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Network Working Group                                 S. Smyshlyaev, Ed.
Internet-Draft                                                 CryptoPro
Intended status: Informational                              V. Nozdrunov
Expires: January 30, 2021                                    V. Shishkin
                                                                   TC 26
                                                           E. Griboedova
                                                               CryptoPro
                                                           July 29, 2020


                     Multilinear Galois Mode (MGM)
                        draft-smyshlyaev-mgm-18

Abstract

   Multilinear Galois Mode (MGM) is an authenticated encryption with
   associated data (AEAD) block cipher mode based on EtM principle.  MGM
   is defined for use with 64-bit and 128-bit block ciphers.

   MGM has been standardized in Russia.  It is used as an AEAD mode for
   the GOST block cipher algorithms in many protocols, e.g.  TLS 1.3 and
   IPsec.  This document provides a reference for MGM to enable review
   of the mechanisms in use and to make MGM available for use with any
   block cipher.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
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   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on January 30, 2021.

Copyright Notice

   Copyright (c) 2020 IETF Trust and the persons identified as the
   document authors.  All rights reserved.





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   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://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.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Conventions Used in This Document . . . . . . . . . . . . . .   3
   3.  Basic Terms and Definitions . . . . . . . . . . . . . . . . .   3
   4.  Specification . . . . . . . . . . . . . . . . . . . . . . . .   4
     4.1.  MGM Encryption and Tag Generation Procedure . . . . . . .   4
     4.2.  MGM Decryption and Tag Verification Check Procedure . . .   7
   5.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .   8
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .   9
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  10
   8.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  10
     8.1.  Normative References  . . . . . . . . . . . . . . . . . .  10
     8.2.  Informative References  . . . . . . . . . . . . . . . . .  10
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .  11
     A.1.  Test Vectors for the Kuznyechik block cipher  . . . . . .  11
     A.2.  Test Vectors for the Magma block cipher . . . . . . . . .  16
   Appendix B.  Contributors . . . . . . . . . . . . . . . . . . . .  22
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  22

1.  Introduction

   Multilinear Galois Mode (MGM) is an authenticated encryption with
   associated data (AEAD) block cipher mode based on EtM principle.  MGM
   is defined for use with 64-bit and 128-bit block ciphers.  The MGM
   design principles can easily be applied to other block sizes.

   MGM has been standardized in Russia [R1323565.1.026-2019].  It is
   used as an AEAD mode for the GOST block cipher algorithms in many
   protocols, e.g.  TLS 1.3 and IPsec.  This document provides a
   reference for MGM to enable review of the mechanisms in use and to
   make MGM available for use with any block cipher.

   This document does not have IETF consensus and does not imply IETF
   support for MGM.






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2.  Conventions Used in This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in BCP
   14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

3.  Basic Terms and Definitions

   This document uses the following terms and definitions for the sets
   and operations on the elements of these sets:

   V*      the set of all bit strings of a finite length (hereinafter
           referred to as strings), including the empty string;
           substrings and string components are enumerated from right to
           left starting from zero;

   V_s     the set of all bit strings of length s, where s is a non-
           negative integer.  For s = 0, the V_0 consists of a single
           empty string;

   |X|     the bit length of the bit string X (if X is an empty string,
           then |X| = 0);

   X || Y  concatenation of strings X and Y both belonging to V*, i.e.,
           a string from V_{|X|+|Y|}, where the left substring from
           V_{|X|} is equal to X, and the right substring from V_{|Y|}
           is equal to Y;

   a^s     the string in V_s that consists of s 'a' bits;

   (xor)   exclusive-or of the two bit strings of the same length;

   Z_{2^s} ring of residues modulo 2^s;

   MSB_i: V_s -> V_i   the transformation that maps the string X =
           (x_{s-1}, ... , x_0) in V_s into the string MSB_i(X) =
           (x_{s-1}, ... , x_{s-i}) in V_i, i <= s, (most significant
           bits);

   Int_s: V_s -> Z_{2^s}    the transformation that maps the string X =
           (x_{s-1}, ... , x_0) in V_s, s > 0, into the integer Int_s(X)
           = 2^{s-1} * x_{s-1} + ... + 2 * x_1 + x_0 (the interpretation
           of the bit string as an integer);

   Vec_s: Z_{2^s} -> V_s  the transformation inverse to the mapping
           Int_s (the interpretation of an integer as a bit string);



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   E_K: V_n -> V_n  the block cipher permutation under the key K in V_k;

   k       the bit length of the block cipher key;

   n       the block size of the block cipher (in bits);

   len: V_s -> V_{n/2}  the transformation that maps a string X in V_s,
           0 <= s <= 2^{n/2} - 1, into the string len(X) =
           Vec_{n/2}(|X|) in V_{n/2}, where n is the block size of the
           used block cipher;

   [+]     the addition operation in Z_{2^{n/2}}, where n is the block
           size of the used block cipher;

   (x)     the transformation that maps two strings X = (x_{n-1}, ... ,
           x_0) in V_n and Y = (y_{n-1}, ... , y_0) in V_n into the
           string Z = X (x) Y = (z_{n-1}, ... , z_0) in V_n; the string
           Z corresponds to the polynomial Z(w) = z_{n-1} * w^{n-1} +
           ... + z_1 * w + z_0 which is a result of the polynomials X(w)
           = x_{n-1} * w^{n-1} + ... + x_1 * w + x_0 and Y(w) = y_{n-1}
           * w^{n-1} + ... + y_1 * w + y_0 multiplication in the field
           GF(2^n), where n is the block size of the used block cipher;
           if n = 64, then the field polynomial is equal to f(w) = w^64
           + w^4 + w^3 + w + 1; if n = 128, then the field polynomial is
           equal to f(w) = w^128 + w^7 + w^2 + w + 1;

   incr_l: V_n -> V_n  the transformation that maps a string L || R,
           where L, R in V_{n/2}, into the string incr_l(L || R) =
           Vec_{n/2}(Int_{n/2}(L) [+] 1) || R;

   incr_r: V_n -> V_n  the transformation that maps a string L || R,
           where L, R in V_{n/2}, into the string incr_r(L || R) = L ||
           Vec_{n/2}(Int_{n/2}(R) [+] 1).

4.  Specification

   An additional parameter that defines the functioning of Multilinear
   Galois Mode (MGM) is the bit length S of the authentication tag, 32
   <= S <= n.  The value of S MUST be fixed for a particular protocol.
   The choice of the value S involves a trade-off between message
   expansion and the forgery probability.

4.1.  MGM Encryption and Tag Generation Procedure

   The MGM encryption and tag generation procedure takes the following
   parameters as inputs:

   1.  Encryption key K in V_k.



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   2.  Initial counter nonce ICN in V_{n-1}.

   3.  Associated authenticated data A, 0 <= |A| < 2^{n/2}. If |A| > 0,
       then A = A_1 || ... || A*_h, A_j in V_n, for j = 1, ... , h - 1,
       A*_h in V_t, 1 <= t <= n.  If |A| = 0, then by definition A*_h is
       empty, and the h and t parameters are set as follows: h = 0, t =
       n.  The associated data is authenticated but is not encrypted.

   4.  Plaintext P, 0 <= |P| < 2^{n/2}. If |P| > 0, then P = P_1 ||
       ... || P*_q, P_i in V_n, for i = 1, ... , q - 1, P*_q in V_u, 1
       <= u <= n.  If |P| = 0, then by definition P*_q is empty, and the
       q and u parameters are set as follows: q = 0, u = n.

   The MGM encryption and tag generation procedure outputs the following
   parameters:

   1.  Initial counter nonce ICN.

   2.  Associated authenticated data A.

   3.  Ciphertext C in V_{|P|}.

   4.  Authentication tag T in V_S.

   The MGM encryption and tag generation procedure consists of the
   following steps:

























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   +----------------------------------------------------------------+
   |  MGM-Encrypt(K, ICN, A, P)                                     |
   |----------------------------------------------------------------|
   |  1. Encryption step:                                           |
   |      - if |P| = 0 then                                         |
   |            - C*_q = P*_q                                       |
   |            - C = P                                             |
   |      - else                                                    |
   |            - Y_1 = E_K(0^1 || ICN),                            |
   |            - For i = 2, 3, ... , q do                          |
   |                    Y_i = incr_r(Y_{i-1}),                      |
   |            - For i = 1, 2, ... , q - 1 do                      |
   |                    C_i = P_i (xor) E_K(Y_i),                   |
   |            - C*_q = P*_q (xor) MSB_u(E_K(Y_q)),                |
   |            - C = C_1 || ... || C*_q.                           |
   |                                                                |
   |  2. Padding step:                                              |
   |      - A_h = A*_h || 0^{n-t},                                  |
   |      - C_q = C*_q || 0^{n-u}.                                  |
   |                                                                |
   |  3. Authentication tag T generation step:                      |
   |      - Z_1 = E_K(1^1 || ICN),                                  |
   |      - sum = 0,                                                |
   |      - For i = 1, 2, ..., h do                                 |
   |              H_i = E_K(Z_i),                                   |
   |              sum = sum (xor) ( H_i (x) A_i ),                  |
   |              Z_{i+1} = incr_l(Z_i),                            |
   |      - For j = 1, 2, ..., q do                                 |
   |              H_{h+j} = E_K(Z_{h+j}),                           |
   |              sum = sum (xor) ( H_{h+j} (x) C_j ),              |
   |              Z_{h+j+1} = incr_l(Z_{h+j}),                      |
   |      - H_{h+q+1} = E_K(Z_{h+q+1}),                             |
   |      - T = MSB_S(E_K(sum (xor) ( H_{h+q+1} (x)                 |
   |                       ( len(A) || len(C) ) ))).                |
   |                                                                |
   |  4. Return (ICN, A, C, T).                                     |
   +----------------------------------------------------------------+


   The ICN value for each message that is encrypted under the given key
   K must be chosen in a unique manner.

   Users who do not wish to encrypt plaintext can provide a string P of
   zero length.  Users who do not wish to authenticate associated data
   can provide a string A of zero length.  The length of the associated
   data A and of the plaintext P MUST be such that 0 < |A| + |P| <
   2^{n/2}.




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4.2.  MGM Decryption and Tag Verification Check Procedure

   The MGM decryption and tag verification procedure takes the following
   parameters as inputs:

   1.  The encryption key K in V_k.

   2.  The initial counter nonce ICN in V_{n-1}.

   3.  The associated authenticated data A, 0 <= |A| < 2^{n/2}. A =
       A_1 || ... || A*_h, A_j in V_n, for j = 1, ... , h - 1, A*_h in
       V_t, 1 <= t <= n.  If |A| = 0, then by definition A*_h is empty,
       and the h and t parameters are set as follows: h = 0, t = n.  The
       associated data is authenticated but is not encrypted.

   4.  The ciphertext C, 0 <= |C| < 2^{n/2}. C = C_1 || ... || C*_q, C_i
       in V_n, for i = 1, ... , q - 1, C*_q in V_u, 1 <= u <= n.  If |C|
       = 0, then by definition C*_q is empty, and the q and u parameters
       are set as follows: q = 0, u = n.

   5.  The authenticated tag T in V_S.

   The MGM decryption and tag verification procedure outputs FAIL or the
   following parameters:

   1.  Associated authenticated data A.

   2.  Plaintext P in V_{|C|}.

   The MGM decryption and tag verification procedure consists of the
   following steps:




















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   +----------------------------------------------------------------+
   |  MGM-Decrypt(K, ICN, A, C, T)                                  |
   |----------------------------------------------------------------|
   |  1. Padding step:                                              |
   |      - A_h = A*_h || 0^{n-t},                                  |
   |      - C_q = C*_q || 0^{n-u}.                                  |
   |                                                                |
   |  2. Authentication tag T verification step:                    |
   |      - Z_1 = E_K(1^1 || ICN),                                  |
   |      - sum = 0,                                                |
   |      - For i = 1, 2, ..., h do                                 |
   |              H_i = E_K(Z_i),                                   |
   |              sum = sum (xor) ( H_i (x) A_i ),                  |
   |              Z_{i+1} = incr_l(Z_i),                            |
   |      - For j = 1,  2, ..., q do                                |
   |              H_{h+j} = E_K(Z_{h+j}),                           |
   |              sum = sum (xor) ( H_{h+j} (x) C_j ),              |
   |              Z_{h+j+1} = incr_l(Z_{h+j}),                      |
   |      - H_{h+q+1} = E_K(Z_{h+q+1}),                             |
   |      - T' = MSB_S(E_K(sum (xor) ( H_{h+q+1} (x)                |
   |                       ( len(A) || len(C) ) ))),                |
   |      - If T' != T then return FAIL.                            |
   |                                                                |
   |  3. Decryption step:                                           |
   |      - if |C| = 0 then                                         |
   |            - P = C                                             |
   |      - else                                                    |
   |            - Y_1 = E_K(0^1 || ICN),                            |
   |            - For i = 2, 3, ... , q do                          |
   |                    Y_i = incr_r(Y_{i-1}),                      |
   |            - For i = 1, 2, ... , q - 1 do                      |
   |                    P_i = C_i (xor) E_K(Y_i),                   |
   |            - P*_q = C*_q (xor) MSB_u(E_K(Y_q)),                |
   |            - P = P_1 || ... || P*_q.                           |
   |                                                                |
   |  4. Return (A, P).                                             |
   +----------------------------------------------------------------+


   The length of the associated data A and of the ciphertext C MUST be
   such that 0 < |A| + |C| < 2^{n/2}.

5.  Rationale

   The MGM was originally proposed in [PDMODE].






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   From the operational point of view the MGM is designed to be
   parallelizable, inverse free, online and to provide availability of
   precomputations.

   Parallelizability of the MGM is achieved due to its counter-type
   structure and the usage of the multilinear function for
   authentication.  Indeed, both encryption blocks E_K(Y_i) and
   authentication blocks H_i are produced in the counter mode manner,
   and the multilinear function determined by H_i is parallelizable in
   itself.  Additionally, the counter-type structure of the mode
   provides the inverse free property.

   The online property means the possibility to process message even if
   it is not completely received (so its length is unknown).  To provide
   this property the MGM uses blocks E_K(Y_i) and H_i which are produced
   basing on two independent source blocks Y_i and Z_i.

   Availability of precomputations for the MGM means the possibility to
   calculate H_i and E_K(Y_i) even before data is retrieved.  It is
   holds due to again the usage of counters for calculating them.

6.  Security Considerations

   The security properties of the MGM are based on the following:

   o  Different functions generating the counter values:
      The functions incr_r and incr_l are chosen to minimize
      intersection (if it happens) of counter values Y_i and Z_i.

   o  Encryption of the multilinear function output:
      It allows to resist attacks based on padding and linear properties
      (see [Ferg05] for details).

   o  Multilinear function for authentication:
      It allows to resist the small subgroup attacks [Saar12].

   o  Encryption of the nonces (0^1 || ICN) and (1^1 || ICN):
      The use of this encryption minimizes the number of plaintext/
      ciphertext pairs of blocks known to an adversary.  It allows to
      resist attacks that need substantial amount of such material
      (e.g., linear and differential cryptanalysis, side-channel
      attacks).

   It is crucial to the security of MGM to use unique ICN values.  Using
   the same ICN values for two different messages encrypted with the
   same key eliminates the security properties of this mode.





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   Security analysis for MGM with E_K being a random permutation was
   performed in [SecMGM].  More precisely, the bounds for
   confidentiality advantage (CA) and integrity advantage (IA) (for
   details see [I-D.wood-cfrg-aead-limits]) were obtained.  According to
   these results, for an adversary making at most q encryption queries
   with the total length of plaintexts and associated data of at most s
   blocks and allowed to output a forgery with the summary length of
   ciphertext and associated data of at most l blocks:

      CA <= ( 3( s + 4q )^2 )/ 2^n,

      IA <= ( 3( s + 4q + l + 3 )^2 )/ 2^n + 2/2^S,

   where n is the block size and S is the authentication tag size.

   These bounds can be used as guidelines on how to calculate
   confidentiality and integrity limits (for details also see
   [I-D.wood-cfrg-aead-limits]).

7.  IANA Considerations

   This document does not require any IANA actions.

8.  References

8.1.  Normative References

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

   [RFC7801]  Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher
              "Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016,
              <https://www.rfc-editor.org/info/rfc7801>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

8.2.  Informative References

   [Ferg05]   Ferguson, N., "Authentication weaknesses in GCM", 2005.

   [GOST3412-2015]
              Federal Agency on Technical Regulating and Metrology,
              "Information technology. Cryptographic data security.
              Block ciphers", GOST R 34.12-2015, 2015.



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   [I-D.dolmatov-magma]
              Dolmatov, V. and D. Eremin-Solenikov, "GOST R 34.12-2015:
              Block Cipher "Magma"", draft-dolmatov-magma-06 (work in
              progress), March 2020.

   [I-D.wood-cfrg-aead-limits]
              Guenther, F., Thomson, M., and C. Wood, "Usage Limits on
              AEAD Algorithms", draft-wood-cfrg-aead-limits-00 (work in
              progress), May 2020.

   [PDMODE]   Nozdrunov, V., "Parallel and double block cipher mode of
              operation (PD-mode) for authenticated encryption", CTCrypt
              2017 proceedings, pp. 36-45, 2017.

   [R1323565.1.026-2019]
              Federal Agency on Technical Regulating and Metrology,
              "Information technology. Cryptographic data security.
              Authenticated encryption block cipher operation modes",
              R 1323565.1.026-2019, 2019.

   [Saar12]   Saarinen, O., "Cycling Attacks on GCM, GHASH and Other
              Polynomial MACs and Hashes", FSE 2012 proceedings, pp.
              216-225, 2012.

   [SecMGM]   Akhmetzyanova, L., Alekseev, E., Karpunin, G. and V.
              Nozdrunov, "Security of Multilinear Galois Mode (MGM).",
              IACR Cryptology ePrint Archive 2019, p. 123, 2019.

Appendix A.  Test Vectors

A.1.  Test Vectors for the Kuznyechik block cipher

   Test vectors for the Kuznyechik block cipher (n = 128, k = 256)
   defined in [GOST3412-2015] (the English version can be found in
   [RFC7801]).


   -------------------------Example 1--------------------------

   Encryption key K:
   00000:   88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
   00010:   FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF

   ICN:
   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Associated authenticated data A:
   00000:   02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01



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   00010:   04 04 04 04 04 04 04 04 03 03 03 03 03 03 03 03
   00020:   EA 05 05 05 05 05 05 05 05

   Plaintext P:
   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88
   00010:   00 11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A
   00020:   11 22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00
   00030:   22 33 44 55 66 77 88 99 AA BB CC EE FF 0A 00 11
   00040:   AA BB CC

   1. Encryption step:

   0^1 || ICN:
   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Y_1:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CD
   E_K(Y_1):
   00000:   B8 57 48 C5 12 F3 19 90 AA 56 7E F1 53 35 DB 74

   Y_2:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CE
   E_K(Y_2):
   00000:   80 64 F0 12 6F AC 9B 2C 5B 6E AC 21 61 2F 94 33

   Y_3:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED CF
   E_K(Y_3):
   00000:   58 58 82 1D 40 C0 CD 0D 0A C1 E6 C2 47 09 8F 1C

   Y_4:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED D0
   E_K(Y_4):
   00000:   E4 3F 50 81 B5 8F 0B 49 01 2F 8E E8 6A CD 6D FA

   Y_5:
   00000:   7F 67 9D 90 BE BC 24 30 5A 46 8D 42 B9 D4 ED D1
   E_K(Y_5):
   00000:   86 CE 9E 2A 0A 12 25 E3 33 56 91 B2 0D 5A 33 48

   C:
   00000:   A9 75 7B 81 47 95 6E 90 55 B8 A3 3D E8 9F 42 FC
   00010:   80 75 D2 21 2B F9 FD 5B D3 F7 06 9A AD C1 6B 39
   00020:   49 7A B1 59 15 A6 BA 85 93 6B 5D 0E A9 F6 85 1C
   00030:   C6 0C 14 D4 D3 F8 83 D0 AB 94 42 06 95 C7 6D EB
   00040:   2C 75 52

   2. Padding step:



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   A_1 || ... || A_h:
   00000:   02 02 02 02 02 02 02 02 01 01 01 01 01 01 01 01
   00010:   04 04 04 04 04 04 04 04 03 03 03 03 03 03 03 03
   00020:   EA 05 05 05 05 05 05 05 05 00 00 00 00 00 00 00

   C_1 || ... || C_q:
   00000:   A9 75 7B 81 47 95 6E 90 55 B8 A3 3D E8 9F 42 FC
   00010:   80 75 D2 21 2B F9 FD 5B D3 F7 06 9A AD C1 6B 39
   00020:   49 7A B1 59 15 A6 BA 85 93 6B 5D 0E A9 F6 85 1C
   00030:   C6 0C 14 D4 D3 F8 83 D0 AB 94 42 06 95 C7 6D EB
   00040:   2C 75 52 00 00 00 00 00 00 00 00 00 00 00 00 00


   3. Authentication tag T generation step:

   1^1 || ICN:
   00000:   91 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Z_1:
   00000:   7F C2 45 A8 58 6E 66 02 A7 BB DB 27 86 BD C6 6F
   H_1:
   00000:   8D B1 87 D6 53 83 0E A4 BC 44 64 76 95 2C 30 0B
   current sum:
   00000:   4C F4 27 F4 AD B7 5C F4 C0 DA 39 D5 AB 48 CF 38

   Z_2:
   00000:   7F C2 45 A8 58 6E 66 03 A7 BB DB 27 86 BD C6 6F
   H_2:
   00000:   7A 24 F7 26 30 E3 76 37 21 C8 F3 CD B1 DA 0E 31
   current sum:
   00000:   94 95 44 0E F6 24 A1 DD C6 F5 D9 77 28 50 C5 73

   Z_3:
   00000:   7F C2 45 A8 58 6E 66 04 A7 BB DB 27 86 BD C6 6F
   H_3:
   00000:   44 11 96 21 17 D2 06 35 C5 25 E0 A2 4D B4 B9 0A
   current sum:
   00000:   A4 9A 8C D8 A6 F2 74 23 DB 79 E4 4A B3 06 D9 42

   Z_4:
   00000:   7F C2 45 A8 58 6E 66 05 A7 BB DB 27 86 BD C6 6F
   H_4:
   00000:   D8 C9 62 3C 4D BF E8 14 CE 7C 1C 0C EA A9 59 DB
   current sum:
   00000:   09 FE 3F 6A 83 3C 21 B3 90 27 D0 20 6A 84 E1 5A

   Z_5:
   00000:   7F C2 45 A8 58 6E 66 06 A7 BB DB 27 86 BD C6 6F



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   H_5:
   00000:   A5 E1 F1 95 33 3E 14 82 96 99 31 BF BE 6D FD 43
   current sum:
   00000:   B5 DA 26 BB 00 EB A8 04 35 D7 97 6B C6 B5 46 4D

   Z_6:
   00000:   7F C2 45 A8 58 6E 66 07 A7 BB DB 27 86 BD C6 6F
   H_6:
   00000:   B4 CA 80 8C AC CF B3 F9 17 24 E4 8A 2C 7E E9 D2
   current sum:
   00000:   DD 1C 0E EE F7 83 C8 EB 2A 33 F3 58 D7 23 0E E5

   Z_7:
   00000:   7F C2 45 A8 58 6E 66 08 A7 BB DB 27 86 BD C6 6F
   H_7:
   00000:   72 90 8F C0 74 E4 69 E8 90 1B D1 88 EA 91 C3 31
   current sum:
   00000:   89 6C E1 08 32 EB EA F9 06 9F 3F 73 76 59 4D 40

   Z_8:
   00000:   7F C2 45 A8 58 6E 66 09 A7 BB DB 27 86 BD C6 6F
   H_8:
   00000:   23 CA 27 15 B0 2C 68 31 3B FD AC B3 9E 4D 0F B8
   current sum:
   00000:   99 1A F5 C9 D0 80 F7 63 87 FE 64 9E 7C 93 C6 42

   Z_9:
   00000:   7F C2 45 A8 58 6E 66 0A A7 BB DB 27 86 BD C6 6F
   H_9:
   00000:   BC BC E6 C4 1A A3 55 A4 14 88 62 BF 64 BD 83 0D
   len(A) || len(C):
   00000:   00 00 00 00 00 00 01 48 00 00 00 00 00 00 02 18
   sum (xor) ( H_9 (x) ( len(A) || len(C) ) ):
   00000:   C0 C7 22 DB 5E 0B D6 DB 25 76 73 83 3D 56 71 28


   Tag T:
   00000:   CF 5D 65 6F 40 C3 4F 5C 46 E8 BB 0E 29 FC DB 4C



   -------------------------Example 2--------------------------

   Encryption key K:
   00000:   99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 FE
   00010:   DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 88

   ICN:



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   00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Associated authenticated data A:
   00000:   01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

   Plaintext P:
   00000:

   1. Encryption step:

   C:
   00000:

   2. Padding step:

   A_1 || ... || A_h:
   00000:   01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

   C_1 || ... || C_q:
   00000:


   3. Authentication tag T generation step:

   1^1 || ICN:
   00000:   91 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

   Z_1:
   00000:   79 32 72 68 96 C4 3E 3F BF D6 50 89 EB F1 E5 B6
   H_1:
   00000:   99 3A 80 66 CC C0 A4 0F AC 4A 14 F7 A2 F6 6D 9B
   current sum:
   00000:   0A C1 1E 2C 1C D6 07 D8 2F E3 55 54 B4 01 02 81

   Z_2:
   00000:   79 32 72 68 96 C4 3E 40 BF D6 50 89 EB F1 E5 B6
   H_2:
   00000:   0C 38 A7 1E E7 93 BF 76 89 81 BF CD 7C DA 78 C8
   len(A) || len(C):
   00000:   00 00 00 00 00 00 00 80 00 00 00 00 00 00 00 00
   sum (xor) ( H_2 (x) ( len(A) || len(C) ) ):
   00000:   CA 1E F8 92 71 EA 60 C4 53 9E 40 EB 26 C2 80 5D

   Tag T:
   00000:   79 01 E9 EA 20 85 CD 24 7E D2 49 69 5F 9F 8A 85






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A.2.  Test Vectors for the Magma block cipher

   Test vectors for the Magma block cipher (n = 64, k = 256) defined in
   [GOST3412-2015] (the English version can be found in
   [I-D.dolmatov-magma]).


   -------------------------Example 1--------------------------

   Encryption key K:
   00000:   FF EE DD CC BB AA 99 88 77 66 55 44 33 22 11 00
   00010:   F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 FA FB FC FD FE FF

   ICN:
   00000:   12 DE F0 6B 3C 13 0A 59

   Associated authenticated data A:
   00000:   01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02
   00010:   03 03 03 03 03 03 03 03 04 04 04 04 04 04 04 04
   00020:   05 05 05 05 05 05 05 05 EA

   Plaintext P:
   00000:   FF EE DD CC BB AA 99 88 11 22 33 44 55 66 77 00
   00010:   88 99 AA BB CC EE FF 0A 00 11 22 33 44 55 66 77
   00020:   99 AA BB CC EE FF 0A 00 11 22 33 44 55 66 77 88
   00030:   AA BB CC EE FF 0A 00 11 22 33 44 55 66 77 88 99
   00040:   AA BB CC

   1. Encryption step:

   0^1 || ICN:
   00000:   12 DE F0 6B 3C 13 0A 59

   Y_1:
   00000:   56 23 89 01 62 DE 31 BF
   E_K(Y_1):
   00000:   38 7B DB A0 E4 34 39 B3

   Y_2:
   00000:   56 23 89 01 62 DE 31 C0
   E_K(Y_2):
   00000:   94 33 00 06 10 F7 F2 AE

   Y_3:
   00000:   56 23 89 01 62 DE 31 C1
   E_K(Y_3):
   00000:   97 B7 AA 6D 73 C5 87 57




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   Y_4:
   00000:   56 23 89 01 62 DE 31 C2
   E_K(Y_4):
   00000:   94 15 52 8B FF C9 E8 0A

   Y_5:
   00000:   56 23 89 01 62 DE 31 C3
   E_K(Y_5):
   00000:   03 F7 68 BF F1 82 D6 70

   Y_6:
   00000:   56 23 89 01 62 DE 31 C4
   E_K(Y_6):
   00000:   FD 05 F8 4E 9B 09 D2 FE

   Y_7:
   00000:   56 23 89 01 62 DE 31 C5
   E_K(Y_7):
   00000:   DA 4D 90 8A 95 B1 75 C4

   Y_8:
   00000:   56 23 89 01 62 DE 31 C6
   E_K(Y_8):
   00000:   65 99 73 96 DA C2 4B D7

   Y_9:
   00000:   56 23 89 01 62 DE 31 C7
   E_K(Y_9):
   00000:   A9 00 50 4A 14 8D EE 26

   C:
   00000:   C7 95 06 6C 5F 9E A0 3B 85 11 33 42 45 91 85 AE
   00010:   1F 2E 00 D6 BF 2B 78 5D 94 04 70 B8 BB 9C 8E 7D
   00020:   9A 5D D3 73 1F 7D DC 70 EC 27 CB 0A CE 6F A5 76
   00030:   70 F6 5C 64 6A BB 75 D5 47 AA 37 C3 BC B5 C3 4E
   00040:   03 BB 9C

   2. Padding step:

   A_1 || ... || A_h:
   00000:   01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02
   00010:   03 03 03 03 03 03 03 03 04 04 04 04 04 04 04 04
   00020:   05 05 05 05 05 05 05 05 EA 00 00 00 00 00 00 00

   C_1 || ... || C_q:
   00000:   C7 95 06 6C 5F 9E A0 3B 85 11 33 42 45 91 85 AE
   00010:   1F 2E 00 D6 BF 2B 78 5D 94 04 70 B8 BB 9C 8E 7D
   00020:   9A 5D D3 73 1F 7D DC 70 EC 27 CB 0A CE 6F A5 76



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   00030:   70 F6 5C 64 6A BB 75 D5 47 AA 37 C3 BC B5 C3 4E
   00040:   03 BB 9C 00 00 00 00 00


   3. Authentication tag T generation step:

   1^1 || ICN:
   00000:   92 DE F0 6B 3C 13 0A 59

   Z_1:
   00000:   2B 07 3F 04 94 F3 72 A0
   H_1:
   00000:   70 8A 78 19 1C DD 22 AA
   current sum:
   00000:   D6 BB 5B EA 81 93 12 62

   Z_2:
   00000:   2B 07 3F 05 94 F3 72 A0
   H_2:
   00000:   6F 02 CC 46 4B 2F A0 A3
   current sum:
   00000:   DD 1C 82 4E 91 78 49 A5

   Z_3:
   00000:   2B 07 3F 06 94 F3 72 A0
   H_3:
   00000:   9F 81 F2 26 FD 19 6F 05
   current sum:
   00000:   05 89 22 17 F6 5A DA C7

   Z_4:
   00000:   2B 07 3F 07 94 F3 72 A0
   H_4:
   00000:   B9 C2 AC 9B E5 B5 DF F9
   current sum:
   00000:   D1 DB 9B 7F C4 9E 7C 97

   Z_5:
   00000:   2B 07 3F 08 94 F3 72 A0
   H_5:
   00000:   74 B5 EC 96 55 1B F8 88
   current sum:
   00000:   56 45 F6 B5 18 5C B7 1A

   Z_6:
   00000:   2B 07 3F 09 94 F3 72 A0
   H_6:
   00000:   7E B0 21 A4 03 5B 04 C3



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   current sum:
   00000:   3F C2 C2 E6 FB EE D0 4D

   Z_7:
   00000:   2B 07 3F 0A 94 F3 72 A0
   H_7:
   00000:   C2 A9 C3 A8 70 4D 9B B0
   current sum:
   00000:   15 47 1F B5 CD 8E 6C 02

   Z_8:
   00000:   2B 07 3F 0B 94 F3 72 A0
   H_8:
   00000:   F5 D5 05 A8 7B 83 83 B5
   current sum:
   00000:   12 56 78 96 1D 40 E0 93

   Z_9:
   00000:   2B 07 3F 0C 94 F3 72 A0
   H_9:
   00000:   F7 95 E7 5F DE B8 93 3C
   current sum:
   00000:   6E F4 0A B0 C1 5F 20 48

   Z_10:
   00000:   2B 07 3F 0D 94 F3 72 A0
   H_10:
   00000:   65 A1 A3 E6 80 F0 81 45
   current sum:
   00000:   A4 64 A7 08 FF 45 14 22

   Z_11:
   00000:   2B 07 3F 0E 94 F3 72 A0
   H_11:
   00000:   1C 74 A5 76 4C B0 D5 95
   current sum:
   00000:   60 94 4E 05 D0 85 75 14

   Z_12:
   00000:   2B 07 3F 0F 94 F3 72 A0
   H_12:
   00000:   DC 84 47 A5 14 E7 83 E7
   current sum:
   00000:   EE 98 B9 B5 0F F7 83 E8

   Z_13:
   00000:   2B 07 3F 10 94 F3 72 A0
   H_13:



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   00000:   A7 E3 AF E0 04 EE 16 E3
   current sum:
   00000:   C0 39 0F A2 28 AF 6D CB

   Z_14:
   00000:   2B 07 3F 11 94 F3 72 A0
   H_14:
   00000:   A5 AA BB 0B 79 80 D0 71
   current sum:
   00000:   73 E0 6E 07 EF 37 CD CC

   Z_15:
   00000:   2B 07 3F 12 94 F3 72 A0
   H_15:
   00000:   6E 10 4C C9 33 52 5C 5D
   current sum:
   00000:   2F 40 69 0A EB 53 F5 39

   Z_16:
   00000:   2B 07 3F 13 94 F3 72 A0
   H_16:
   00000:   83 11 B6 02 4A A9 66 C1
   len(A) || len(C):
   00000:   00 00 01 48 00 00 02 18
   sum (xor) ( H_16 (x) ( len(A) || len(C) ) ):
   00000:   73 CE F4 4B AE 6B DB 61


   Tag T:
   00000:   A7 92 80 69 AA 10 FD 10



   -------------------------Example 2--------------------------

   Encryption key K:
   00000:   99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 FE
   00010:   DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 88

   ICN:
   00000:   00 77 66 55 44 33 22 11

   Associated authenticated data A:
   00000:

   Plaintext P:
   00000:   22 33 44 55 66 77 00 FF




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   1. Encryption step:

   0^1 || ICN:
   00000:   00 77 66 55 44 33 22 11

   Y_1:
   00000:   5B 2A 7E 60 4F 9F BB 95
   E_K(Y_1):
   00000:   48 A6 A5 17 0D 52 9D B1

   C:
   00000:   6A 95 E1 42 6B 25 9D 4E

   2. Padding step:

   A_1 || ... || A_h:
   00000:

   C_1 || ... || C_q:
   00000:   6A 95 E1 42 6B 25 9D 4E


   3. Authentication tag T generation step:

   1^1 || ICN:
   00000:   80 77 66 55 44 33 22 11

   Z_1:
   00000:   59 73 54 78 7E 52 E6 EB
   H_1:
   00000:   EC E3 F9 DA 11 8C 7D 95
   current sum:
   00000:   25 D0 E4 20 7B 6B F6 3D

   Z_2:
   00000:   59 73 54 79 7E 52 E6 EB
   H_2:
   00000:   31 0C 0D AC C9 D0 4D 93
   len(A) || len(C):
   00000:   00 00 00 00 00 00 00 40
   sum (xor) ( H_2 (x) ( len(A) || len(C) ) ):
   00000:   66 D3 8F 12 0F 78 92 49


   Tag T:
   00000:   33 4E E2 70 45 0B EC 9E





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Appendix B.  Contributors

   o  Evgeny Alekseev
      CryptoPro
      alekseev@cryptopro.ru

   o  Alexandra Babueva
      CryptoPro
      babueva@cryptopro.ru

   o  Lilia Akhmetzyanova
      CryptoPro
      lah@cryptopro.ru

   o  Grigory Marshalko
      TC 26
      marshalko_gb@tc26.ru

   o  Vladimir Rudskoy
      TC 26
      rudskoy_vi@tc26.ru

   o  Alexey Nesterenko
      National Research University Higher School of Economics
      anesterenko@hse.ru

   o  Lidia Nikiforova
      CryptoPro
      nikiforova@cryptopro.ru

Authors' Addresses

   Stanislav Smyshlyaev (editor)
   CryptoPro

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


   Vladislav Nozdrunov
   TC 26

   Email: nozdrunov_vi@tc26.ru








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   Vasily Shishkin
   TC 26

   Email: shishkin_va@tc26.ru


   Ekaterina Griboedova
   CryptoPro

   Email: griboedova.e.s@gmail.com









































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