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This is a modified version of the Internet RFC suitable for machine-translating. Original version is available here: RFC8133




Independent Submission                                S. Smyshlyaev, Ed.
Request for Comments: 8133                                   E. Alekseev
Category: Informational                                        I. Oshkin
ISSN: 2070-1721                                                 V. Popov
                                                              CRYPTO-PRO
                                                              March 2017


The Security Evaluated Standardized Password-Authenticated Key Exchange
(SESPAKE) Protocol

Abstract

This document describes the Security Evaluated Standardized Password- Authenticated Key Exchange (SESPAKE) protocol. The SESPAKE protocol provides password-authenticated key exchange for usage in systems for protection of sensitive information. The security proofs of the protocol were made for situations involving an active adversary in the channel, including man-in-the-middle (MitM) attacks and attacks based on the impersonation of one of the subjects.

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 7841.

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/rfc8133.

Copyright Notice

Copyright (c) 2017 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.

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RFC 8133                         SESPAKE                      March 2017
Table of Contents

1. Introduction ....................................................2
2. Conventions Used in This Document ...............................2
3. Notations .......................................................3
4. Protocol Description ............................................4
4.1. Protocol Parameters ........................................5
4.2. Initial Values of the Protocol Counters ....................7
4.3. Protocol Steps .............................................7
5. Construction of Points {Q_1,...,Q_N} ...........................11
6. Security Considerations ........................................13
7. IANA Considerations ............................................13
8. References .....................................................14
8.1. Normative References ......................................14
8.2. Informative References ....................................15
Appendix A. Test Examples for GOST-Based Protocol Implementation ..16
A.1. Examples of Points .........................................16
A.2. Test Examples of SESPAKE ...................................17
Appendix B. Point Verification Script .............................33
Acknowledgments ...................................................51 Authors' Addresses ................................................51


1. Introduction

This document describes the Security Evaluated Standardized Password- Authenticated Key Exchange (SESPAKE) protocol. The SESPAKE protocol provides password-authenticated key exchange for usage in systems for protection of sensitive information. The protocol is intended to be used to establish keys that are then used to organize a secure channel for protection of sensitive information. The security proofs of the protocol were made for situations involving an active adversary in the channel, including man-in-the-middle (MitM) attacks and attacks based on the impersonation of one of the subjects.


2. Conventions Used in This Document

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].

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RFC 8133                         SESPAKE                      March 2017

3. Notations

This document uses the following parameters of elliptic curves in accordance with [RFC6090]:

E an elliptic curve defined over a finite prime field GF(p),
where p > 3;

p the characteristic of the underlying prime field;

a, b the coefficients of the equation of the elliptic curve in the
canonical form;

m the elliptic curve group order;

q the elliptic curve subgroup order;

P a generator of the subgroup of order q;

X, Y the coordinates of the elliptic curve point in the canonical
form;

O zero point (point at infinity) of the elliptic curve.

This memo uses the following functions:

HASH the underlying hash function;

HMAC the function for calculating a message authentication code
(MAC), based on a HASH function in accordance with [RFC2104];

F(PW, salt, n)
the value of the function PBKDF2(PW, salt, n, len), where PBKDF2(PW, salt, n, len) is calculated according to [RFC8018]. The parameter len is considered equal to the minimum integer that is a multiple of 8 and satisfies the following condition:

len >= floor(log_2(q)). This document uses the following terms and definitions for the sets and operations on the elements of these sets:

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B_n the set of byte strings of size n, n >= 0; for n = 0, the B_n set consists of a single empty string of size 0; if b is an element of B_n, then b = (b_1,...,b_n), where b_1,...,b_n are elements of {0,...,255};
   ||      concatenation of byte strings A and C, i.e., if A in B_n1,
           C in B_n2, A = (a_1,a_2,...,a_n1) and C = (c_1,c_2,...,c_n2),
           then A || C = (a_1,a_2,...,a_n1,c_1,c_2,...,c_n2) is an
           element of B_(n1 + n2);

int(A) for the byte string A = (a_1,...,a_n) in B_n, an integer
int(A) = 256^(n - 1)a_n +...+ 256^(0)a_1;

bytes_n(X)
the byte string A in B_n, such that int(A) = X, where X is an integer and 0 <= X < 256^n;
   BYTES(Q)
           for Q in E, the byte string bytes_n(X) || bytes_n(Y), where
           X, Y are standard Weierstrass coordinates of point Q and
n = ceil(log_{256}(p)).

4. Protocol Description

The main point of the SESPAKE protocol is that parties sharing a weak key (a password) generate a strong common key. An active adversary who has access to a channel is not able to obtain any information that can be used to find a key in offline mode, i.e., without interaction with legitimate participants.

The protocol is used by subjects A (client) and B (server) that share some secret parameter that was established in an out-of-band mechanism: a client is a participant who stores a password as a secret parameter, and a server is a participant who stores a password-based computed point of the elliptic curve.

The SESPAKE protocol consists of two steps: the key-agreement step and the key-confirmation step. During the first step (the key-agreement step), the parties exchange keys using Diffie-Hellman with public components masked by an element that depends on the password -- one of the predefined elliptic curve points multiplied by the password-based coefficient. This approach provides an implicit key authentication, which means that after this step, one party is assured that no other party, aside from a specifically identified second party, may gain access to the generated secret key. During the second step (the key-confirmation step), the parties exchange strings that strongly depend on the generated key. After this step, the parties are assured that a legitimate party, and no one else, actually has possession of the secret key.

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To protect against online guessing attacks, counters that indicate the number of failed connections were introduced in the SESPAKE protocol. There is also a special technique for small-order point processing and a mechanism that provides protection against reflection attacks by using different operations for different sides.
4.1.  Protocol Parameters

Various elliptic curves can be used in the protocol. For each elliptic curve supported by clients, the following values MUST be defined:

o the protocol parameters identifier, ID_ALG (which can also define a HASH function, a pseudorandom function (PRF) used in the PBKDF2 function, etc.), which is a byte string of an arbitrary length;

o the point P, which is a generator point of the subgroup of order q of the curve;

o the set of distinct curve points {Q_1,Q_2,...,Q_N} of order q, where the total number of points, N, is defined for the protocol instance.

The method of generation of the points {Q_1,Q_2,...,Q_N} is described in Section 5.

The following protocol parameters are used by subject A:

1. The secret password value PW, which is a byte string that is uniformly randomly chosen from a subset of cardinality 10^10 or greater of the set B_k, where k >= 6 is the password length.

2. The list of curve identifiers supported by A.

3. Sets of points {Q_1,Q_2,...,Q_N}, corresponding to curves supported by A.

4. The C_1^A counter, which tracks the total number of unsuccessful authentication trials in a row, and a value of CLim_1 that stores the maximum possible number of such events.

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RFC 8133                         SESPAKE                      March 2017
5. The C_2^A counter, which tracks the total number of unsuccessful authentication events during the period of usage of the specific PW, and a value of CLim_2 that stores the maximum possible number of such events.

6. The C_3^A counter, which tracks the total number of authentication events (successful and unsuccessful) during the period of usage of the specific PW, and a value of CLim_3 that stores the maximum possible number of such events.

7. The unique identifier, ID_A, of subject A (OPTIONAL), which is a byte string of an arbitrary length.

The following protocol parameters are used by subject B:

1. The values ind and salt, where ind is in {1,...,N} and salt is in {1,...,2^128-1}.

2. The point Q_PW, satisfying the following equation:

Q_PW = int(F(PW, salt, 2000))*Q_ind.

It is possible that the point Q_PW is not stored and is calculated using PW in the beginning of the protocol. In that case, B has to store PW and points {Q_1,Q_2,...,Q_N}.

3. The ID_ALG identifier.

4. The C_1^B counter, which tracks the total number of unsuccessful authentication trials in a row, and a value of CLim_1 that stores the maximum possible number of such events.

5. The C_2^B counter, which tracks the total number of unsuccessful authentication events during the period of usage of the specific PW, and a value of CLim_2 that stores the maximum possible number of such events.

6. The C_3^B counter, which tracks the total number of authentication events (successful and unsuccessful) during the period of usage of the specific PW, and a value of CLim_3 that stores the maximum possible number of such events.

7. The unique identifier, ID_B, of subject B (OPTIONAL), which is a byte string of an arbitrary length.

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RFC 8133                         SESPAKE                      March 2017
4.2.  Initial Values of the Protocol Counters

After the setup of a new password value PW, the values of the counters MUST be assigned as follows:

o C_1^A = C_1^B = CLim_1, where CLim_1 is in {3,...,5};

o C_2^A = C_2^B = CLim_2, where CLim_2 is in {7,...,20};

o C_3^A = C_3^B = CLim_3, where CLim_3 is in {10^3,10^3+1,...,10^5}.
4.3.  Protocol Steps

The basic SESPAKE steps are shown in the scheme below:
   +--------------------------+------------+---------------------------+
   |       A [A_ID, PW]       |            | B [B_ID, Q_PW, ind, salt] |
   +--------------------------+------------+---------------------------+
   |   if C_1^A or C_2^A or   |            |                           |
   |    C_3^A = 0 ==> quit    |            |                           |
   | decrement C_1^A, C_2^A,  | A_ID --->  |    if C_1^B or C_2^B or   |
   |        C_3^A by 1        |            |     C_3^B = 0 ==> quit    |
   |         z_A = 0          |    <---    |  decrement C_1^B, C_2^B,  |
   |                          |  ID_ALG,   |         C_3^B by 1        |
   |                          | B_ID       |                           |
   |                          | (OPTIONAL),|                           |
   |                          | ind, salt  |                           |
   | Q_PW^A = int(F(PW, salt, |            |                           |
   |       2000))*Q_ind       |            |                           |
   |  choose alpha randomly   |            |                           |
   |     from {1,...,q-1}     |            |                           |
   |  u_1 = alpha*P - Q_PW^A  |  u_1 --->  |  if u_1 not in E ==> quit |
   |                          |            |          z_B = 0          |
   |                          |            |      Q_B = u_1 + Q_PW     |
   |                          |            | choose beta randomly from |
   |                          |            |        {1,...,q-1}        |
   |                          |            |  if m/q*Q_B = O ==> Q_B = |
   |                          |            |      beta*P, z_B = 1      |
   |                          |            |           K_B =           |
   |                          |            | HASH(BYTES(( m/q*beta*    |
   |                          |            |       (mod q))*Q_B ))     |
   | if u_2 not in E ==> quit |  <--- u_2  |    u_2 = beta*P + Q_PW    |
   |    Q_A = u_2 - Q_PW^A    |            |                           |
   | if m/q*Q_A = O ==> Q_A = |            |                           |
   |     alpha*P, z_A = 1     |            |                           |
   | K_A = HASH(BYTES(( m/q*  |            |                           |
   |   alpha(mod q))*Q_A ))   |            |                           |
   |                          |            |                           |



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RFC 8133                         SESPAKE                      March 2017


   | U_1 = BYTES(u_1), U_2 =  |            |                           |
   |        BYTES(u_2)        |            |                           |
   |  MAC_A = HMAC(K_A, 0x01  |  DATA_A,   |  U_1 = BYTES(u_1), U_2 =  |
   |  || ID_A || ind || salt  | MAC_A ---> |         BYTES(u_2)        |
   | || U_1 || U_2 || ID_ALG  |            |                           |
   |  (OPTIONAL) || DATA_A)   |            |                           |
   |                          |            |   if MAC_A != HMAC(K_B,   |
   |                          |            |   0x01 || ID_A || ind ||  |
   |                          |            |   salt || U_1 || U_2 ||   |
   |                          |            |    ID_ALG (OPTIONAL) ||   |
   |                          |            |      DATA_A) ==> quit     |
   |                          |            |    if z_B = 1 ==> quit    |
   |                          |            | C_1^B = CLim_1, increment |
   |                          |            |         C_2^B by 1        |
   |  if MAC_B != HMAC(K_A,   |    <---    | MAC_B = HMAC(K_B, 0x02 || |
   |  0x02 || ID_B || ind ||  |  DATA_B,   |   ID_B || ind || salt ||  |
   |  salt || U_1 || U_2 ||   |   MAC_B    |    U_1 || U_2 || ID_ALG   |
   |   ID_ALG (OPTIONAL) ||   |            |  (OPTIONAL) || DATA_A ||  |
   |  DATA_A || DATA_B) ==>   |            |          DATA_B)          |
   |           quit           |            |                           |
   |   if z_A = 1 ==> quit    |            |                           |
   |     C_1^A = CLim_1,      |            |                           |
   |   increment C_2^A by 1   |            |                           |
   +--------------------------+------------+---------------------------+

Table 1: SESPAKE Protocol Steps

The full description of the protocol consists of the following steps:

1. If any of the counters C_1^A, C_2^A, or C_3^A is equal to 0, A finishes the protocol with an informational error regarding exceeding the number of trials that is controlled by the corresponding counter.

2. A decrements each of the counters C_1^A, C_2^A, and C_3^A by 1, requests open authentication information from B, and sends the ID_A identifier.

3. If any of the counters C_1^B, C_2^B, or C_3^B is equal to 0, B finishes the protocol with an informational error regarding exceeding the number of trials that is controlled by the corresponding counter.

4. B decrements each of the counters C_1^B, C_2^B, and C_3^B by 1.

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RFC 8133                         SESPAKE                      March 2017
5. B sends the values of ind, salt, and the ID_ALG identifier to A. B also can OPTIONALLY send the ID_B identifier to A. All subsequent calculations are done by B in the elliptic curve group defined by the ID_ALG identifier.

6. A sets the curve defined by the received ID_ALG identifier as the used elliptic curve. All subsequent calculations are done by A in this elliptic curve group.

7. A calculates the point Q_PW^A = int(F(PW, salt, 2000))*Q_ind.

8. A chooses randomly (according to the uniform distribution) the value alpha; alpha is in {1,...,q-1}; then A assigns z_A = 0.

9. A sends the value u_1 = alpha*P - Q_PW^A to B.

10. After receiving u_1, B checks to see if u_1 is in E. If it is not, B finishes with an error and considers the authentication process unsuccessful.

11. B calculates Q_B = u_1 + Q_PW, assigns z_B = 0, and chooses randomly (according to the uniform distribution) the value beta; beta is in {1,...,q-1}.

12. If m/q*Q_B = O, B assigns Q_B = beta*P and z_B = 1.

13. B calculates K_B = HASH(BYTES(( m/q*beta*(mod q))*Q_B )).

14. B sends the value u_2 = beta*P + Q_PW to A.

15. After receiving u_2, A checks to see if u_2 is in E. If it is not, A finishes with an error and considers the authentication process unsuccessful.

16. A calculates Q_A = u_2 - Q_PW^A.

17. If m/q*Q_A = O, then A assigns Q_A = alpha*P and z_A = 1.

18. A calculates K_A = HASH(BYTES(( m/q*alpha(mod q))*Q_A )).

19. A calculates U_1 = BYTES(u_1), U_2 = BYTES(u_2).
   20.  A calculates MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt ||
        U_1 || U_2 || ID_ALG (OPTIONAL) || DATA_A), where DATA_A is an
        OPTIONAL string that is authenticated with MAC_A (if it is not
used, then DATA_A is considered to be of zero length).

21. A sends DATA_A, MAC_A to B. 22. B calculates U_1 = BYTES(u_1), U_2 = BYTES(u_2).


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RFC 8133                         SESPAKE                      March 2017



   23.  B checks to see if the values MAC_A and HMAC(K_B, 0x01 || ID_A
        || ind || salt || U_1 || U_2 || ID_ALG (OPTIONAL) || DATA_A) are
        equal.  If they are not, it finishes with an error and considers
the authentication process unsuccessful.

24. If z_B = 1, B finishes with an error and considers the authentication process unsuccessful.

25. B sets the value of C_1^B to CLim_1 and increments C_2^B by 1.
   26.  B calculates MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt ||
        U_1 || U_2 || ID_ALG (OPTIONAL) || DATA_A || DATA_B), where
        DATA_B is an OPTIONAL string that is authenticated with MAC_B
(if it is not used, then DATA_B is considered to be of zero length).

27. B sends DATA_B, MAC_B to A.
   28.  A checks to see if the values MAC_B and HMAC(K_A, 0x02 || ID_B
        || ind || salt || U_1 || U_2 || ID_ALG (OPTIONAL) || DATA_A ||
        DATA_B) are equal.  If they are not, it finishes with an error
and considers the authentication process unsuccessful.

29. If z_A = 1, A finishes with an error and considers the authentication process unsuccessful.

30. A sets the value of C_1^A to CLim_1 and increments C_2^A by 1.

After the procedure finishes successfully, subjects A and B are mutually authenticated, and each subject has an explicitly authenticated value of K = K_A = K_B.

Notes:

1. In cases where the interaction process can be initiated by any subject (client or server), the ID_A and ID_B options MUST be used, and the receiver MUST check to see if the identifier he had received is not equal to his own; otherwise, it finishes the protocol. If an OPTIONAL parameter ID_A (or ID_B) is not used in the protocol, it SHOULD be considered equal to a fixed byte string (a zero-length string is allowed) defined by a specific implementation.

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RFC 8133                         SESPAKE                      March 2017
2. The ind, ID_A, ID_B, and salt parameters can be agreed upon in advance. If some parameter is agreed upon in advance, it is possible not to send it during a corresponding step. Nevertheless, all parameters MUST be used as corresponding inputs to the HMAC function during Steps 20, 23, 26, and 28.

3. The ID_ALG parameter can be fixed or agreed upon in advance.

4. It is RECOMMENDED that the ID_ALG parameter be used in HMAC during Steps 20, 23, 26, and 28.

5. Continuation of protocol interaction in a case where any of the counters C_1^A or C_1^B is equal to zero MAY be done without changing the password. In this case, these counters can be used for protection against denial-of-service attacks. For example, continuation of interaction can be allowed after a certain delay period.

6. Continuation of protocol interaction in a case where any of the counters C_2^A, C_3^A, C_2^B, or C_3^B is equal to zero MUST be done only after changing the password.

7. It is RECOMMENDED that during Steps 9 and 14 the points u_1 and u_2 be sent in a non-compressed format (BYTES(u_1) and BYTES(u_2)). However, point compression MAY be used.

8. The use of several Q points can reinforce the independence of the data streams when working with several applications -- for example, when two high-level protocols can use two different points. However, the use of more than one point is OPTIONAL.


5. Construction of Points {Q_1,...,Q_N}

This section provides an example of a possible algorithm for the generation of each point Q_i in the set {Q_1,...,Q_N} that corresponds to the given elliptic curve E.

The algorithm is based on choosing points with coordinates with known preimages of a cryptographic hash function H, which is the GOST R 34.11-2012 hash function (see [RFC6986]) with 256-bit output if 2^254 < q < 2^256, and the GOST R 34.11-2012 hash function (see [RFC6986]) with 512-bit output if 2^508 < q < 2^512. The algorithm consists of the following steps:

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RFC 8133                         SESPAKE                      March 2017

1. Set i = 1, SEED = 0, s = 4.
   2.  Calculate X = int(HASH(BYTES(P) || bytes_s(SEED))) mod p.

3. Check to see if the value of X^3 + aX + b is a quadratic residue in the field F_p. If it is not, set SEED = SEED + 1 and return to Step 2.

4. Choose the value of Y = min{r1, r2}, where r1, r2 from {0,1,...,p-1} are such that r1 != r2 and r1^2 = r2^2 = R mod p for R = X^3 + aX + b.

5. Check to see if the following relations hold for the point Q = (X, Y): Q != O and q*Q = O. If they do, go to Step 6; if not, set SEED = SEED + 1 and return to Step 2.

6. Set Q_i = Q. If i < N, then set i = i + 1 and go to Step 2; otherwise, finish.

With the defined algorithm for any elliptic curve E, point sets {Q_1,...,Q_N} are constructed. Constructed points in one set MUST have distinct X-coordinates.

Note: The knowledge of a hash function preimage prevents knowledge of the multiplicity of any point related to generator point P. It is of primary importance, because such knowledge could be used to implement an attack against the protocol with an exhaustive search for the password.

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6. Security Considerations

Any cryptographic algorithms -- particularly HASH functions and HMAC functions -- that are used in the SESPAKE protocol MUST be carefully designed and MUST be able to withstand all known types of cryptanalytic attacks.

It is RECOMMENDED that the HASH function satisfy the following condition:

o hashlen <= log_2(q) + 4, where hashlen is the length of the HASH function output.

It is RECOMMENDED that the output length of hash functions used in the SESPAKE protocol be greater than or equal to 256 bits.

The points {Q_1,Q_2,...,Q_N} and P MUST be chosen in such a way that they are provably pseudorandom. As a practical matter, this means that the algorithm for generation of each point Q_i in the set {Q_1,...,Q_N} (see Section 5) ensures that the multiplicity of any point under any other point is unknown.

Using N = 1 is RECOMMENDED.

Note: The specific adversary models for the protocol discussed in this document can be found in [SESPAKE-SECURITY], which contains the security proofs.


7. IANA Considerations

This document does not require any IANA actions.

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8. References
8.1.  Normative References

[GOST3410-2012]
"Information technology. Cryptographic data security. Signature and verification processes of [electronic] digital signature", GOST R 34.10-2012, Federal Agency on Technical Regulating and Metrology (in Russian), 2012.

[GOST3411-2012]
"Information technology. Cryptographic Data Security. Hashing function", GOST R 34.11-2012, Federal Agency on Technical Regulating and Metrology (in Russian), 2012.

[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC:
Keyed-Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997, <http://www.rfc-editor.org/info/rfc2104>.

[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <http://www.rfc-editor.org/info/rfc2119>.

[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
Curve Cryptography Algorithms", RFC 6090, DOI 10.17487/RFC6090, February 2011, <http://www.rfc-editor.org/info/rfc6090>.

[RFC6986] Dolmatov, V., Ed., and A. Degtyarev, "GOST R 34.11-2012:
Hash Function", RFC 6986, DOI 10.17487/RFC6986, August 2013, <http://www.rfc-editor.org/info/rfc6986>.

[RFC7091] Dolmatov, V., Ed., and A. Degtyarev, "GOST R 34.10-2012:
Digital Signature Algorithm", RFC 7091, DOI 10.17487/RFC7091, December 2013, <http://www.rfc-editor.org/info/rfc7091>.

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RFC 8133                         SESPAKE                      March 2017
[RFC7836] Smyshlyaev, S., Ed., Alekseev, E., Oshkin, I., Popov, V.,
Leontiev, S., Podobaev, V., and D. Belyavsky, "Guidelines on the Cryptographic Algorithms to Accompany the Usage of Standards GOST R 34.10-2012 and GOST R 34.11-2012", RFC 7836, DOI 10.17487/RFC7836, March 2016, <http://www.rfc-editor.org/info/rfc7836>.

[RFC8018] Moriarty, K., Ed., Kaliski, B., and A. Rusch, "PKCS #5:
Password-Based Cryptography Specification Version 2.1", RFC 8018, DOI 10.17487/RFC8018, January 2017, <http://www.rfc-editor.org/info/rfc8018>.
8.2.  Informative References

[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, DOI 10.17487/RFC4357, January 2006, <http://www.rfc-editor.org/info/rfc4357>.

[SESPAKE-SECURITY]
Smyshlyaev, S., Oshkin, I., Alekseev, E., and L. Ahmetzyanova, "On the Security of One Password Authenticated Key Exchange Protocol", 2015, <http://eprint.iacr.org/2015/1237.pdf>.

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RFC 8133                         SESPAKE                      March 2017
Appendix A.  Test Examples for GOST-Based Protocol Implementation

The following test examples are made for the protocol implementation that is based on the Russian national standards GOST R 34.10-2012 [GOST3410-2012] and GOST R 34.11-2012 [GOST3411-2012]. The English versions of these standards can be found in [RFC7091] and [RFC6986].
A.1.  Examples of Points

There is one point Q_1 for each of the elliptic curves below. These points were constructed using the method described in Section 5 for N = 1 and the GOST R 34.11-2012 hash function (see [RFC6986]). If 2^254 < q < 2^256, the GOST R 34.11-2012 hash function with 256-bit output is used, and if 2^508 < q < 2^512, the GOST R 34.11-2012 hash function with 512-bit output is used.

Each of the points complies with the GOST R 34.10-2012 [GOST3410-2012] standard and is represented by a pair of (X, Y) coordinates in the canonical form and also by a pair of (U, V) coordinates in the twisted Edwards form in accordance with [RFC7836] for the curves that have equivalent representations in this form. There is a SEED value for each point, by which it was generated.

id-GostR3410-2001-CryptoPro-A-ParamSet, id-GostR3410-2001-CryptoPro-B-ParamSet, etc. are defined in [RFC4357]. id-tc26-gost-3410-2012-512-paramSetA, id-tc26-gost-3410-2012-512-paramSetB, etc. are defined in [RFC7836].
A.1.1.  Curve id-GostR3410-2001-CryptoPro-A-ParamSet

Point Q_1 X = 0xa69d51caf1a309fa9e9b66187759b0174c274e080356f23cfcbfe84d396ad7bb Y = 0x5d26f29ecc2e9ac0404dcf7986fa55fe94986362170f54b9616426a659786dac SEED = 0x0001
A.1.2.  Curve id-GostR3410-2001-CryptoPro-B-ParamSet

Point Q_1 X = 0x3d715a874a4b17cb3b517893a9794a2b36c89d2ffc693f01ee4cc27e7f49e399 Y = 0x1c5a641fcf7ce7e87cdf8cea38f3db3096eace2fad158384b53953365f4fe7fe SEED = 0x0000
A.1.3.  Curve id-GostR3410-2001-CryptoPro-C-ParamSet

Point Q_1 X = 0x1e36383e43bb6cfa2917167d71b7b5dd3d6d462b43d7c64282ae67dfbec2559d Y = 0x137478a9f721c73932ea06b45cf72e37eb78a63f29a542e563c614650c8b6399 SEED = 0x0006


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RFC 8133                         SESPAKE                      March 2017


A.1.4.  Curve id-tc26-gost-3410-2012-512-paramSetA

Point Q_1 X = 0x2a17f8833a32795327478871b5c5e88aefb91126c64b4b8327289bea62559425
d18198f133f400874328b220c74497cd240586cb249e158532cb8090776cd61c
Y = 0x728f0c4a73b48da41ce928358fad26b47a6e094e9362bae82559f83cddc4ec3a
4676bd3707edeaf4cd85e99695c64c241edc622be87dc0cf87f51f4367f723c5
SEED = 0x0001
A.1.5.  Curve id-tc26-gost-3410-2012-512-paramSetB

Point Q_1 X = 0x7e1fae8285e035bec244bef2d0e5ebf436633cf50e55231dea9c9cf21d4c8c33
df85d4305de92971f0a4b4c07e00d87bdbc720eb66e49079285aaf12e0171149
Y = 0x2cc89998b875d4463805ba0d858a196592db20ab161558ff2f4ef7a85725d209
53967ae621afdeae89bb77c83a2528ef6fce02f68bda4679d7f2704947dbc408
SEED = 0x0000
A.1.6.  Curve id-tc26-gost-3410-2012-256-paramSetA

Point Q_1 X = 0xb51adf93a40ab15792164fad3352f95b66369eb2a4ef5efae32829320363350e Y = 0x74a358cc08593612f5955d249c96afb7e8b0bb6d8bd2bbe491046650d822be18 U = 0xebe97afffe0d0f88b8b0114b8de430ac2b34564e4420af24728e7305bc48aeaa V = 0x828f2dcf8f06612b4fea4da72ca509c0f76dd37df424ea22bfa6f4f65748c1e4 SEED = 0x0001
A.1.7.  Curve id-tc26-gost-3410-2012-512-paramSetC

Point Q_1 X = 0x489c91784e02e98f19a803abca319917f37689e5a18965251ce2ff4e8d8b298f
5ba7470f9e0e713487f96f4a8397b3d09a270c9d367eb5e0e6561adeeb51581d
Y = 0x684ea885aca64eaf1b3fee36c0852a3be3bd8011b0ef18e203ff87028d6eb5db
2c144a0dcc71276542bfd72ca2a43fa4f4939da66d9a60793c704a8c94e16f18
U = 0x3a3496f97e96b3849a4fa7db60fd93858bde89958e4beebd05a6b3214216b37c
9d9a560076e7ea59714828b18fbfef996ffc98bf3dc9f2d3cb0ed36a0d6ace88
V = 0x52d884c8bf0ad6c5f7b3973e32a668daa1f1ed092eff138dae6203b2ccdec561
47464d35fec4b727b2480eb143074712c76550c7a54ff3ea26f70059480dcb50
SEED = 0x0013
A.2.  Test Examples of SESPAKE

This protocol implementation uses the GOST R 34.11-2012 hash function (see [RFC6986]) with 256-bit output as the H function and the HMAC_GOSTR3411_2012_512 function defined in [RFC7836] as a PRF for the F function. The parameter len is considered equal to 256 if 2^254 < q < 2^256, and equal to 512 if 2^508 < q < 2^512. The test examples for the point of each curve in Appendix A.1 are given below.



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RFC 8133                         SESPAKE                      March 2017



A.2.1.  Curve id-GostR3410-2001-CryptoPro-A-ParamSet

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0xA69D51CAF1A309FA9E9B66187759B0174C274E080356F23CFCBFE84D396AD7BB Y = 0x5D26F29ECC2E9AC0404DCF7986FA55FE94986362170F54B9616426A659786DAC

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67

The coordinates of the point Q_PW are:

X = 0x59495655D1E7C7424C622485F575CCF121F3122D274101E8AB734CC9C9A9B45E Y = 0x48D1C311D33C9B701F3B03618562A4A07A044E3AF31E3999E67B487778B53C62

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x1F2538097D5A031FA68BBB43C84D12B3DE47B7061C0D5E24993E0C873CDBA6B3 alpha*P: X = 0xBBC77CF42DC1E62D06227935379B4AA4D14FEA4F565DDF4CB4FA4D31579F9676 Y = 0x8E16604A4AFDF28246684D4996274781F6CB80ABBBA1414C1513EC988509DABF
u_1:
X = 0x204F564383B2A76081B907F3FCA8795E806BE2C2ED228730B5B9E37074229E8D Y = 0xE84F9E442C61DDE37B601A7F37E7CA11C56183FA071DFA9320EDE3E7521F9D41 When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

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RFC 8133                         SESPAKE                      March 2017

beta=0xDC497D9EF6324912FD367840EE509A2032AEDB1C0A890D133B45F596FCCBD45D src: 2E 01 A3 D8 4F DB 7E 94 7B B8 92 9B E9 36 3D F5 F7 25 D6 40 1A A5 59 D4 1A 67 24 F8 D5 F1 8E 2C A0 DB A9 31 05 CD DA F4 BF AE A3 90 6F DD 71 9D BE B2 97 B6 A1 7F 4F BD 96 DC C7 23 EA 34 72 A9
K_B:
1A 62 65 54 92 1D C2 E9 2B 4D D8 D6 7D BE 5A 56 62 E5 62 99 37 3F 06 79 95 35 AD 26 09 4E CA A3
beta*P:
X = 0x6097341C1BE388E83E7CA2DF47FAB86E2271FD942E5B7B2EB2409E49F742BC29 Y = 0xC81AA48BDB4CA6FA0EF18B9788AE25FE30857AA681B3942217F9FED151BAB7D0
u_2:
X = 0xDC137A2F1D4A35AEBC0ECBF6D3486DEF8480BFDC752A86DD4F207D7D1910E22D Y = 0x7532F0CE99DCC772A4D77861DAE57C138F07AE304A727907FB0AAFDB624ED572

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: 1A 62 65 54 92 1D C2 E9 2B 4D D8 D6 7D BE 5A 56 62 E5 62 99 37 3F 06 79 95 35 AD 26 09 4E CA A3

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: 23 7A 03 C3 5F 49 17 CE 86 B3 58 94 45 F1 1E 1A 6F 10 8B 2F DD 0A A9 E8 10 66 4B 25 59 60 B5 79

The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

MAC_B: 9E E0 E8 73 3B 06 98 50 80 4D 97 98 73 1D CD 1C FF E8 7A 3B 15 1F 0A E8 3E A9 6A FB 4F FC 31 E4

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RFC 8133                         SESPAKE                      March 2017
A.2.2.  Curve id-GostR3410-2001-CryptoPro-B-ParamSet

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0x3D715A874A4B17CB3B517893A9794A2B36C89D2FFC693F01EE4CC27E7F49E399 Y = 0x1C5A641FCF7CE7E87CDF8CEA38F3DB3096EACE2FAD158384B53953365F4FE7FE

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67

The coordinates of the point Q_PW are:

X = 0x6DC2AE26BC691FCA5A73D9C452790D15E34BA5404D92955B914C8D2662ABB985 Y = 0x3B02AAA9DD65AE30C335CED12F3154BBAC059F66B088306747453EDF6E5DB077

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x499D72B90299CAB0DA1F8BE19D9122F622A13B32B730C46BD0664044F2144FAD alpha*P: X = 0x61D6F916DB717222D74877F179F7EBEF7CD4D24D8C1F523C048E34A1DF30F8DD Y = 0x3EC48863049CFCFE662904082E78503F4973A4E105E2F1B18C69A5E7FB209000
u_1:
X = 0x21F5437AF33D2A1171A070226B4AE82D3765CD0EEBFF1ECEFE158EBC50C63AB1 Y = 0x5C9553B5D11AAAECE738AD9A9F8CB4C100AD4FA5E089D3CBCCEA8C0172EB7ECC When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

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RFC 8133                         SESPAKE                      March 2017

beta=0x0F69FF614957EF83668EDC2D7ED614BE76F7B253DB23C5CC9C52BF7DF8F4669D src: 50 14 0A 5D ED 33 43 EF C8 25 7B 79 E6 46 D9 F0 DF 43 82 8C 04 91 9B D4 60 C9 7A D1 4B A3 A8 6B 00 C4 06 B5 74 4D 8E B1 49 DC 8E 7F C8 40 64 D8 53 20 25 3E 57 A9 B6 B1 3D 0D 38 FE A8 EE 5E 0A
K_B:
A6 26 DE 01 B1 68 0F F7 51 30 09 12 2B CE E1 89 68 83 39 4F 96 03 01 72 45 5C 9A E0 60 CC E4 4A
beta*P:
X = 0x33BC6F7E9C0BA10CFB2B72546C327171295508EA97F8C8BA9F890F2478AB4D6C Y = 0x75D57B396C396F492F057E9222CCC686437A2AAD464E452EF426FC8EEED1A4A6
u_2:
X = 0x089DDEE718EE8A224A7F37E22CFFD731C25FCBF58860364EE322412CDCEF99AC Y = 0x0ECE03D4E395A6354C571871BEF425A532D5D463B0F8FD427F91A43E20CDA55C

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: A6 26 DE 01 B1 68 0F F7 51 30 09 12 2B CE E1 89 68 83 39 4F 96 03 01 72 45 5C 9A E0 60 CC E4 4A

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: B9 1F 43 90 2A FA 90 D3 E5 C6 91 CB DC 43 8A 1E BF 54 7F 4C 2C B4 14 43 CC 38 79 7B E2 47 A7 D0

The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

MAC_B: 79 D5 54 83 FD 99 B1 2B CC A5 ED C6 BB E1 D7 B9 15 CE 04 51 B0 89 1E 77 5D 4A 61 CB 16 E3 3F CC

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RFC 8133                         SESPAKE                      March 2017
A.2.3.  Curve id-GostR3410-2001-CryptoPro-C-ParamSet

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0x1E36383E43BB6CFA2917167D71B7B5DD3D6D462B43D7C64282AE67DFBEC2559D Y = 0x137478A9F721C73932EA06B45CF72E37EB78A63F29A542E563C614650C8B6399

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67

The coordinates of the point Q_PW are:

X = 0x945821DAF91E158B839939630655A3B21FF3E146D27041E86C05650EB3B46B59 Y = 0x3A0C2816AC97421FA0E879605F17F0C9C3EB734CFF196937F6284438D70BDC48

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x3A54AC3F19AD9D0B1EAC8ACDCEA70E581F1DAC33D13FEAFD81E762378639C1A8 alpha*P: X = 0x96B7F09C94D297C257A7DA48364C0076E59E48D221CBA604AE111CA3933B446A Y = 0x54E4953D86B77ECCEB578500931E822300F7E091F79592CA202A020D762C34A6
u_1:
X = 0x81BBD6FCA464D2E2404A66D786CE4A777E739A89AEB68C2DAC99D53273B75387 Y = 0x6B6DBD922EA7E060998F8B230AB6EF07AD2EC86B2BF66391D82A30612EADD411 When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

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RFC 8133                         SESPAKE                      March 2017

beta=0x448781782BF7C0E52A1DD9E6758FD3482D90D3CFCCF42232CF357E59A4D49FD4 src: 16 A1 2D 88 54 7E 1C 90 06 BA A0 08 E8 CB EC C9 D1 68 91 ED C8 36 CF B7 5F 8E B9 56 FA 76 11 94 D2 8E 25 DA D3 81 8D 16 3C 49 4B 05 9A 8C 70 A5 A1 B8 8A 7F 80 A2 EE 35 49 30 18 46 54 2C 47 0B
K_B:
BE 7E 7E 47 B4 11 16 F2 C7 7E 3B 8F CE 40 30 72 CA 82 45 0D 65 DE FC 71 A9 56 49 E4 DE EA EC EE
beta*P:
X = 0x4B9C0AB55A938121F282F48A2CC4396EB16E7E0068B495B0C1DD4667786A3EB7 Y = 0x223460AA8E09383E9DF9844C5A0F2766484738E5B30128A171B69A77D9509B96
u_2:
X = 0x2ED9B903254003A672E89EBEBC9E31503726AD124BB5FC0A726EE0E6FCCE323E Y = 0x4CF5E1042190120391EC8DB62FE25E9E26EC60FB0B78B242199839C295FCD022

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: BE 7E 7E 47 B4 11 16 F2 C7 7E 3B 8F CE 40 30 72 CA 82 45 0D 65 DE FC 71 A9 56 49 E4 DE EA EC EE

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: D3 B4 1A E2 C9 43 11 36 06 3E 6D 08 A6 1B E9 63 BD 5E D6 A1 FF F9 37 FA 8B 09 0A 98 E1 62 BF ED

The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

MAC_B: D6 B3 9A 44 99 BE D3 E0 4F AC F9 55 50 2D 16 B2 CB 67 4A 20 5F AC 3C D8 3D 54 EC 2F D5 FC E2 58

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RFC 8133                         SESPAKE                      March 2017
A.2.4.  Curve id-tc26-gost-3410-2012-512-paramSetA

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0x2A17F8833A32795327478871B5C5E88AEFB91126C64B4B8327289BEA62559425
D18198F133F400874328B220C74497CD240586CB249E158532CB8090776CD61C
Y = 0x728F0C4A73B48DA41CE928358FAD26B47A6E094E9362BAE82559F83CDDC4EC3A
4676BD3707EDEAF4CD85E99695C64C241EDC622BE87DC0CF87F51F4367F723C5

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67 1C 62 13 E3 93 0E FD DA 26 45 17 92 C6 20 81 22 EE 60 D2 00 52 0D 69 5D FD 9F 5F 0F D5 AB A7 02

The coordinates of the point Q_PW are:

X = 0x0C0AB53D0E0A9C607CAD758F558915A0A7DC5DC87B45E9A58FDDF30EC3385960
283E030CD322D9E46B070637785FD49D2CD711F46807A24C40AF9A42C8E2D740
Y = 0xDF93A8012B86D3A3D4F8A4D487DA15FC739EB31B20B3B0E8C8C032AAF8072C63
37CF7D5B404719E5B4407C41D9A3216A08CA69C271484E9ED72B8AAA52E28B8B

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RFC 8133                         SESPAKE                      March 2017
During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x3CE54325DB52FE798824AEAD11BB16FA766857D04A4AF7D468672F16D90E7396
046A46F815693E85B1CE5464DA9270181F82333B0715057BBE8D61D400505F0E
alpha*P:
X = 0xB93093EB0FCC463239B7DF276E09E592FCFC9B635504EA4531655D76A0A3078E
2B4E51CFE2FA400CC5DE9FBE369DB204B3E8ED7EDD85EE5CCA654C1AED70E396
Y = 0x809770B8D910EA30BD2FA89736E91DC31815D2D9B31128077EEDC371E9F69466
F497DC64DD5B1FADC587F860EE256109138C4A9CD96B628E65A8F590520FC882
u_1:
X = 0xE7510A9EDD37B869566C81052E2515E1563FDFE79F1D782D6200F33C3CC2764D
40D0070B73AD5A47BAE9A8F2289C1B07DAC26A1A2FF9D3ECB0A8A94A4F179F13
Y = 0xBA333B912570777B626A5337BC7F727952460EEBA2775707FE4537372E902DF5
636080B25399751BF48FB154F3C2319A91857C23F39F89EF54A8F043853F82DE

When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

beta=0xB5C286A79AA8E97EC0E19BC1959A1D15F12F8C97870BA9D68CC12811A56A3BB1
1440610825796A49D468CDC9C2D02D76598A27973D5960C5F50BCE28D8D345F4
src:
84 59 C2 0C B5 C5 32 41 6D B9 28 EB 50 C0 52 0F B2 1B 9C D3 9A 4E 76 06 B2 21 BE 15 CA 1D 02 DA 08 15 DE C4 49 79 C0 8C 7D 23 07 AF 24 7D DA 1F 89 EC 81 20 69 F5 D9 CD E3 06 AF F0 BC 3F D2 6E D2 01 B9 53 52 A2 56 06 B6 43 E8 88 30 2E FC 8D 3E 95 1E 3E B4 68 4A DB 5C 05 7B 8F 8C 89 B6 CC 0D EE D1 00 06 5B 51 8A 1C 71 7F 76 82 FF 61 2B BC 79 8E C7 B2 49 0F B7 00 3F 94 33 87 37 1C 1D
K_B:
53 24 DE F8 48 B6 63 CC 26 42 2F 5E 45 EE C3 4C 51 D2 43 61 B1 65 60 CA 58 A3 D3 28 45 86 CB 7A
beta*P:
X = 0x238B38644E440452A99FA6B93D9FD7DA0CB83C32D3C1E3CFE5DF5C3EB0F9DB91
E588DAEDC849EA2FB867AE855A21B4077353C0794716A6480995113D8C20C7AF
Y = 0xB2273D5734C1897F8D15A7008B862938C8C74CA7E877423D95243EB7EBD02FD2
C456CF9FC956F078A59AA86F19DD1075E5167E4ED35208718EA93161C530ED14
u_2:
X = 0xC33844126216E81B372001E77C1FE9C7547F9223CF7BB865C4472EC18BE0C79A
678CC5AE4028E3F3620CCE355514F1E589F8A0C433CEAFCBD2EE87884D953411
Y = 0x8B520D083AAF257E8A54EC90CBADBAF4FEED2C2D868C82FF04FCBB9EF6F38E56
F6BAF9472D477414DA7E36F538ED223D2E2EE02FAE1A20A98C5A9FCF03B6F30D

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RFC 8133                         SESPAKE                      March 2017
When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: 53 24 DE F8 48 B6 63 CC 26 42 2F 5E 45 EE C3 4C 51 D2 43 61 B1 65 60 CA 58 A3 D3 28 45 86 CB 7A

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: E8 EF 9E A8 F1 E6 B1 26 68 E5 8C D2 2D D8 EE C6 4A 16 71 00 39 FA A6 B6 03 99 22 20 FA FE 56 14

The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

MAC_B: 61 14 34 60 83 6B 23 5C EC D0 B4 9B 58 7E A4 5D 51 3C 3A 38 78 3F 1C 9D 3B 05 97 0A 95 6A 55 BA
A.2.5.  Curve id-tc26-gost-3410-2012-512-paramSetB

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0x7E1FAE8285E035BEC244BEF2D0E5EBF436633CF50E55231DEA9C9CF21D4C8C33
DF85D4305DE92971F0A4B4C07E00D87BDBC720EB66E49079285AAF12E0171149
Y = 0x2CC89998B875D4463805BA0D858A196592DB20AB161558FF2F4EF7A85725D209
53967AE621AFDEAE89BB77C83A2528EF6FCE02F68BDA4679D7F2704947DBC408

Smyshlyaev, et al. Informational [Page 26]


RFC 8133                         SESPAKE                      March 2017
The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67 1C 62 13 E3 93 0E FD DA 26 45 17 92 C6 20 81 22 EE 60 D2 00 52 0D 69 5D FD 9F 5F 0F D5 AB A7 02

The coordinates of the point Q_PW are:

X = 0x7D03E65B8050D1E12CBB601A17B9273B0E728F5021CD47C8A4DD822E4627BA5F
9C696286A2CDDA9A065509866B4DEDEDC4A118409604AD549F87A60AFA621161
Y = 0x16037DAD45421EC50B00D50BDC6AC3B85348BC1D3A2F85DB27C3373580FEF87C
2C743B7ED30F22BE22958044E716F93A61CA3213A361A2797A16A3AE62957377

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x715E893FA639BF341296E0623E6D29DADF26B163C278767A7982A989462A3863
FE12AEF8BD403D59C4DC4720570D4163DB0805C7C10C4E818F9CB785B04B9997
alpha*P:
X = 0x10C479EA1C04D3C2C02B0576A9C42D96226FF033C1191436777F66916030D87D
02FB93738ED7669D07619FFCE7C1F3C4DB5E5DF49E2186D6FA1E2EB5767602B9
Y = 0x039F6044191404E707F26D59D979136A831CCE43E1C5F0600D1DDF8F39D0CA3D
52FBD943BF04DDCED1AA2CE8F5EBD7487ACDEF239C07D015084D796784F35436
u_1:
X = 0x45C05CCE8290762F2470B719B4306D62B2911CEB144F7F72EF11D10498C7E921
FF163FE72044B4E7332AD8CBEC3C12117820F53A60762315BCEB5BC6DA5CF1E0
Y = 0x5BE483E382D0F5F0748C4F6A5045D99E62755B5ACC9554EC4A5B2093E121A2DD
5C6066BC9EDE39373BA19899208BB419E38B39BBDEDEB0B09A5CAAEAA984D02E

Smyshlyaev, et al. Informational [Page 27]


RFC 8133                         SESPAKE                      March 2017
When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

beta=0x30FA8C2B4146C2DBBE82BED04D7378877E8C06753BD0A0FF71EBF2BEFE8DA8F3
DC0836468E2CE7C5C961281B6505140F8407413F03C2CB1D201EA1286CE30E6D
src:
3F 04 02 E4 0A 9D 59 63 20 5B CD F4 FD 89 77 91 9B BA F4 80 F8 E4 FB D1 25 5A EC E6 ED 57 26 4B D0 A2 87 98 4F 59 D1 02 04 B5 F4 5E 4D 77 F3 CF 8A 63 B3 1B EB 2D F5 9F 8A F7 3C 20 9C CA 8B 50 B4 18 D8 01 E4 90 AE 13 3F 04 F4 F3 F4 D8 FE 8E 19 64 6A 1B AF 44 D2 36 FC C2 1B 7F 4D 8F C6 A1 E2 9D 6B 69 AC CE ED 4E 62 AB B2 0D AD 78 AC F4 FE B0 ED 83 8E D9 1E 92 12 AB A3 89 71 4E 56 0C
K_B:
D5 90 E0 5E F5 AE CE 8B 7C FB FC 71 BE 45 5F 29 A5 CC 66 6F 85 CD B1 7E 7C C7 16 C5 9F F1 70 E9
beta*P:
X = 0x34C0149E7BB91AE377B02573FCC48AF7BFB7B16DEB8F9CE870F384688E3241A3
A868588CC0EF4364CCA67D17E3260CD82485C202ADC76F895D5DF673B1788E67
Y = 0x608E944929BD643569ED5189DB871453F13333A1EAF82B2FE1BE8100E775F13D
D9925BD317B63BFAF05024D4A738852332B64501195C1B2EF789E34F23DDAFC5
u_2:
X = 0x0535F95463444C4594B5A2E14B35760491C670925060B4BEBC97DE3A3076D1A5
81F89026E04282B040925D9250201024ACA4B2713569B6C3916A6F3344B840AD
Y = 0x40E6C2E55AEC31E7BCB6EA0242857FC6DFB5409803EDF4CA20141F72CC3C7988
706E076765F4F004340E5294A7F8E53BA59CB67502F0044558C854A7D63FE900

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: D5 90 E0 5E F5 AE CE 8B 7C FB FC 71 BE 45 5F 29 A5 CC 66 6F 85 CD B1 7E 7C C7 16 C5 9F F1 70 E9

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: DE 46 BB 4C 8C E0 8A 6E F3 B8 DF AC CC 1A 39 B0 8D 8C 27 B6 CB 0F CF 59 23 86 A6 48 F4 E5 BD 8C The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

Smyshlyaev, et al. Informational [Page 28]


RFC 8133                         SESPAKE                      March 2017

MAC_B: EC B1 1D E2 06 1C 55 F1 D1 14 59 CB 51 CE 31 40 99 99 99 2F CA A1 22 2F B1 4F CE AB 96 EE 7A AC
A.2.6.  Curve id-tc26-gost-3410-2012-256-paramSetA

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0xB51ADF93A40AB15792164FAD3352F95B66369EB2A4EF5EFAE32829320363350E Y = 0x74A358CC08593612F5955D249C96AFB7E8B0BB6D8BD2BBE491046650D822BE18

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67

The coordinates of the point Q_PW are:

X = 0xDBF99827078956812FA48C6E695DF589DEF1D18A2D4D35A96D75BF6854237629 Y = 0x9FDDD48BFBC57BEE1DA0CFF282884F284D471B388893C48F5ECB02FC18D67589

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x147B72F6684FB8FD1B418A899F7DBECAF5FCE60B13685BAA95328654A7F0707F alpha*P: X = 0x33FBAC14EAE538275A769417829C431BD9FA622B6F02427EF55BD60EE6BC2888 Y = 0x22F2EBCF960A82E6CDB4042D3DDDA511B2FBA925383C2273D952EA2D406EAE46
u_1:
X = 0xE569AB544E3A13C41077DE97D659A1B7A13F61DDD808B633A5621FE2583A2C43 Y = 0xA21A743A08F4D715661297ECD6F86553A808925BF34802BF7EC34C548A40B2C0 When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:



Smyshlyaev, et al.            Informational                    [Page 29]


RFC 8133                         SESPAKE                      March 2017



beta=0x30D5CFADAA0E31B405E6734C03EC4C5DF0F02F4BA25C9A3B320EE6453567B4CB src: A3 39 A0 B8 9C EF 1A 6F FD 4C A1 28 04 9E 06 84 DF 4A 97 75 B6 89 A3 37 84 1B F7 D7 91 20 7F 35 11 86 28 F7 28 8E AA 0F 7E C8 1D A2 0A 24 FF 1E 69 93 C6 3D 9D D2 6A 90 B7 4D D1 A2 66 28 06 63
K_B:
7D F7 1A C3 27 ED 51 7D 0D E4 03 E8 17 C6 20 4B C1 91 65 B9 D1 00 2B 9F 10 88 A6 CD A6 EA CF 27
beta*P:
X = 0x2B2D89FAB735433970564F2F28CFA1B57D640CB902BC6334A538F44155022CB2 Y = 0x10EF6A82EEF1E70F942AA81D6B4CE5DEC0DDB9447512962874870E6F2849A96F
u_2:
X = 0x190D2F283F7E861065DB53227D7FBDF429CEBF93791262CB29569BDF63C86CA4 Y = 0xB3F1715721E9221897CCDE046C9B843A8386DBF7818A112F15A02BC820AC8F6D

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: 7D F7 1A C3 27 ED 51 7D 0D E4 03 E8 17 C6 20 4B C1 91 65 B9 D1 00 2B 9F 10 88 A6 CD A6 EA CF 27

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: F9 29 B6 1A 3C 83 39 85 B8 29 F2 68 55 7F A8 11 00 9F 82 0A B1 A7 30 B5 AA 33 4C 3E 6B A3 17 7F

The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:

MAC_B: A2 92 8A 5C F6 20 BB C4 90 0D E4 03 F7 FC 59 A5 E9 80 B6 8B E0 46 D0 B5 D9 B4 AE 6A BF A8 0B D6

Smyshlyaev, et al. Informational [Page 30]


RFC 8133                         SESPAKE                      March 2017
A.2.7.  Curve id-tc26-gost-3410-2012-512-paramSetC

The input protocol parameters in this example take the following values:

N = 1 ind = 1 ID_A:
00 00 00 00
ID_B:
00 00 00 00
PW:
31 32 33 34 35 36 ('123456')
salt:
29 23 BE 84 E1 6C D6 AE 52 90 49 F1 F1 BB E9 EB
Q_ind:
X = 0x489C91784E02E98F19A803ABCA319917F37689E5A18965251CE2FF4E8D8B298F
5BA7470F9E0E713487F96F4A8397B3D09A270C9D367EB5E0E6561ADEEB51581D
Y = 0x684EA885ACA64EAF1B3FEE36C0852A3BE3BD8011B0EF18E203FF87028D6EB5DB
2C144A0DCC71276542BFD72CA2A43FA4F4939DA66D9A60793C704A8C94E16F18

The function F(PW, salt, 2000) takes the following values:

F(PW, salt, 2000): BD 04 67 3F 71 49 B1 8E 98 15 5B D1 E2 72 4E 71 D0 09 9A A2 51 74 F7 92 D3 32 6C 6F 18 12 70 67 1C 62 13 E3 93 0E FD DA 26 45 17 92 C6 20 81 22 EE 60 D2 00 52 0D 69 5D FD 9F 5F 0F D5 AB A7 02

The coordinates of the point Q_PW are:

X = 0x0185AE6271A81BB7F236A955F7CAA26FB63849813C0287D96C83A15AE6B6A864
67AB13B6D88CE8CD7DC2E5B97FF5F28FAC2C108F2A3CF3DB5515C9E6D7D210E8
Y = 0xED0220F92EF771A71C64ECC77986DB7C03D37B3E2AB3E83F32CE5E074A762EC0
8253C9E2102B87532661275C4B1D16D2789CDABC58ACFDF7318DE70AB64F09B8

During the calculation of u_1 on subject A, the parameter alpha, the point alpha*P, and u_1 take the following values:

alpha=0x332F930421D14CFE260042159F18E49FD5A54167E94108AD80B1DE60B13DE799
9A34D611E63F3F870E5110247DF8EC7466E648ACF385E52CCB889ABF491EDFF0
alpha*P:
X = 0x561655966D52952E805574F4281F1ED3A2D498932B00CBA9DECB42837F09835B
FFBFE2D84D6B6B242FE7B57F92E1A6F2413E12DDD6383E4437E13D72693469AD
Y = 0xF6B18328B2715BD7F4178615273A36135BC0BF62F7D8BB9F080164AD36470AD0
3660F51806C64C6691BADEF30F793720F8E3FEAED631D6A54A4C372DCBF80E82




Smyshlyaev, et al.            Informational                    [Page 31]


RFC 8133                         SESPAKE                      March 2017


u_1:
X = 0x40645B4B9A908D74DEF98886A336F98BAE6ADA4C1AC9B7594A33D5E4A16486C5
533C7F3C5DD84797AB5B4340BFC70CAF1011B69A01A715E5B9B5432D5151CBD7
Y = 0x267FBB18D0B79559D1875909F2A15F7B49ECD8ED166CF7F4FCD1F44891550483
5E80D52BE8D34ADA5B5E159CF52979B1BCFE8F5048DC443A0983AA19192B8407

When processing u_1, calculating the K_B key, and calculating u_2 on subject B, the parameters beta, src, K_B = HASH(src), beta*P, and u_2 take the following values:

beta=0x38481771E7D054F96212686B613881880BD8A6C89DDBC656178F014D2C093432
A033EE10415F13A160D44C2AD61E6E2E05A7F7EC286BCEA3EA4D4D53F8634FA2
src:
4F 4D 64 B5 D0 70 08 E9 E6 85 87 4F 88 2C 3E 1E 60 A6 67 5E ED 42 1F C2 34 16 3F DE B4 4C 69 18 B7 BC CE AB 88 A0 F3 FB 78 8D A8 DB 10 18 51 FF 1A 41 68 22 BA 37 C3 53 CE C4 C5 A5 23 95 B7 72 AC 93 C0 54 E3 F4 05 5C ED 6F F0 BE E4 A6 A2 4E D6 8B 86 FE FA 70 DE 4A 2B 16 08 51 42 A4 DF F0 5D 32 EC 7D DF E3 04 F5 C7 04 FD FA 06 0F 64 E9 E8 32 14 00 25 F3 92 E5 03 50 77 0E 3F B6 2C AC
K_B:
A0 83 84 A6 2F 4B E1 AE 48 98 FC A3 6D AA 3F AA 45 1B 3E C5 B5 9C E3 75 F8 9E 92 9F 4B 13 25 8C
beta*P:
X = 0xB7C5818687083433BC1AFF61CB5CA79E38232025E0C1F123B8651E62173CE687
3F3E6FFE7281C2E45F4F524F66B0C263616ED08FD210AC4355CA3292B51D71C3
Y = 0x497F14205DBDC89BDDAF50520ED3B1429AD30777310186BE5E68070F016A44E0
C766DB08E8AC23FBDFDE6D675AA4DF591EB18BA0D348DF7AA40973A2F1DCFA55
u_2:
X = 0xB772FD97D6FDEC1DA0771BC059B3E5ADF9858311031EAE5AEC6A6EC8104B4105
C45A6C65689A8EE636C687DB62CC0AFC9A48CA66E381286CC73F374C1DD8F445
Y = 0xC64F69425FFEB2995130E85A08EDC3A686EC28EE6E8469F7F09BD3BCBDD843AC
573578DA6BA1CB3F5F069F205233853F06255C4B28586C9A1643537497B1018C

When processing u_2 and calculating the key on subject A, the K_A key takes the following values:

K_A: A0 83 84 A6 2F 4B E1 AE 48 98 FC A3 6D AA 3F AA 45 1B 3E C5 B5 9C E3 75 F8 9E 92 9F 4B 13 25 8C

The message MAC_A = HMAC(K_A, 0x01 || ID_A || ind || salt || u_1 || u_2) from subject A takes the following values:

MAC_A: 12 63 F2 89 0E 90 EE 42 6B 9B A0 8A B9 EA 7F 1F FF 26 E1 60 5C C6 5D E2 96 96 91 15 E5 31 76 87 The message MAC_B = HMAC(K_B, 0x02 || ID_B || ind || salt || u_1 || u_2) from subject B takes the following values:


Smyshlyaev, et al.            Informational                    [Page 32]


RFC 8133                         SESPAKE                      March 2017



MAC_B: 6D FD 06 04 5D 6D 97 A0 E4 19 B0 0E 00 35 B9 D2 E3 AB 09 8B 7C A4 AD 52 54 60 FA B6 21 85 AA 57
Appendix B.  Point Verification Script

The points from Appendix A.1 were generated with the following point verification script in Python:

curvesParams = [ { "OID":"id-GostR3410-2001-CryptoPro-A-ParamSet", "p":0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD97, "a":0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD94, "b":166, "m":0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6C611070995AD10045841B09B761B893, "q":0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF6C611070995AD10045841B09B761B893, "x":1, "y":0x8D91E471E0989CDA27DF505A453F2B7635294F2DDF23E3B122ACC99C9E9F1E14, "n":32 }, { "OID":"id-GostR3410-2001-CryptoPro-B-ParamSet", "p":0x8000000000000000000000000000000000000000000000000000000000000C99, "a":0x8000000000000000000000000000000000000000000000000000000000000C96, "b":0x3E1AF419A269A5F866A7D3C25C3DF80AE979259373FF2B182F49D4CE7E1BBC8B, "m":0x800000000000000000000000000000015F700CFFF1A624E5E497161BCC8A198F, "q":0x800000000000000000000000000000015F700CFFF1A624E5E497161BCC8A198F, "x":1, "y":0x3FA8124359F96680B83D1C3EB2C070E5C545C9858D03ECFB744BF8D717717EFC, "n":32 }, { "OID":"id-GostR3410-2001-CryptoPro-C-ParamSet", "p":0x9B9F605F5A858107AB1EC85E6B41C8AACF846E86789051D37998F7B9022D759B, "a":0x9B9F605F5A858107AB1EC85E6B41C8AACF846E86789051D37998F7B9022D7598, "b":32858, "m":0x9B9F605F5A858107AB1EC85E6B41C8AA582CA3511EDDFB74F02F3A6598980BB9, "q":0x9B9F605F5A858107AB1EC85E6B41C8AA582CA3511EDDFB74F02F3A6598980BB9, "x":0, "y":0x41ECE55743711A8C3CBF3783CD08C0EE4D4DC440D4641A8F366E550DFDB3BB67, "n":32 }, { "OID":"id-tc26-gost-3410-2012-512-paramSetA", "p":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\ (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<80)+\ 0xFFFFFFFFFFFFFFFFFDC7L,




Smyshlyaev, et al.            Informational                    [Page 33]


RFC 8133                         SESPAKE                      March 2017


"a":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\
(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<80)+\ 0xFFFFFFFFFFFFFFFFFDC4L,
"b":(0xE8C2505DEDFC86DDC1BD0B2B6667F1DA34B82574761CB0E879BD08L<<296)+\
(0x1CFD0B6265EE3CB090F30D27614CB4574010DA90DD862EF9D4EBEEL<<80)+\ 0x4761503190785A71C760L,
"m":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\
(0xFFFFFFFFFF27E69532F48D89116FF22B8D4E0560609B4B38ABFAD2L<<80)+\ 0xB85DCACDB1411F10B275L,
"q":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\
(0xFFFFFFFFFF27E69532F48D89116FF22B8D4E0560609B4B38ABFAD2L<<80)+\ 0xB85DCACDB1411F10B275L, "x":3, "y":(0x7503CFE87A836AE3A61B8816E25450E6CE5E1C93ACF1ABC1778064L<<296)+\ (0xFDCBEFA921DF1626BE4FD036E93D75E6A50E3A41E98028FE5FC235L<<80)+\ 0xF5B889A589CB5215F2A4L, "n":64 }, { "OID":"id-tc26-gost-3410-2012-512-paramSetB", "p":(0x800000000000000000000000000000000000000000000000000000L<<296)+\ (0x000000000000000000000000000000000000000000000000000000L<<80)+\ 0x0000000000000000006FL,
"a":(0x800000000000000000000000000000000000000000000000000000L<<296)+\
(0x000000000000000000000000000000000000000000000000000000L<<80)+\ 0x0000000000000000006CL,
"b":(0x687D1B459DC841457E3E06CF6F5E2517B97C7D614AF138BCBF85DCL<<296)+\
(0x806C4B289F3E965D2DB1416D217F8B276FAD1AB69C50F78BEE1FA3L<<80)+\ 0x106EFB8CCBC7C5140116L,
"m":(0x800000000000000000000000000000000000000000000000000000L<<296)+\
(0x000000000149A1EC142565A545ACFDB77BD9D40CFA8B996712101BL<<80)+\ 0xEA0EC6346C54374F25BDL,
"q":(0x800000000000000000000000000000000000000000000000000000L<<296)+\
(0x000000000149A1EC142565A545ACFDB77BD9D40CFA8B996712101BL<<80)+\ 0xEA0EC6346C54374F25BDL, "x":2, "y":(0x1A8F7EDA389B094C2C071E3647A8940F3C123B697578C213BE6DD9L<<296)+\ (0xE6C8EC7335DCB228FD1EDF4A39152CBCAAF8C0398828041055F94CL<<80)+\ 0xEEEC7E21340780FE41BDL, "n":64 }, { "OID":"id-tc26-gost-3410-2012-256-paramSetA", "p":0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD97, "a":0xC2173F1513981673AF4892C23035A27CE25E2013BF95AA33B22C656F277E7335, "b":0x295F9BAE7428ED9CCC20E7C359A9D41A22FCCD9108E17BF7BA9337A6F8AE9513, "m":0x1000000000000000000000000000000003F63377F21ED98D70456BD55B0D8319C, "q":0x400000000000000000000000000000000FD8CDDFC87B6635C115AF556C360C67, "x":0x91E38443A5E82C0D880923425712B2BB658B9196932E02C78B2582FE742DAA28, "y":0x32879423AB1A0375895786C4BB46E9565FDE0B5344766740AF268ADB32322E5C, "n":32 }, { "OID":"id-tc26-gost-3410-2012-512-paramSetC", "p":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\ (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<80)+\ 0xFFFFFFFFFFFFFFFFFDC7L,




Smyshlyaev, et al.            Informational                    [Page 34]


RFC 8133                         SESPAKE                      March 2017


"a":(0xDC9203E514A721875485A529D2C722FB187BC8980EB866644DE41CL<<296)+\
(0x68E143064546E861C0E2C9EDD92ADE71F46FCF50FF2AD97F951FDAL<<80)+\ 0x9F2A2EB6546F39689BD3L,
"b":(0xB4C4EE28CEBC6C2C8AC12952CF37F16AC7EFB6A9F69F4B57FFDA2EL<<296)+\
(0x4F0DE5ADE038CBC2FFF719D2C18DE0284B8BFEF3B52B8CC7A5F5BFL<<80)+\ 0x0A3C8D2319A5312557E1L,
"m":(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\
(0xFFFFFFFFFF26336E91941AAC0130CEA7FD451D40B323B6A79E9DA6L<<80)+\ 0x849A5188F3BD1FC08FB4L,
"q":(0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL<<296)+\
(0xFFFFFFFFFFC98CDBA46506AB004C33A9FF5147502CC8EDA9E7A769L<<80)+\ 0xA12694623CEF47F023EDL,
"x":(0xE2E31EDFC23DE7BDEBE241CE593EF5DE2295B7A9CBAEF021D385F7L<<296)+\
(0x074CEA043AA27272A7AE602BF2A7B9033DB9ED3610C6FB85487EAEL<<80)+\ 0x97AAC5BC7928C1950148L,
"y":(0xF5CE40D95B5EB899ABBCCFF5911CB8577939804D6527378B8C108CL<<296)+\
(0x3D2090FF9BE18E2D33E3021ED2EF32D85822423B6304F726AA854BL<<80)+\ 0xAE07D0396E9A9ADDC40FL, "n":64 } ] def str2list( s ): res = [] for c in s:

Smyshlyaev, et al. Informational [Page 35]


RFC 8133                         SESPAKE                      March 2017
res += [ ord( c ) ]
return res

def list2str( l ):
r = "" for k in l:
r += chr( k )
return r

def hprint( data ):
r = "" for i in range( len( data ) ): r += "%02X " % data[ i ] if i % 16 == 15:
r += "\n"
print( r )

class Stribog:
  __A = [
    0x8e20faa72ba0b470, 0x47107ddd9b505a38, 0xad08b0e0c3282d1c,
0xd8045870ef14980e, 0x6c022c38f90a4c07, 0x3601161cf205268d, 0x1b8e0b0e798c13c8, 0x83478b07b2468764, 0xa011d380818e8f40, 0x5086e740ce47c920, 0x2843fd2067adea10, 0x14aff010bdd87508, 0x0ad97808d06cb404, 0x05e23c0468365a02, 0x8c711e02341b2d01, 0x46b60f011a83988e, 0x90dab52a387ae76f, 0x486dd4151c3dfdb9, 0x24b86a840e90f0d2, 0x125c354207487869, 0x092e94218d243cba, 0x8a174a9ec8121e5d, 0x4585254f64090fa0, 0xaccc9ca9328a8950, 0x9d4df05d5f661451, 0xc0a878a0a1330aa6, 0x60543c50de970553, 0x302a1e286fc58ca7, 0x18150f14b9ec46dd, 0x0c84890ad27623e0, 0x0642ca05693b9f70, 0x0321658cba93c138, 0x86275df09ce8aaa8, 0x439da0784e745554, 0xafc0503c273aa42a, 0xd960281e9d1d5215, 0xe230140fc0802984, 0x71180a8960409a42, 0xb60c05ca30204d21, 0x5b068c651810a89e, 0x456c34887a3805b9, 0xac361a443d1c8cd2, 0x561b0d22900e4669, 0x2b838811480723ba, 0x9bcf4486248d9f5d, 0xc3e9224312c8c1a0, 0xeffa11af0964ee50, 0xf97d86d98a327728, 0xe4fa2054a80b329c, 0x727d102a548b194e, 0x39b008152acb8227, 0x9258048415eb419d, 0x492c024284fbaec0, 0xaa16012142f35760, 0x550b8e9e21f7a530, 0xa48b474f9ef5dc18, 0x70a6a56e2440598e, 0x3853dc371220a247, 0x1ca76e95091051ad, 0x0edd37c48a08a6d8, 0x07e095624504536c, 0x8d70c431ac02a736, 0xc83862965601dd1b, 0x641c314b2b8ee083
]




Smyshlyaev, et al.            Informational                    [Page 36]


RFC 8133                         SESPAKE                      March 2017


  __Sbox = [
    0xFC, 0xEE, 0xDD, 0x11, 0xCF, 0x6E, 0x31, 0x16, 0xFB, 0xC4, 0xFA,
0xDA, 0x23, 0xC5, 0x04, 0x4D, 0xE9, 0x77, 0xF0, 0xDB, 0x93, 0x2E, 0x99, 0xBA, 0x17, 0x36, 0xF1, 0xBB, 0x14, 0xCD, 0x5F, 0xC1, 0xF9, 0x18, 0x65, 0x5A, 0xE2, 0x5C, 0xEF, 0x21, 0x81, 0x1C, 0x3C, 0x42, 0x8B, 0x01, 0x8E, 0x4F, 0x05, 0x84, 0x02, 0xAE, 0xE3, 0x6A, 0x8F, 0xA0, 0x06, 0x0B, 0xED, 0x98, 0x7F, 0xD4, 0xD3, 0x1F, 0xEB, 0x34, 0x2C, 0x51, 0xEA, 0xC8, 0x48, 0xAB, 0xF2, 0x2A, 0x68, 0xA2, 0xFD, 0x3A, 0xCE, 0xCC, 0xB5, 0x70, 0x0E, 0x56, 0x08, 0x0C, 0x76, 0x12, 0xBF, 0x72, 0x13, 0x47, 0x9C, 0xB7, 0x5D, 0x87, 0x15, 0xA1, 0x96, 0x29, 0x10, 0x7B, 0x9A, 0xC7, 0xF3, 0x91, 0x78, 0x6F, 0x9D, 0x9E, 0xB2, 0xB1, 0x32, 0x75, 0x19, 0x3D, 0xFF, 0x35, 0x8A, 0x7E, 0x6D, 0x54, 0xC6, 0x80, 0xC3, 0xBD, 0x0D, 0x57, 0xDF, 0xF5, 0x24, 0xA9, 0x3E, 0xA8, 0x43, 0xC9, 0xD7, 0x79, 0xD6, 0xF6, 0x7C, 0x22, 0xB9, 0x03, 0xE0, 0x0F, 0xEC, 0xDE, 0x7A, 0x94, 0xB0, 0xBC, 0xDC, 0xE8, 0x28, 0x50, 0x4E, 0x33, 0x0A, 0x4A, 0xA7, 0x97, 0x60, 0x73, 0x1E, 0x00, 0x62, 0x44, 0x1A, 0xB8, 0x38, 0x82, 0x64, 0x9F, 0x26, 0x41, 0xAD, 0x45, 0x46, 0x92, 0x27, 0x5E, 0x55, 0x2F, 0x8C, 0xA3, 0xA5, 0x7D, 0x69, 0xD5, 0x95, 0x3B, 0x07, 0x58, 0xB3, 0x40, 0x86, 0xAC, 0x1D, 0xF7, 0x30, 0x37, 0x6B, 0xE4, 0x88, 0xD9, 0xE7, 0x89, 0xE1, 0x1B, 0x83, 0x49, 0x4C, 0x3F, 0xF8, 0xFE, 0x8D, 0x53, 0xAA, 0x90, 0xCA, 0xD8, 0x85, 0x61, 0x20, 0x71, 0x67, 0xA4, 0x2D, 0x2B, 0x09, 0x5B, 0xCB, 0x9B, 0x25, 0xD0, 0xBE, 0xE5, 0x6C, 0x52, 0x59, 0xA6, 0x74, 0xD2, 0xE6, 0xF4, 0xB4, 0xC0, 0xD1, 0x66, 0xAF, 0xC2, 0x39, 0x4B, 0x63, 0xB6
]
  __Tau = [
    0,  8, 16, 24, 32, 40, 48, 56,
1, 9, 17, 25, 33, 41, 49, 57, 2, 10, 18, 26, 34, 42, 50, 58, 3, 11, 19, 27, 35, 43, 51, 59, 4, 12, 20, 28, 36, 44, 52, 60, 5, 13, 21, 29, 37, 45, 53, 61, 6, 14, 22, 30, 38, 46, 54, 62, 7, 15, 23, 31, 39, 47, 55, 63
]

Smyshlyaev, et al. Informational [Page 37]


RFC 8133                         SESPAKE                      March 2017
  __C = [
    [
0xb1, 0x08, 0x5b, 0xda, 0x1e, 0xca, 0xda, 0xe9, 0xeb, 0xcb, 0x2f, 0x81, 0xc0, 0x65, 0x7c, 0x1f, 0x2f, 0x6a, 0x76, 0x43, 0x2e, 0x45, 0xd0, 0x16, 0x71, 0x4e, 0xb8, 0x8d, 0x75, 0x85, 0xc4, 0xfc, 0x4b, 0x7c, 0xe0, 0x91, 0x92, 0x67, 0x69, 0x01, 0xa2, 0x42, 0x2a, 0x08, 0xa4, 0x60, 0xd3, 0x15, 0x05, 0x76, 0x74, 0x36, 0xcc, 0x74, 0x4d, 0x23, 0xdd, 0x80, 0x65, 0x59, 0xf2, 0xa6, 0x45, 0x07 ], [ 0x6f, 0xa3, 0xb5, 0x8a, 0xa9, 0x9d, 0x2f, 0x1a, 0x4f, 0xe3, 0x9d, 0x46, 0x0f, 0x70, 0xb5, 0xd7, 0xf3, 0xfe, 0xea, 0x72, 0x0a, 0x23, 0x2b, 0x98, 0x61, 0xd5, 0x5e, 0x0f, 0x16, 0xb5, 0x01, 0x31, 0x9a, 0xb5, 0x17, 0x6b, 0x12, 0xd6, 0x99, 0x58, 0x5c, 0xb5, 0x61, 0xc2, 0xdb, 0x0a, 0xa7, 0xca, 0x55, 0xdd, 0xa2, 0x1b, 0xd7, 0xcb, 0xcd, 0x56, 0xe6, 0x79, 0x04, 0x70, 0x21, 0xb1, 0x9b, 0xb7 ], [ 0xf5, 0x74, 0xdc, 0xac, 0x2b, 0xce, 0x2f, 0xc7, 0x0a, 0x39, 0xfc, 0x28, 0x6a, 0x3d, 0x84, 0x35, 0x06, 0xf1, 0x5e, 0x5f, 0x52, 0x9c, 0x1f, 0x8b, 0xf2, 0xea, 0x75, 0x14, 0xb1, 0x29, 0x7b, 0x7b, 0xd3, 0xe2, 0x0f, 0xe4, 0x90, 0x35, 0x9e, 0xb1, 0xc1, 0xc9, 0x3a, 0x37, 0x60, 0x62, 0xdb, 0x09, 0xc2, 0xb6, 0xf4, 0x43, 0x86, 0x7a, 0xdb, 0x31, 0x99, 0x1e, 0x96, 0xf5, 0x0a, 0xba, 0x0a, 0xb2 ], [ 0xef, 0x1f, 0xdf, 0xb3, 0xe8, 0x15, 0x66, 0xd2, 0xf9, 0x48, 0xe1, 0xa0, 0x5d, 0x71, 0xe4, 0xdd, 0x48, 0x8e, 0x85, 0x7e, 0x33, 0x5c, 0x3c, 0x7d, 0x9d, 0x72, 0x1c, 0xad, 0x68, 0x5e, 0x35, 0x3f, 0xa9, 0xd7, 0x2c, 0x82, 0xed, 0x03, 0xd6, 0x75, 0xd8, 0xb7, 0x13, 0x33, 0x93, 0x52, 0x03, 0xbe, 0x34, 0x53, 0xea, 0xa1, 0x93, 0xe8, 0x37, 0xf1, 0x22, 0x0c, 0xbe, 0xbc, 0x84, 0xe3, 0xd1, 0x2e ], [ 0x4b, 0xea, 0x6b, 0xac, 0xad, 0x47, 0x47, 0x99, 0x9a, 0x3f, 0x41, 0x0c, 0x6c, 0xa9, 0x23, 0x63, 0x7f, 0x15, 0x1c, 0x1f, 0x16, 0x86, 0x10, 0x4a, 0x35, 0x9e, 0x35, 0xd7, 0x80, 0x0f, 0xff, 0xbd, 0xbf, 0xcd, 0x17, 0x47, 0x25, 0x3a, 0xf5, 0xa3, 0xdf, 0xff, 0x00, 0xb7, 0x23, 0x27, 0x1a, 0x16, 0x7a, 0x56, 0xa2, 0x7e, 0xa9, 0xea, 0x63, 0xf5, 0x60, 0x17, 0x58, 0xfd, 0x7c, 0x6c, 0xfe, 0x57 ], [ 0xae, 0x4f, 0xae, 0xae, 0x1d, 0x3a, 0xd3, 0xd9, 0x6f, 0xa4, 0xc3, 0x3b, 0x7a, 0x30, 0x39, 0xc0, 0x2d, 0x66, 0xc4, 0xf9, 0x51, 0x42, 0xa4, 0x6c, 0x18, 0x7f, 0x9a, 0xb4, 0x9a, 0xf0, 0x8e, 0xc6, 0xcf, 0xfa, 0xa6, 0xb7, 0x1c, 0x9a, 0xb7, 0xb4, 0x0a, 0xf2, 0x1f, 0x66, 0xc2, 0xbe, 0xc6, 0xb6, 0xbf, 0x71, 0xc5, 0x72, 0x36, 0x90, 0x4f, 0x35, 0xfa, 0x68, 0x40, 0x7a, 0x46, 0x64, 0x7d, 0x6e ], [ 0xf4, 0xc7, 0x0e, 0x16, 0xee, 0xaa, 0xc5, 0xec, 0x51, 0xac, 0x86, 0xfe, 0xbf, 0x24, 0x09, 0x54, 0x39, 0x9e, 0xc6, 0xc7, 0xe6, 0xbf, 0x87, 0xc9, 0xd3, 0x47, 0x3e, 0x33, 0x19, 0x7a, 0x93, 0xc9, 0x09, 0x92, 0xab, 0xc5, 0x2d, 0x82, 0x2c, 0x37, 0x06, 0x47, 0x69, 0x83, 0x28, 0x4a, 0x05, 0x04, 0x35, 0x17, 0x45, 0x4c, 0xa2, 0x3c, 0x4a, 0xf3, 0x88, 0x86, 0x56, 0x4d, 0x3a, 0x14, 0xd4, 0x93 ], [ 0x9b, 0x1f, 0x5b, 0x42, 0x4d, 0x93, 0xc9, 0xa7, 0x03, 0xe7, 0xaa, 0x02, 0x0c, 0x6e, 0x41, 0x41, 0x4e, 0xb7, 0xf8, 0x71, 0x9c, 0x36, 0xde, 0x1e, 0x89, 0xb4, 0x44, 0x3b, 0x4d, 0xdb, 0xc4, 0x9a, 0xf4, 0x89, 0x2b, 0xcb, 0x92, 0x9b, 0x06, 0x90, 0x69, 0xd1, 0x8d, 0x2b, 0xd1, 0xa5, 0xc4, 0x2f, 0x36, 0xac, 0xc2, 0x35, 0x59, 0x51, 0xa8, 0xd9, 0xa4, 0x7f, 0x0d, 0xd4, 0xbf, 0x02, 0xe7, 0x1e ], [ 0x37, 0x8f, 0x5a, 0x54, 0x16, 0x31, 0x22, 0x9b, 0x94, 0x4c, 0x9a, 0xd8, 0xec, 0x16, 0x5f, 0xde, 0x3a, 0x7d, 0x3a, 0x1b, 0x25, 0x89, 0x42, 0x24, 0x3c, 0xd9, 0x55, 0xb7, 0xe0, 0x0d, 0x09, 0x84, 0x80, 0x0a, 0x44, 0x0b, 0xdb, 0xb2, 0xce, 0xb1, 0x7b, 0x2b, 0x8a, 0x9a, 0xa6, 0x07, 0x9c, 0x54, 0x0e, 0x38, 0xdc, 0x92, 0xcb, 0x1f, 0x2a, 0x60, 0x72, 0x61, 0x44, 0x51, 0x83, 0x23, 0x5a, 0xdb ], [ 0xab, 0xbe, 0xde, 0xa6, 0x80, 0x05, 0x6f, 0x52, 0x38, 0x2a, 0xe5, 0x48, 0xb2, 0xe4, 0xf3, 0xf3, 0x89, 0x41, 0xe7, 0x1c, 0xff, 0x8a, 0x78, 0xdb, 0x1f, 0xff, 0xe1, 0x8a, 0x1b, 0x33, 0x61, 0x03, 0x9f, 0xe7, 0x67, 0x02, 0xaf, 0x69, 0x33, 0x4b, 0x7a, 0x1e, 0x6c, 0x30, 0x3b, 0x76, 0x52, 0xf4, 0x36, 0x98, 0xfa, 0xd1, 0x15, 0x3b, 0xb6, 0xc3, 0x74, 0xb4, 0xc7, 0xfb, 0x98, 0x45, 0x9c, 0xed ], [ 0x7b, 0xcd, 0x9e, 0xd0, 0xef, 0xc8, 0x89, 0xfb, 0x30, 0x02, 0xc6, 0xcd, 0x63, 0x5a, 0xfe, 0x94, 0xd8, 0xfa, 0x6b, 0xbb, 0xeb, 0xab, 0x07, 0x61, 0x20, 0x01, 0x80, 0x21, 0x14, 0x84, 0x66, 0x79, 0x8a, 0x1d, 0x71, 0xef, 0xea, 0x48, 0xb9, 0xca, 0xef, 0xba, 0xcd, 0x1d, 0x7d, 0x47, 0x6e, 0x98, 0xde, 0xa2, 0x59, 0x4a, 0xc0, 0x6f, 0xd8, 0x5d, 0x6b, 0xca, 0xa4, 0xcd, 0x81, 0xf3, 0x2d, 0x1b ], [ 0x37, 0x8e, 0xe7, 0x67, 0xf1, 0x16, 0x31, 0xba, 0xd2, 0x13, 0x80, 0xb0, 0x04, 0x49, 0xb1, 0x7a, 0xcd, 0xa4, 0x3c, 0x32, 0xbc, 0xdf, 0x1d, 0x77, 0xf8, 0x20, 0x12, 0xd4, 0x30, 0x21, 0x9f, 0x9b, 0x5d, 0x80, 0xef, 0x9d, 0x18, 0x91, 0xcc, 0x86, 0xe7, 0x1d, 0xa4, 0xaa, 0x88, 0xe1, 0x28, 0x52, 0xfa, 0xf4, 0x17, 0xd5, 0xd9, 0xb2, 0x1b, 0x99, 0x48, 0xbc, 0x92, 0x4a, 0xf1, 0x1b, 0xd7, 0x20

Smyshlyaev, et al. Informational [Page 38]


RFC 8133                         SESPAKE                      March 2017

Smyshlyaev, et al. Informational [Page 39]


RFC 8133                         SESPAKE                      March 2017
]
]

Smyshlyaev, et al. Informational [Page 40]


RFC 8133                         SESPAKE                      March 2017
  def __AddModulo(self, A, B):
    result = [0] * 64
t = 0 for i in reversed(range(0, 64)): t = A[i] + B[i] + (t >> 8) result[i] = t & 0xFF
return result
  def __AddXor(self, A, B):
    result = [0] * 64
for i in range(0, 64):
result[i] = A[i] ^ B[i]
return result
  def __S(self, state):
    result = [0] * 64
    for i in range(0, 64):
      result[i] = self.__Sbox[state[i]]
    return result

  def __P(self, state):
    result = [0] * 64
    for i in range(0, 64):
      result[i] = state[self.__Tau[i]]
    return result

  def __L(self, state):
    result = [0] * 64
for i in range(0, 8):
t = 0 for k in range(0, 8):
for j in range(0, 8):
          if ((state[i * 8 + k] & (1 << (7 - j))) != 0):
            t ^= self.__A[k * 8 + j]
      for k in range(0, 8):
result[i * 8 + k] = (t & (0xFF << (7 - k) * 8)) >> (7 - k) * 8
return result
  def __KeySchedule(self, K, i):
    K = self.__AddXor(K, self.__C[i])
    K = self.__S(K)
    K = self.__P(K)
    K = self.__L(K)
    return K

Smyshlyaev, et al. Informational [Page 41]


RFC 8133                         SESPAKE                      March 2017
  # E(K, m)
  def __E(self, K, m):
    state = self.__AddXor(K, m)
    for i in range(0, 12):
      state = self.__S(state)
      state = self.__P(state)
      state = self.__L(state)
      K = self.__KeySchedule(K, i)
      state = self.__AddXor(state, K)
    return state

  def __G_n(self, N, h, m):
    K = self.__AddXor(h, N)
    K = self.__S(K)
    K = self.__P(K)
    K = self.__L(K)
    t = self.__E(K, m)
    t = self.__AddXor(t, h)
    return self.__AddXor(t, m)

  def __Padding(self, last, N, h, Sigma):
    if (len(last) < 64):
padding = [0] * (64 - len(last)) padding[-1] = 1
      padded_message = padding + last
    h = self.__G_n(N, h, padded_message)
    N_len = [0] * 64
N_len[63] = (len(last) * 8) & 0xff
    N_len[62] = (len(last) * 8) >> 8
    N = self.__AddModulo(N, N_len)
    Sigma = self.__AddModulo(Sigma, padded_message)
    return (h, N, Sigma)

def digest( self, message, out=512 ):
return list2str( self.GetHash( str2list( message ), out ) )

def GetHash(self, message, out=512, no_pad=False):
N = [0] * 64 Sigma = [0] * 64 if out == 512:
h = [0] * 64
elif out == 256:
h = [0x01] * 64
else:
print("Wrong hash out length!")

N_512 = [0] * 64 N_512[62] = 0x02 # 512 = 0x200 length_bits = len(message) * 8 length = len(message)


Smyshlyaev, et al.            Informational                    [Page 42]


RFC 8133                         SESPAKE                      March 2017



i = 0 asd = message[::-1] while (length_bits >= 512):
      tmp = (message[i * 64: (i + 1) * 64])[::-1]
      h = self.__G_n(N, h, tmp)
      N = self.__AddModulo(N, N_512)
      Sigma = self.__AddModulo(Sigma, tmp)
      length_bits -= 512
i += 1

last = (message[i * 64: length])[::-1]

if (len(last) == 0 and no_pad):
pass
    else:
      h, N, Sigma = self.__Padding(last, N, h, Sigma)

    N_0 = [0] * 64
    h = self.__G_n(N_0, h, N)
    h = self.__G_n(N_0, h, Sigma)

if out == 512:
return h[::-1]
elif out == 256:
return (h[0:32])[::-1]

def hash(self, str_message, out=512, no_pad=False):
return list2str(self.GetHash(str2list(str_message), out, no_pad))

def H256(msg):
S = Stribog() return S.hash(msg, out=256)

def H512(msg):
S = Stribog() return S.hash(msg)

def num2le( s, n ):
res = "" for i in range(n): res += chr(s & 0xFF) s >>= 8
return res




Smyshlyaev, et al.            Informational                    [Page 43]


RFC 8133                         SESPAKE                      March 2017


def le2num( s ):
res = 0 for i in range(len(s) - 1, -1, -1):
res = (res << 8) + ord(s[i])
return res

def XGCD(a,b):
"""XGCD(a,b) returns a list of form [g,x,y], where g is GCD(a,b) and x,y satisfy the equation g = ax + by.""" a1=1; b1=0; a2=0; b2=1; aneg=1; bneg=1; swap = False if(a < 0):
a = -a; aneg=-1
if(b < 0):
b = -b; bneg=-1
if(b > a):
swap = True [a,b] = [b,a]
while (1):
quot = -(a / b) a = a % b a1 = a1 + quot*a2; b1 = b1 + quot*b2 if(a == 0):
if(swap):
return [b, b2*bneg, a2*aneg]
else:
return [b, a2*aneg, b2*bneg]
quot = -(b / a) b = b % a a2 = a2 + quot*a1; b2 = b2 + quot*b1 if(b == 0):
if(swap):
return [a, b1*bneg, a1*aneg]
else:
return [a, a1*aneg, b1*bneg]

def getMultByMask( elems, mask ):
n = len( elems ) r = 1 for i in range( n ):
if mask & 1:
r *= elems[ n - 1 - i ]
mask = mask >> 1
return r

Smyshlyaev, et al. Informational [Page 44]


RFC 8133                         SESPAKE                      March 2017
def subF(P, other, p):
return (P - other) % p

def divF(P, other, p):
return mulF(P, invF(other, p), p)

def addF(P, other, p):
return (P + other) % p

def mulF(P, other, p):
return (P * other) % p

def invF(R, p):
assert (R != 0) return XGCD(R, p)[1] % p

def negF(R, p):
return (-R) % p

def powF(R, m, p):
assert R != None assert type(m) in (int, long)

if m == 0: assert R != 0 return 1
elif m < 0:
t = invF(R, p) return powF(t, (-m), p)
else:
i = m.bit_length() - 1 r = 1 while i > 0:
if (m >> i) & 1:
r = (r * R) % p
r = (r * r) % p i -= 1
if m & 1:
r = (r * R) % p
return r

Smyshlyaev, et al. Informational [Page 45]


RFC 8133                         SESPAKE                      March 2017
def add(Px, Py, Qx, Qy, p, a, b):
if Qx == Qy == None:
return [Px, Py]

if Px == Py == None:
return [Qx, Qy]

if (Px == Qx) and (Py == negF(Qy, p)):
return [None, None]

if (Px == Qx) and (Py == Qy):
assert Py != 0 return duplicate(Px, Py, p, a)
else:
l = divF( subF( Qy, Py, p ), subF( Qx, Px, p ), p ) resX = subF( subF( powF( l, 2, p ), Px, p ), Qx, p ) resY = subF( mulF( l, subF( Px, resX, p ), p ), Py, p ) return [resX, resY]

def duplicate(Px, Py, p, a):
if (Px == None) and (Py == None):
return [None, None]

if Py == 0:
return [None, None]

l = divF(addF(mulF(powF(Px, 2, p), 3, p), a, p), mulF(Py, 2, p), p) resX = subF(powF(l, 2, p), mulF(Px, 2, p), p) resY = subF(mulF(l, subF(Px, resX, p), p), Py, p) return [resX, resY]

Smyshlyaev, et al. Informational [Page 46]


RFC 8133                         SESPAKE                      March 2017
def mul(Px, Py, s, p, a, b):
assert type(s) in (int, long) assert Px != None and Py != None

X = Px Y = Py

i = s.bit_length() - 1 resX = None resY = None while i > 0:
if (s >> i) & 1:
resX, resY = add(resX, resY, X, Y, p, a, b)
resX, resY = duplicate(resX, resY, p, a) i -= 1
if s & 1:
resX, resY = add(resX, resY, X, Y, p, a, b)
return [resX, resY]

def Ord(Px, Py, m, q, p, a, b):
assert Px != None and Py != None assert (m != None) and (q != None) assert mul(Px, Py, m, p, a, b) == [None, None]

X = Px Y = Py r = m for mask in range(1 << len(q)): t = getMultByMask(q, mask) Rx, Ry = mul(X, Y, t, p, a, b) if (Rx == None) and (Ry == None):
r = min(r, t)
return r

def isQuadraticResidue( R, p ):
if R == 0:
assert False
temp = powF(R, ((p - 1) / 2), p) if temp == (p - 1):
return False
else:
assert temp == 1 return True

Smyshlyaev, et al. Informational [Page 47]


RFC 8133                         SESPAKE                      March 2017
def getRandomQuadraticNonresidue(p):
from random import randint r = (randint(2, p - 1)) % p while isQuadraticResidue(r, p):
r = (randint(2, p - 1)) % p
return r

def ModSqrt( R, p ):
assert R != None assert isQuadraticResidue(R, p)

if p % 4 == 3: res = powF(R, (p + 1) / 4, p) if powF(res, 2, p) != R:
res = None
return [res, negF(res, p)]
else:
ainvF = invF(R, p)

s = p - 1 alpha = 0 while (s % 2) == 0: alpha += 1 s = s / 2

b = powF(getRandomQuadraticNonresidue(p), s, p) r = powF(R, (s + 1) / 2, p)

bj = 1 for k in range(0, alpha - 1): # alpha >= 2 because p % 4 = 1 d = 2 ** (alpha - k - 2) x = powF(mulF(powF(mulF(bj, r, p), 2, p), ainvF, p), d, p) if x != 1:
bj = mulF(bj, powF(b, (2 ** k), p), p)
res = mulF(bj, r, p) return [res, negF(res, p)]

Smyshlyaev, et al. Informational [Page 48]


RFC 8133                         SESPAKE                      March 2017
def generateQs( p, pByteSize, a, b, m, q, orderDivisors, Px, Py, N ):
assert pByteSize in ( 256 / 8, 512 / 8 ) PxBytes = num2le( Px, pByteSize ) PyBytes = num2le( Py, pByteSize ) Qs = [] S = [] Hash_src = [] Hash_res = [] co_factor = m / q

seed = 0 while len( Qs ) != N: hashSrc = PxBytes + PyBytes + num2le( seed, 4 ) if pByteSize == ( 256 / 8 ):
QxBytes = H256( hashSrc )
else:
QxBytes = H512( hashSrc )

Qx = le2num( QxBytes ) % p

R = addF( addF( powF(Qx, 3, p ), mulF(Qx, a, p), p), b, p ) if ( R == 0 ) or ( not isQuadraticResidue( R, p ) ): seed += 1 continue

Qy_sqrt = ModSqrt( R, p ) Qy = min(Qy_sqrt) if co_factor * Ord(Qx, Qy, m, orderDivisors, p, a, b) != m: seed += 1 continue

Qs += [(Qx, Qy)] S += [seed] Hash_src += [hashSrc] Hash_res += [QxBytes] seed += 1

return Qs, S, Hash_src, Hash_res

Smyshlyaev, et al. Informational [Page 49]


RFC 8133                         SESPAKE                      March 2017
if __name__ == "__main__":
for i, curve in enumerate(curvesParams):
print "A.1." + str(i+1) + ". Curve " + curve["OID"] if "3410-2012-256-paramSetA" in curve["OID"] or \
"3410-2012-512-paramSetC" in curve["OID"]:
Q, S, Hash_src, Hash_res = generateQs(curve["p"],\
curve["n"],\ curve["a"],\ curve["b"],\ curve["m"],\ curve["q"],\ [ 2, 2, curve["q"]],\ curve["x"],\ curve["y"],\ 1)
else:
Q, S, Hash_src, Hash_res = generateQs(curve["p"],\
curve["n"],\ curve["a"],\ curve["b"],\ curve["m"],\ curve["q"],\ [curve["q"]],\ curve["x"],\ curve["y"],\ 1)

j = 1 for q, s, hash_src, hash_res in zip(Q, S, Hash_src, Hash_res): print "Point Q_" + str(j) j += 1

print "X=", hex(q[0])[:-1] print "Y=", hex(q[1])[:-1]

print "SEED=","{0:#0{1}x}".format(s,6) print

Smyshlyaev, et al. Informational [Page 50]


RFC 8133                         SESPAKE                      March 2017
Acknowledgments

We thank Lolita Sonina, Georgiy Borodin, Sergey Agafin, and Ekaterina Smyshlyaeva for their careful readings and useful comments.

Authors' Addresses

Stanislav Smyshlyaev (editor) CRYPTO-PRO 18, Suschevsky val Moscow 127018 Russian Federation

Phone: +7 (495) 995-48-20 Email: svs@cryptopro.ru


Evgeny Alekseev CRYPTO-PRO 18, Suschevsky val Moscow 127018 Russian Federation

Phone: +7 (495) 995-48-20 Email: alekseev@cryptopro.ru


Igor Oshkin CRYPTO-PRO 18, Suschevsky val Moscow 127018 Russian Federation

Phone: +7 (495) 995-48-20 Email: oshkin@cryptopro.ru


Vladimir Popov CRYPTO-PRO 18, Suschevsky val Moscow 127018 Russian Federation

Phone: +7 (495) 995-48-20 Email: vpopov@cryptopro.ru






Smyshlyaev, et al. Informational [Page 51]


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