Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-09

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Network Working Group                                 S. Smyshlyaev, Ed.
Internet-Draft                                                 CryptoPro
Intended status: Informational                              V. Nozdrunov
Expires: April 22, 2019                                      V. Shishkin
                                                                   TC 26
                                                        October 19, 2018

                     Multilinear Galois Mode (MGM)
                        draft-smyshlyaev-mgm-09

Abstract

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

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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Existing Constructions  . . . . . . . . . . . . . . . . .   2
   2.  Conventions Used in This Document . . . . . . . . . . . . . .   2
   3.  Basic Terms and Definitions . . . . . . . . . . . . . . . . .   2
   4.  Specification . . . . . . . . . . . . . . . . . . . . . . . .   4
     4.1.  MGM Encryption and Authentication Procedure . . . . . . .   4
     4.2.  MGM Decryption and Authentication Check Procedure . . . .   6
   5.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .   7
   6.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     6.1.  Normative References  . . . . . . . . . . . . . . . . . .   8
     6.2.  Informative References  . . . . . . . . . . . . . . . . .   8
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .   8
   Appendix B.  Contributors . . . . . . . . . . . . . . . . . . . .  12
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  12

1.  Introduction

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

1.1.  Existing Constructions

   The text will be added in the future versions of the draft.

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].

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;

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   |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: a^s = (a, a,
           ... , a), 'a' in V_1;

   (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 a string X =
           (x_{s-1}, ... , x_0) in V_s 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);

   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)     multiplication in 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 = x^64 + x^4 + x^3 + x + 1; if n = 128, then the
           field polynomial is equal to f = x^128 + x^7 + x^2 + x + 1;

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   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 MGM mode is
   the size S of the authentication field (in bits).  The value of S
   MUST be fixed for a particular protocol, 32 <= S <= 128.  The choice
   of the value S involves a trade-off between message expansion and the
   probability that an attacker can modify a message undetectably.

4.1.  MGM Encryption and Authentication Procedure

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

   1.  Encryption key K in V_k.

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

   3.  Plaintext P, 0 <= |P| < 2^{n/2}. If |P| > 0, then P = P_1 ||
       ... || P*_q, P_i in V_n, i = 1, ... , q - 1, P*_q in V_u, 1 <= u
       <= n.  If |P| = 0, then by definition P*_q is empty, q = 0, and u
       = n.

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

   The MGM encryption and authentication 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.

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   The MGM encryption and authentication procedure consists of the
   following steps:

   +----------------------------------------------------------------+
   |  MGM-Encrypt(K, ICN, P, A)                                     |
   |----------------------------------------------------------------|
   |  1. Encryption step:                                           |
   |      - 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.  Using the same ICN values for
   two different messages encrypted with the same key eliminates the
   security properties of this mode.

   Users who do not wish to encrypt plaintext can provide a string P of
   length zero.  Users who do not wish to authenticate associated data
   can provide a string A of length zero.  The length of the associated

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   data A and of the plaintext P MUST be such that 0 < |A| + |P| <
   2^{n/2}.

4.2.  MGM Decryption and Authentication Check Procedure

   The MGM decryption and authentication 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, j = 1, ... , h - 1, A*_h in V_t,
       1 <= t <= n.

   4.  The ciphertext C, 0 <= |C| < 2^{n/2}. C = C_1 || ... || C*_q, C_i
       in V_n, i = 1, ... , q - 1, C*_q in V_u, 1 <= u <= n.

   5.  The authenticated tag T in V_S.

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

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

   2.  Associated authenticated data A.

   The MGM decryption and authentication 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' generation step:                     |
   |      - Z_1 = E_K(1^1 || ICN),                                  |
   |      - sum1 = 0, sum2 = 0,                                     |
   |      - For i = 1, 2, ..., h do                                 |
   |              H_i = E_K(Z_i),                                   |
   |              sum1 = sum1 (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}),                           |
   |              sum2 = sum2 (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(sum1 (xor) sum2 (xor)                    |
   |                       H_{h+q+1} (x) (len(A) || len(C)))),      |
   |      - If T' != T then return FAIL                             |
   |             return FAIL.                                       |
   |                                                                |
   |  3. Decryption step:                                           |
   |      - 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 (P, A).                                             |
   |----------------------------------------------------------------+

5.  Rationale

   The MGM mode was originally proposed in [PDMODE].

   The MGM mode is designed to be fast, parallelizable, inverse free,
   online and secure.

   The MGM is based on counters for the reasons of performance.  The
   first counter (Y_i, see Section 4.1) is used for message encryption,
   the second counter (H_i, see Section 4.1) is used for authentication.
   The second counter is encrypted eliminating the chance of obtaining

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   any information about the H_k value in case when the H_l value is
   known to the adversary ( here l is not equal to k ).

   To provide parallelizable authentication a multilinear function is
   used.

   To avoid attacks based on padding and linear properties of
   multilinear function the lengths of associated data A, encrypted
   message C, and encrypting authentication tag is added.

   A collision of "usual" counters leads to obtaining the information
   about the H_i values and possible authentication vulnerabilities.  To
   minimize the probability of this event we change the principle of
   counters operating by using the functions incr_l and incr_r.  To
   counteract finding collisions we encrypt initial values of both
   counters.

6.  References

6.1.  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>.

6.2.  Informative References

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

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

Appendix A.  Test Vectors

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

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   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

   Associated authenticated data A:
   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

   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

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   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:

   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

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   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
   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

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Appendix B.  Contributors

   o  Evgeny Alekseev
      CryptoPro
      alekseev@cryptopro.ru

   o  Ekaterina Smyshlyaeva
      CryptoPro
      ess@cryptopro.ru

   o  Lilia Ahmetzyanova
      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

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

   Vasily Shishkin
   TC 26

   Email: shishkin_va@tc26.ru

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