Network Working Group S. Smyshlyaev, Ed.
Internet-Draft CryptoPro
Intended status: Informational V. Nozdrunov
Expires: September 19, 2020 V. Shishkin
TC 26
E. Smyshlyaeva
CryptoPro
March 18, 2020
Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-17
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
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Copyright (c) 2020 IETF Trust and the persons identified as the
document authors. All rights reserved.
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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 Authentication Procedure . . . . . . . 4
4.2. MGM Decryption and Authentication Check Procedure . . . . 7
5. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6. Security Considerations . . . . . . . . . . . . . . . . . . . 9
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 10
8.1. Normative References . . . . . . . . . . . . . . . . . . 10
8.2. Informative References . . . . . . . . . . . . . . . . . 10
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 10
Appendix B. Contributors . . . . . . . . . . . . . . . . . . . . 14
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 14
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. 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: 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);
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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;
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 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}.
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3. 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.
4. 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.
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.
The MGM encryption and authentication procedure consists of the
following steps:
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+----------------------------------------------------------------+
| MGM-Encrypt(K, ICN, P, A) |
|----------------------------------------------------------------|
| 1. Encryption step: |
| - Y_1 = E_K(0 || 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 || 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 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, for 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, for 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 verification step: |
| - Z_1 = E_K(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: |
| - Y_1 = E_K(0 || 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 was originally proposed in [PDMODE].
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,
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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 || ICN) and (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.
A security analysis for MGM regarding privacy and authenticity was
performed in [SecMGM].
7. IANA Considerations
This document does not require any IANA actions.
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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,
.
[RFC7801] Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher
"Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
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.
[PDMODE] Nozdrunov, V., "Parallel and double block cipher mode of
operation (PD-mode) for authenticated encryption", CTCrypt
2017 proceedings, pp. 36-45, 2017.
[Saar12] Saarinen, O., "Cycling Attacks on GCM, GHASH and Other
Polynomial MACs and Hashes", FSE 2012 proceedings, pp.
216-225, 2012.
[SecMGM] Akhmetzyanova, Liliya R and Alekseev, Evgeny K and
Karpunin, Grigory and Nozdrunov, Vladislav, "Security of
Multilinear Galois Mode (MGM).", IACR Cryptology ePrint
Archive 2019, p. 123, 2019.
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]).
Encryption key K:
00000: 88 99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77
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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
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
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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
H_4:
00000: D8 C9 62 3C 4D BF E8 14 CE 7C 1C 0C EA A9 59 DB
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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 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
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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Ekaterina Smyshlyaeva
CryptoPro
Email: ess@cryptopro.ru
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