Multilinear Galois Mode (MGM)
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General
Network Working Groupauthenticated encryption, mode of operation, AEAD
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.
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.
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
.
This document uses the following terms and definitions for the sets and operations
on the elements of these sets:
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;
the set of all bit strings of length s, where s is a non-negative integer;
the bit length of the bit string X (if X is an empty string, then |X| = 0);
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;
the string in V_s that consists of s 'a' bits: a^s = (a, a, ... , a), 'a' in V_1;
exclusive-or of the two bit strings of the same length,
ring of residues modulo 2^s;
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);
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);
the transformation inverse to the mapping Int_s (the interpretation of an integer as a bit string);
the block cipher permutation under the key K in V_k;
the bit length of the block cipher key;
the block size of the block cipher (in bits);
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;
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;
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;
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).
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.
The MGM encryption and authentication procedure takes the following parameters as inputs:
Encryption key K in V_k.
Initial counter nonce ICN in V_{n-1}.
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.
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:
Initial counter nonce ICN.Associated authenticated data A.Ciphertext C in V_{|P|}.Authentication tag T in V_S.
The MGM encryption and authentication procedure consists of the following steps:
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}.
The MGM decryption and authentication procedure takes the following parameters as inputs:
The encryption key K in V_k.The initial counter nonce ICN in V_{n-1}.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.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.The authenticated tag T in V_S.
The MGM decryption and authentication procedure outputs FAIL or the following parameters:
Plaintext P in V_{|C|}.Associated authenticated data A.
The MGM decryption and authentication procedure consists of the following steps:
The MGM mode was originally proposed in .
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, 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:
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.
This procedure allows to resist attacks based on padding
and linear properties (see for details).
It allows to resist the small subgroup attacks .
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).
Parallel and double block cipher mode of operation (PD-mode) for authenticated encryption
Nozdrunov, V.
Information technology. Cryptographic data security. Block ciphers
Federal Agency on Technical Regulating and Metrology
Authentication weaknesses in GCM
Ferguson, N.
Cycling Attacks on GCM, GHASH and Other Polynomial MACs and Hashes
Saarinen, O.
Test vectors for the Kuznyechik block cipher (n = 128, k = 256) defined in (the English version can be found in ).
Evgeny Alekseev
CryptoPro
alekseev@cryptopro.ru
Alexandra Babueva
CryptoPro
babueva@cryptopro.ru
Lilia Akhmetzyanova
CryptoPro
lah@cryptopro.ru
Grigory Marshalko
TC 26
marshalko_gb@tc26.ru
Vladimir Rudskoy
TC 26
rudskoy_vi@tc26.ru
Alexey Nesterenko
National Research University Higher School of Economics
anesterenko@hse.ru