Network Working Group S. Smyshlyaev, Ed.
Internet-Draft CryptoPro
Intended status: Informational V. Nozdrunov
Expires: December 31, 2018 V. Shishkin
TC 26
June 29, 2018
Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-08
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 . . . . . . . . . . . . . . . . . . . . 11
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}. P = P_1 || ... || P*_q, P_i in
V_n, i = 1, ... , q - 1, P*_q in V_u, 1 <= u <= n.
4. 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. 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(0^1 || 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
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, 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 mode was originally proposed in [PDMODE].
During the construction of the MGM mode our task was to create a
fast, parallelizable, inverse free, online and secure block cipher
mode. The MGM is based on counters for reasons of performance. The
first counter is used for message encryption, the second counter is
used for authentication.
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For providing parallelizable authentication a multilinear function is
used. By encrypting the second counter we produce elements H_i with
the property that if one knows any information about value H_k he/she
can't obtain any information about value H_l ( l is not equal to k )
besides that H_k is not equal to H_l.
By adding the length of associated data A and encrypted message C and
encrypting authentication tag we avoid attacks based on padding and
linear properties of multilinear function.
A collision of "usual" counters leads to obtaining information about
values H_i, that could be dangerous to authentication. For
minimizing probability of this event we change the principle of
counters operating by using the functions incr_l and incr_l. To
counteract finding collisions we encrypt initialization 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,
.
6.2. Informative References
[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)
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
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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:
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
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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
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
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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
Appendix B. Contributors
o Evgeny Alekseev
CryptoPro
alekseev@cryptopro.ru
o Ekaterina Smyshlyaeva
CryptoPro
ess@cryptopro.ru
o Lilia Ahmetzyanova
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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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