| Rfc | 5832 |
|---|
| Title | GOST R 34.10-2001: Digital Signature Algorithm |
|---|
| Author | V. Dolmatov, Ed. |
|---|
| Date | March 2010 |
|---|
| Format: | TXT, HTML |
|---|
| Updated by | RFC7091 |
|---|
| Status: | INFORMATIONAL |
|---|
|
Independent Submission V. Dolmatov, Ed.
Request for Comments: 5832 Cryptocom, Ltd.
Category: Informational March 2010
ISSN: 2070-1721
GOST R 34.10-2001:
Digital Signature Algorithm
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
information for developers and users of GOST R 34.10-2001 for digital
signature generation and verification.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor are not a candidate for any level of Internet
Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc5832.
Copyright Notice
Copyright (c) 2010 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document.
This document may not be modified, and derivative works of it may not
be created, except to format it for publication as an RFC or to
translate it into languages other than English.
Table of Contents
1. Introduction ....................................................3
1.1. General Information ........................................3
1.2. The Purpose of GOST R 34.10-2001 ...........................3
2. Applicability ...................................................4
3. Definitions and Notations .......................................4
3.1. Definitions ................................................4
3.2. Notations ..................................................6
4. General Statements ..............................................7
5. Mathematical Conventions ........................................8
5.1. Mathematical Definitions ...................................9
5.2. Digital Signature Parameters ..............................10
5.3. Binary Vectors ............................................11
6. Main Processes .................................................12
6.1. Digital Signature Generation Process ......................12
6.2. Digital Signature Verification ............................13
7. Test Examples (Appendix to GOST R 34.10-2001) ..................14
7.1. The Digital Signature Scheme Parameters ...................14
7.2. Digital Signature Process (Algorithm I) ...................16
7.3. Verification Process of Digital Signature (Algorithm II) ..17
8. Security Considerations ........................................19
9. References .....................................................19
9.1. Normative References ......................................19
9.2. Informative References ....................................19
Appendix A. Extra Terms in the Digital Signature Area .............21
Appendix B. Contributors ..........................................22
1. Introduction
1.1. General Information
1. GOST R 34.10-2001 [GOST3410] was developed by the Federal Agency
for Government Communication and Information under the President
of the Russian Federation with the 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 the Russian
Federation.
2. GOST R 34.10-2001 was accepted and activated by the Act 380-st of
12.09.2001 issued by the Government Committee of Russia for
Standards.
3. GOST R 34.10-2001 was developed in accordance with the terminology
and concepts of international standards ISO 2382-2:1976 "Data
processing - Vocabulary - Part 2: Arithmetic and logic
operations"; ISO/IEC 9796:1991 "Information technology -- Security
techniques -- Digital signature schemes giving message recovery";
ISO/IEC 14888 "Information technology - Security techniques -
Digital signatures with appendix"; and ISO/IEC 10118 "Information
technology - Security techniques - Hash-functions".
4. GOST R 34.10-2001 replaces GOST R 34.10-94.
1.2. The Purpose of GOST R 34.10-2001
GOST R 34.10-2001 describes the generation and verification processes
for digital signatures, based on operations with an elliptic curve
points group, defined over a prime finite field.
GOST R 34.10-2001 has been developed to replace GOST R 34.10-94.
Necessity for this development is caused by the need to increase
digital signature security against unauthorized modification.
Digital signature security is based on the complexity of discrete
logarithm calculation in an 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 accordance
with international standards ISO 2382-2 [ISO2382-2], ISO/IEC 9796
[ISO9796-1991], ISO/IEC 14888 Parts 1-3 [ISO14888-1]-[ISO14888-3],
and ISO/IEC 10118 Parts 1-4 [ISO10118-1]-[ISO10118-4].
Note: the main part of GOST R 34.10-2001 is supplemented with three
appendixes:
"Extra Terms in the Digital Signature Area" (Appendix A of this
memo);
"Test Examples" (Section 7 of this memo);
"A Bibliography in the Digital Signature Area" (Section 9.2 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 a digital signature based on GOST R 34.10-2001 makes
transmitted messages more resistant to forgery and loss of integrity,
in comparison with the digital signature scheme prescribed by the
previous standard.
GOST R 34.10-2001 is obligatory to use in the 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:
Appendix: Bit string, formed by a digital signature and by the
arbitrary text field [ISO14888-1].
Signature key: Element of secret data, specific to the subject and
used only by this subject during the signature generation process
[ISO14888-1].
Verification key: Element of data mathematically linked to the
signature key data element, used by the verifier during the digital
signature verification process [ISO14888-1].
Domain parameter: Element of data that is common for all the subjects
of the digital signature scheme, known or accessible to all the
subjects [ISO14888-1].
Signed message: A set of data elements, which consists of the message
and the appendix, which is a part of the message.
Pseudo-random number sequence: A sequence of numbers, which is
obtained during some arithmetic (calculation) process, used in a
specific case instead of a true random number sequence [ISO2382-2].
Random number sequence: A sequence of numbers none of which can be
predicted (calculated) using only the preceding numbers of the same
sequence [ISO2382-2].
Verification process: A process that uses the signed message, the
verification key, and the digital signature scheme parameters as
initial data and that gives the conclusion about digital signature
validity or invalidity as a result [ISO14888-1].
Signature generation process: A process that uses the message, the
signature key, and the digital signature scheme parameters as initial
data and that generates the digital signature as the result
[ISO14888-1].
Witness: Element of data (resulting from the verification process)
that states to the verifier whether the digital signature is valid or
invalid [ISO148881-1]).
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 [ISO2382-2].
Message: String of bits of a limited length [ISO9796-1991].
Hash code: String of bits that is a result of the hash function
[ISO148881-1].
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;
3) it is difficult to find a pair of different input data,
producing the same hash function value.
Note: 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 (falsified) 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, which both produce the same
signature.
(Electronic) Digital signature: String of bits obtained as a result
of the 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
digital 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 a 256-bit length
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 non-negative 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
^ - 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 Section 6
of [ISO/IEC14888-1]) 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 the realization of two
main processes (see Section 7):
- signature generation (see Section 6.1) and
- signature verification (see Section 6.2).
The digital signature is meant for the authentication of the
signatory of the electronic message. Besides, 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
Figure 1.
appendix
|
+-------------------------------+
| |
+-----------+ +------------------------+- - - +
| message M |---| digital signature zeta | text |
+-----------+ +------------------------+- - - +
Figure 1: Signed message scheme
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 the 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 the complexity of solving the problem of the calculation of the
discrete logarithm in the 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 Section 5.2.
GOST R 34.10-2001 does not determine the process of generating
parameters needed for the digital signature scheme. Possible sets of
these parameters are defined, for example, in [RFC4357].
The digital signature represented as a binary vector of a 512-bit
length must be calculated using a definite set of rules, as stated in
Section 6.1.
The digital signature of the received message is accepted or denied
in accordance with the set of rules, as stated in Section 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
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 congruencies:
| 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 the 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)
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 the
introduced addition operation. For m, the following inequalities
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 the specific realization of the 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 a 256-bit length. 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.
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 a 256-bit length.
Consider the following binary vector of a 256-bit length 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. If:
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 512-bit length, 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 512-bit length into two binary vectors of 256-bit
length, 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.
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 if:
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 a
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 signatures for the received message M belonging to
V_all, it is necessary to perform the following actions (steps)
according to Algorithm II:
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. Otherwise, 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 if:
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 if:
R = x_C (mod q), (23)
where x_C is x-coordinate of the point.
Step 7 - if the equality R = r holds, then the signature is accepted.
Otherwise, 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.
7. Test Examples (Appendix to GOST R 34.10-2001)
This section is included in GOST R 34.10-2001 as a reference appendix
but is not officially mentioned as 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 the 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,
respectively.
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
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
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 steps 1-3, according to Algorithm I (Section 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:
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 steps 1-3, according to Algorithm II (Section
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
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.
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. References
9.1. 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 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 Russia for Standards, 1994.
(In Russian)
[RFC4357] Popov, V., Kurepkin, I., and 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", RFC 4357, January
2006.
9.2. Informative References
[ISO2382-2] ISO 2382-2 (1976), "Data processing - Vocabulary -
Part 2: Arithmetic and logic operations".
[ISO9796-1991] ISO/IEC 9796:1991, "Information technology --
Security techniques -- Digital signature schemes
giving message recovery."
[ISO14888-1] ISO/IEC 14888-1 (1998), "Information technology -
Security techniques - Digital signatures with
appendix - Part 1: General".
[ISO14888-2] ISO/IEC 14888-2 (1999), "Information technology -
Security techniques - Digital signatures with
appendix - Part 2: Identity-based mechanisms".
[ISO14888-3] ISO/IEC 14888-3 (1998), "Information technology -
Security techniques - Digital signatures with
appendix - Part 3: Certificate-based mechanisms".
[ISO10118-1] ISO/IEC 10118-1 (2000), "Information technology -
Security techniques - Hash-functions - Part 1:
General".
[ISO10118-2] ISO/IEC 10118-2 (2000), "Information technology -
Security techniques - Hash-functions - Part 2: Hash-
functions using an n-bit block cipher algorithm".
[ISO10118-3] ISO/IEC 10118-3 (2004), "Information technology -
Security techniques - Hash-functions - Part 3:
Dedicated hash-functions".
[ISO10118-4] ISO/IEC 10118-4 (1998), "Information technology -
Security techniques - Hash-functions - Part 4: Hash-
functions using modular arithmetic".
Appendix A. Extra Terms in the Digital Signature Area
The appendix gives extra international terms applied in the
considered and allied areas.
1. Padding: Extending a data string with extra bits [ISO10118-1].
2. Identification data: A list of data elements, including specific
object identifier, that belongs to the object and is used for its
denotation [ISO14888-1].
3. Signature equation: An equation, defined by the digital signature
function [ISO14888-1].
4. Verification function: A verification process function, defined by
the verification key, which outputs a witness of the signature
authenticity [ISO14888-1].
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 pseudo-random number sequence generator
(randomizer), and outputs the second part of the digital
signature.
Appendix B. Contributors
Dmitry Kabelev
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: kdb@cryptocom.ru
Igor Ustinov
Cryptocom, Ltd.
14 Kedrova St., Bldg. 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
Author's Address
Vasily Dolmatov, Ed.
Cryptocom, Ltd.
14 Kedrova St., Bldg. 2
Moscow, 117218
Russian Federation
EMail: dol@cryptocom.ru