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Table of Contents

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 [AUTH-ENC-BLOCK-CIPHER]. 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.¶

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.¶

(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 the string X = (x_{s-1}, ... , x_0) in V_s, s > 0, 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)

The transformation that maps two strings, X = (x_{n-1}, ... , x_0) in V_n and Y = (y_{n-1}, ... , y_0), in V_n into the string Z = X (x) Y = (z_{n-1}, ... , z_0) in V_n; the string Z corresponds to the polynomial Z(w) = z_{n-1} * w^{n-1} + ... + z_1 * w + z_0, which is the result of multiplying the polynomials X(w) = x_{n-1} * w^{n-1} + ... + x_1 * w + x_0 and Y(w) = y_{n-1} * w^{n-1} + ... + y_1 * w + y_0 in the field 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(w) = w^64 + w^4 + w^3 + w + 1; if n = 128, then the field polynomial is equal to f(w) = w^128 + w^7 + w^2 + w + 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).¶

5. Rationale

MGM was originally proposed in [PDMODE].¶

From the operational point of view, MGM is designed to be parallelizable, inverse free, and online and is also designed to provide availability of precomputations.¶

Parallelizability of 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, 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 of processing messages even if it is not completely received (so its length is unknown). To provide this property, MGM uses blocks E_K(Y_i) and H_i, which are produced based on two independent source blocks Y_i and Z_i.¶

Availability of precomputations for MGM means the possibility of calculating H_i and E_K(Y_i) even before data is retrieved. It holds again due to the usage of counters for calculating them.¶

6. Security Considerations

The security properties of MGM are based on the following:¶

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.¶

Encryption of the multilinear function output:

It allows resisting attacks based on padding and linear properties (see [FERG05] for details).¶

Multilinear function for authentication:

It allows resisting the small subgroup attacks [SAAR12].¶

Encryption of the nonces (0^1 || ICN) and (1^1 || ICN):

The use of this encryption minimizes the number of plaintext/ciphertext pairs of blocks known to an adversary. It allows resisting attacks that need a substantial amount of such material (e.g., linear and differential cryptanalysis and 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.¶

It is crucial for the security of MGM not to process empty plaintext and empty associated data at the same time. Otherwise, a tag becomes independent from a nonce value, leading to vulnerability to forgery attacks.¶

Security analysis for MGM with E_K being a random permutation was performed in [SEC-MGM]. More precisely, the bounds for confidentiality advantage (CA) and integrity advantage (IA) (for details, see [AEAD-LIMITS]) were obtained. According to these results, for an adversary making at most q encryption queries with the total length of plaintexts and associated data of at most s blocks, and allowed to output a forgery with the summary length of ciphertext and associated data of at most l blocks:¶

CA <= ( 3( s + 4q )^2 )/ 2^n,¶

IA <= ( 3( s + 4q + l + 3 )^2 )/ 2^n + 2/2^S,¶

where n is the block size and S is the authentication tag size.¶

These bounds can be used as guidelines on how to calculate confidentiality and integrity limits (for details, also see [AEAD-LIMITS]).¶

7. IANA Considerations

This document has no IANA actions.¶

Appendix A. Test Vectors

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

ICN:
00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

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

Encryption key K:
00000:   99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 FE
00010:   DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 88

ICN:
00000:   11 22 33 44 55 66 77 00 FF EE DD CC BB AA 99 88

Associated authenticated data A:
00000:   01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

Plaintext P:
00000:

Encryption key K:
00000:   FF EE DD CC BB AA 99 88 77 66 55 44 33 22 11 00
00010:   F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 FA FB FC FD FE FF

ICN:
00000:   12 DE F0 6B 3C 13 0A 59

Associated authenticated data A:
00000:   01 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02
00010:   03 03 03 03 03 03 03 03 04 04 04 04 04 04 04 04
00020:   05 05 05 05 05 05 05 05 EA

Plaintext P:
00000:   FF EE DD CC BB AA 99 88 11 22 33 44 55 66 77 00
00010:   88 99 AA BB CC EE FF 0A 00 11 22 33 44 55 66 77
00020:   99 AA BB CC EE FF 0A 00 11 22 33 44 55 66 77 88
00030:   AA BB CC EE FF 0A 00 11 22 33 44 55 66 77 88 99
00040:   AA BB CC

Encryption key K:
00000:   99 AA BB CC DD EE FF 00 11 22 33 44 55 66 77 FE
00010:   DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF 88

ICN:
00000:   00 77 66 55 44 33 22 11

Associated authenticated data A:
00000:

Plaintext P:
00000:   22 33 44 55 66 77 00 FF

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