Multilinear Galois Mode (MGM)
draft-smyshlyaev-mgm-07

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
• 1.1. Existing Constructions
• 2. Conventions Used in This Document
• 3. Basic Terms and Definitions
• 4. Specification
• 4.1. MGM Encryption and Authentication Procedure
• 4.2. MGM Decryption and Authentication Check Procedure
• 5. Rationale
• 6. References
• 6.1. Normative References
• 6.2. Informative References
• Appendix A. Test Vectors
• Appendix B. Contributors
• Authors' Addresses

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;

|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;

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:

                            
+----------------------------------------------------------------+
|  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),                                  |
|      - 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)))).       |
|                                                                |
|  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}.

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:

                            
+----------------------------------------------------------------+
|  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 MGM mode our task was to create a fast, parallelizable, inverse free, online and secure block cipher mode. MGM is based on counters for reasons of performance. The first counter is used for message encryption, the second counter is used for authentication.

For providing parallelizable authentication we use multilinear function. By encrypting 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

6.2. Informative References

Appendix A. Test Vectors

The text will be added in the future versions of the draft.

Appendix B. Contributors

• Evgeny Alekseev
  CryptoPro
  alekseev@cryptopro.ru
• Ekaterina Smyshlyaeva
  CryptoPro
  ess@cryptopro.ru
• Lilia Ahmetzyanova
  CryptoPro
  lah@cryptopro.ru
• Grigory Marshalko
  TC 26
  marshalko_gb@tc26.ru
• Vladimir Rudskoy
  TC 26
  rudskoy_vi@tc26.ru

Authors' Addresses