8CF6FCC\\
   7B956DE33814E95B7FE64FED924594DCEAB3
   
   And the multiple point C = k * P has the coordinates: 
   
   x_C = 297009809158179528743712049839382569\\
   90422752107994319651632687982059210933395
   
   x_C = 0x41AA28D2F1AB148280CD9ED56FED\\
   A41974053554A42767B83AD043FD39DC0493
   
   y[C] = 328425352786846634770946653225170845\\
   06804721032454543268132854556539274060910
   
   y[C] = 0x489C375A9941A3049E33B34361DD\\
   204172AD98C3E5916DE27695D22A61FAE46E 

   Parameter r = x_C(mod q) takes the value: 

V.Dolmatov                Expires    June 21, 2010            [Page 15]


   r = 297009809158179528743712049839382569\\
   90422752107994319651632687982059210933395 
   
   r = 0x41AA28D2F1AB148280CD9ED56FED\\
   A41974053554A42767B83AD043FD39DC0493
   
   Parameter s = (r*d + k*e)(mod q) takes the value: 
   
   s = 57497340027008465417892531001914703\\
   8455227042649098563933718999175515839552
   
   s = 0x1456C64BA4642A1653C235A98A602\\
   49BCD6D3F746B631DF928014F6C5BF9C40

7.3 Verification process of digital signature (Algorithm II) 

   Suppose that after the steps 1-3 according to the Algorithm II (6.2)
   are performed the following numerical value is obtained: 
   
   e = 2079889367447645201713406156150827013\\
   0637142515379653289952617252661468872421
   
   e = 0x2DFBC1B372D89A1188C09C52E0EE\\ 
   C61FCE52032AB1022E8E67ECE6672B043EE5
   
   And the parameter v = e^(-1) (mod q) takes the value: 
   
   v = 176866836059344686773017138249002685\\ 
   62746883080675496715288036572431145718978
   
   v = 0x271A4EE429F84EBC423E388964555BB\\ 
   29D3BA53C7BF945E5FAC8F381706354C2
 
   The parameters z1 = s*v(mod q) and z2 = -r*v(mod q) take the values:

   z1 = 376991675009019385568410572935126561\\ 
   08841345190491942619304532412743720999759
   
   z1 = 0x5358F8FFB38F7C09ABC782A2DF2A\\ 
   3927DA4077D07205F763682F3A76C9019B4F
   
   z2 = 141719984273434721125159179695007657\\ 
   6924665583897286211449993265333367109221
   
   z2 = 0x3221B4FBBF6D101074EC14AFAC2D4F7\\ 
   EFAC4CF9FEC1ED11BAE336D27D527665

   The point C = z1*P + z2*Q has the coordinates: 
   
   x_C = 2970098091581795287437120498393825699\\ 
   0422752107994319651632687982059210933395
   
   x_C = 0x41AA28D2F1AB148280CD9ED56FED\\ 
   A41974053554A42767B83AD043FD39DC0493
   
V.Dolmatov                Expires    June 21, 2010            [Page 16]

   
   y[C] = 3284253527868466347709466532251708450\\ 
   6804721032454543268132854556539274060910 
   
   y[C] = 0x489C375A9941A3049E33B34361DD\\ 
   204172AD98C3E5916DE27695D22A61FAE46E 
   
   Then the parameter R = x_C (mod q) takes the value: 
   
   R = 2970098091581795287437120498393825699\\ 
   0422752107994319651632687982059210933395

   R = 0x41AA28D2F1AB148280CD9ED56FED\\ 
   A41974053554A42767B83AD043FD39DC0493

  Since the equality R = r holds, the digital signature is accepted.

Appendix A. Extra terms in digital signature area

   The appendix gives extra international terms applied in the
   considered and allied areas.
   
   1. Padding: Extending of a data string with extra bits (ISO/IEC
   10118-1 [6]). 
   
   2. Identification data: A list of data elements, including specific
   object identifier, that belongs to the object and is used for its
   denotation (ISO/IEC 148881-1 [3]). 
 
   3. Signature equation: An equation, defined by the digital signature
   function (ISO/IEC 148881-1 [3]).

   4. Verification function: A verification process function, defined by
   the verification key, which outputs a witness of the signature
   authenticity (ISO/IEC 148881-1 [3]).

   5. Signature function: A function within a signature generation
   process, defined by the signature key and by the digital signature
   scheme parameters. This function inputs a part of initial data and,
   probably, a pseudorandom number sequence generator (randomizer), and
   outputs the second part of the digital signature.

8. Security considerations

   This entire document is about security considerations.

   Current cryptographic resistance of GOST R 34.10-2001 digital 
   signature algorithm is estimated as 2**128 operations of multiple 
   elliptic curve point computations on prime modulus of order 2**256.

9. IANA Considerations

   This document has no actions for IANA.

V.Dolmatov                Expires    June 21, 2010            [Page 17]


10. Normative references

   [GOST3410] "Information technology. Cryptographic data security. 
               Signature and verification processes of [electronic] 
               digital signature. GOST R 34.10-2001, Gosudarstvennyi 
               Standard of Russian Federation, Government Committee of
               the Russia for Standards, 2001. (In Russian)

   [GOST3411] "Information technology. Cryptographic Data Security.
              Hashing function.", GOST R 34.10-94, Gosudarstvennyi
              Standard of Russian Federation, Government Committee of
              the Russia for Standards, 1994. (In Russian)

   [RFC4357] RFC 4357. V.Popov, I.Kurepkin, S.Leontiev. Additional
             Cryptographic Algorithms for Use with GOST 28147-89, 
             GOST R 34.10-94, GOST R 34.10-2001, and GOST R 34.11-94 
             Algorithms

11. Informative references

   [1] ISO 2382-2-76 Data processing. Dictionary. Part 2. Arithmetic and
   logic operations 

   [2] ISO/IEC 9796-91 Information technology. Secure methods. Digital
   signature scheme with message recovering 
   
   [3] ISO/IEC 14888-1-98 Information technology. Secure methods.
   Digital signatures and application. Part 1. General statements 
   
   [4] ISO/IEC 14888-2-99 Information technology. Secure methods.
   Digital signatures and application. Part 2. Mechanisms on
   authentication base 
   
   [5] ISO/IEC 14888-3-99 Information technology. Secure methods.
   Digital signatures and application. Part 3. Mechanisms on certificate
   base 
   
   [6] ISO/IEC 10118-1-94 Information technology. Secure methods. Hash
   functions Part 1. General statements. 

   [7] ISO/IEC 10118-2-94 Information technology. Secure methods. Hash
   functions Part 2. Hash functions using n-bit block encryption
   algorithm 
  
   [8] ISO/IEC 10118-3-98 Information technology. Secure methods. Hash
   functions Part 3. Decimal hash functions 
   
   [9] ISO/IEC 10118-4-98 Information technology. Secure methods. Hash
   functions Part 4. Hash functions using modular arithmetic.

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Authors' Addresses

Vasily Dolmatov, Ed.
Cryptocom Ltd.
Kedrova st., 14, bld.2
Moscow, 117218, Russian Federation

EMail: dol@cryptocom.ru

Dmitry Kabelev
Cryptocom Ltd.
Kedrova st., 14, bld.2
Moscow, 117218, Russian Federation

EMail: kdb@cryptocom.ru

Igor Ustinov
Cryptocom Ltd.
Kedrova st., 14, bld.2
Moscow, 117218, Russian Federation

EMail: igus@cryptocom.ru

Sergey Vyshensky
Moscow State University
Leninskie gory, 1
Moscow, 119991, Russian Federation

EMail: svysh@pn.sinp.msu.ru