Network Working Group S. Smyshlyaev, Ed.
Internet-Draft CryptoPro
Intended status: Informational V. Nozdrunov
Expires: December 20, 2019 V. Shishkin
TC 26
E. Smyshlyaeva
CryptoPro
June 18, 2019
Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-11
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
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 . . . . . . . . . . . . . . . . . 9
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 9
Appendix B. Contributors . . . . . . . . . . . . . . . . . . . . 12
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 13
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.
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;
|X| the bit length of the bit string X (if X is an empty string,
then |X| = 0);
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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;
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;
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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 bit length S of the authentication tag, 32 <= S <= 128. 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}.
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. 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
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 mode was originally proposed in [PDMODE].
From the operational point of view the MGM mode is designed to be
parallelizable, inverse free, online and to provide availability of
precomputations.
Parallelizability of the MGM mode 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 mode 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 mode 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.
The MGM mode incorporates some mechanisms for advancing cryptographic
properties. Further we note the main ones:
Different functions generating the counter values: The functions
incr_r and incr_l are chosen to minimize intersection (if it
happens) between the sets of counter values Y_i and Z_i.
Ciphering of the multilinear function output: This procedure allows
to resist attacks based on padding and linear properties (see
[Ferg05] for details).
Multilinear function for authentication: It allows to resist the
small subgroup attacks [Saar12].
Ciphering of the nonces (0 || ICN) and (1 || ICN): The aim of this
ciphering is to minimize the number of plaintext/ciphertext pairs
of blocks known to an adversary. Small number of these pairs
allows to resist attacks that need substantial amount of such
material (e.g., linear and differential cryptanalysis, side-
channel attacks).
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,
.
[RFC7801] Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher
"Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016,
.
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6.2. 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.
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
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):
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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:
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
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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
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
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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
Appendix B. Contributors
o Evgeny Alekseev
CryptoPro
alekseev@cryptopro.ru
o Ekaterina Smyshlyaeva
CryptoPro
ess@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
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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
Ekaterina Smyshlyaeva
CryptoPro
Email: ess@cryptopro.ru
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