Network Working Group S. Smyshlyaev, Ed. Internet-Draft CryptoPro Intended status: Informational V. Nozdrunov Expires: April 16, 2020 V. Shishkin TC 26 E. Smyshlyaeva CryptoPro October 14, 2019 Multilinear Galois Mode (MGM) draft-smyshlyaev-mgm-12 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. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on April 16, 2020. Copyright Notice Copyright (c) 2019 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of Smyshlyaev, et al. Expires April 16, 2020 [Page 1]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. 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); Smyshlyaev, et al. Expires April 16, 2020 [Page 2]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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; Smyshlyaev, et al. Expires April 16, 2020 [Page 3]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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: Smyshlyaev, et al. Expires April 16, 2020 [Page 4]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 +----------------------------------------------------------------+ | 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}. Smyshlyaev, et al. Expires April 16, 2020 [Page 5]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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: Smyshlyaev, et al. Expires April 16, 2020 [Page 6]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 +----------------------------------------------------------------+ | 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, Smyshlyaev, et al. Expires April 16, 2020 [Page 7]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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, <https://www.rfc-editor.org/info/rfc2119>. [RFC7801] Dolmatov, V., Ed., "GOST R 34.12-2015: Block Cipher "Kuznyechik"", RFC 7801, DOI 10.17487/RFC7801, March 2016, <https://www.rfc-editor.org/info/rfc7801>. Smyshlyaev, et al. Expires April 16, 2020 [Page 8]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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): Smyshlyaev, et al. Expires April 16, 2020 [Page 9]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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 Smyshlyaev, et al. Expires April 16, 2020 [Page 10]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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 Smyshlyaev, et al. Expires April 16, 2020 [Page 11]

Internet-Draft Multilinear Galois Mode (MGM) October 2019 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 Alexandra Babueva CryptoPro babueva@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 Smyshlyaev, et al. Expires April 16, 2020 [Page 12]

```
Internet-Draft Multilinear Galois Mode (MGM) October 2019
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
Smyshlyaev, et al. Expires April 16, 2020 [Page 13]
```