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Keywords: authenticated encryption, mode of operation, AEAD

Independent Submission S. Smyshlyaev, Ed.
Request for Comments: 9058 CryptoPro
Category: Informational V. Nozdrunov
ISSN: 2070-1721 V. Shishkin
TC 26
E. Griboedova
CryptoPro
June 2021
Multilinear Galois Mode (MGM)
Abstract
Multilinear Galois Mode (MGM) is an Authenticated Encryption with
Associated Data (AEAD) block cipher mode based on the Encrypt-then-
MAC (EtM) principle. MGM is defined for use with 64-bit and 128-bit
block ciphers.
MGM has been standardized in Russia. 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.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This is a contribution to the RFC Series, independently of any other
RFC stream. The RFC Editor has chosen to publish this document at
its discretion and makes no statement about its value for
implementation or deployment. Documents approved for publication by
the RFC Editor are not candidates for any level of Internet Standard;
see Section 2 of RFC 7841.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
https://www.rfc-editor.org/info/rfc9058.
Copyright Notice
Copyright (c) 2021 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.
Table of Contents
1. Introduction
2. Conventions Used in This Document
3. Basic Terms and Definitions
4. Specification
4.1. MGM Encryption and Tag Generation Procedure
4.2. MGM Decryption and Tag Verification Check Procedure
5. Rationale
6. Security Considerations
7. IANA Considerations
8. References
8.1. Normative References
8.2. Informative References
Appendix A. Test Vectors
A.1. Test Vectors for the Kuznyechik Block Cipher
A.1.1. Example 1
A.1.2. Example 2
A.2. Test Vectors for the Magma Block Cipher
A.2.1. Example 1
A.2.2. Example 2
Contributors
Authors' Addresses
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.
2. Conventions Used in This Document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
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 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 an n-byte string A = L || R
into the n-byte string incr_l(A) = Vec_{n/2}(Int_{n/2}(L)
[+] 1) || R, where L and R are n/2-byte strings.
incr_r V_n -> V_n
The transformation that maps an n-byte string A = L || R
into the n-byte string incr_r(A) = L ||
Vec_{n/2}(Int_{n/2}(R) [+] 1), where L and R are n/2-byte
strings.
4. Specification
An additional parameter that defines the functioning of MGM is the
bit length S of the authentication tag, 32 <= S <= n. 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 Tag Generation Procedure
The MGM encryption and tag generation procedure takes the following
parameters as inputs:
1. Encryption key K in V_k.
2. Initial counter nonce ICN in V_{n-1}.
3. 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.
4. 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.
The MGM encryption and tag generation 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 tag generation procedure consists of the
following steps:
+----------------------------------------------------------------+
| MGM-Encrypt(K, ICN, A, P) |
|----------------------------------------------------------------|
| 1. Encryption step: |
| - if |P| = 0 then |
| - C*_q = P*_q |
| - C = P |
| - else |
| - 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^n, |
| - 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.
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}.
4.2. MGM Decryption and Tag Verification Check Procedure
The MGM decryption and tag verification procedure takes the following
parameters as inputs:
1. Encryption key K in V_k.
2. Initial counter nonce ICN in V_{n-1}.
3. 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.
4. Ciphertext C, 0 <= |C| < 2^{n/2}. If |C| > 0, then C = C_1 ||
... || C*_q, C_i in V_n, for i = 1, ... , q - 1, C*_q in V_u, 1
<= u <= n. If |C| = 0, then by definition C*_q is empty, and the
q and u parameters are set as follows: q = 0, u = n.
5. Authentication tag T in V_S.
The MGM decryption and tag verification procedure outputs FAIL or the
following parameters:
1. Associated authenticated data A.
2. Plaintext P in V_{|C|}.
The MGM decryption and tag verification 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 verification step: |
| - Z_1 = E_K(1^1 || ICN), |
| - sum = 0^n, |
| - 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: |
| - if |C| = 0 then |
| - P = C |
| - else |
| - 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 (A, P). |
+----------------------------------------------------------------+
The length of the associated data A and of the ciphertext C MUST be
such that 0 < |A| + |C| < 2^{n/2}.
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 attacks based on padding and linear properties to be
resisted (see [FERG05] for details).
Multilinear function for authentication:
It allows the small subgroup attacks to be resisted [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 prevents
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.
8. References
8.1. 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>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>.
[RFC8891] Dolmatov, V., Ed. and D. Baryshkov, "GOST R 34.12-2015:
Block Cipher "Magma"", RFC 8891, DOI 10.17487/RFC8891,
September 2020, <https://www.rfc-editor.org/info/rfc8891>.
8.2. Informative References
[AEAD-LIMITS]
Günther, F., Thomson, M., and C. A. Wood, "Usage Limits on
AEAD Algorithms", Work in Progress, Internet-Draft, draft-
irtf-cfrg-aead-limits-02, 22 February 2021,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
aead-limits-02>.
[AUTH-ENC-BLOCK-CIPHER]
Federal Agency on Technical Regulating and Metrology,
"Information technology. Cryptographic data security.
Authenticated encryption block cipher operation modes", R
1323565.1.026-2019, 2019.
[FERG05] Ferguson, N., "Authentication weaknesses in GCM", May
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, June 2017.
[SAAR12] Saarinen, M-J., "Cycling Attacks on GCM, GHASH and Other
Polynomial MACs and Hashes", FSE 2012 proceedings, pp.
216-225, DOI 10.1007/978-3-642-34047-5_13, 2012,
<https://doi.org/10.1007/978-3-642-34047-5_13>.
[SEC-MGM] Akhmetzyanova, L., Alekseev, E., Karpunin, G., and V.
Nozdrunov, "Security of Multilinear Galois Mode (MGM)",
IACR Cryptology ePrint Archive 2019, pp. 123, 2019.
Appendix A. Test Vectors
A.1. Test Vectors for the Kuznyechik Block Cipher
Test vectors for the Kuznyechik block cipher (n = 128, k = 256) are
defined in [GOST3412-2015] (the English version can be found in
[RFC7801]).
A.1.1. Example 1
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
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
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
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
A.1.2. Example 2
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:
1. Encryption step:
C:
00000:
2. Padding step:
A_1 || ... || A_h:
00000: 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01
C_1 || ... || C_q:
00000:
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: 79 32 72 68 96 C4 3E 3F BF D6 50 89 EB F1 E5 B6
H_1:
00000: 99 3A 80 66 CC C0 A4 0F AC 4A 14 F7 A2 F6 6D 9B
current sum:
00000: 0A C1 1E 2C 1C D6 07 D8 2F E3 55 54 B4 01 02 81
Z_2:
00000: 79 32 72 68 96 C4 3E 40 BF D6 50 89 EB F1 E5 B6
H_2:
00000: 0C 38 A7 1E E7 93 BF 76 89 81 BF CD 7C DA 78 C8
len(A) || len(C):
00000: 00 00 00 00 00 00 00 80 00 00 00 00 00 00 00 00
sum (xor) ( H_2 (x) ( len(A) || len(C) ) ):
00000: CA 1E F8 92 71 EA 60 C4 53 9E 40 EB 26 C2 80 5D
Tag T:
00000: 79 01 E9 EA 20 85 CD 24 7E D2 49 69 5F 9F 8A 85
A.2. Test Vectors for the Magma Block Cipher
Test vectors for the Magma block cipher (n = 64, k = 256) are defined
in [GOST3412-2015] (the English version can be found in [RFC8891]).
A.2.1. Example 1
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
1. Encryption step:
0^1 || ICN:
00000: 12 DE F0 6B 3C 13 0A 59
Y_1:
00000: 56 23 89 01 62 DE 31 BF
E_K(Y_1):
00000: 38 7B DB A0 E4 34 39 B3
Y_2:
00000: 56 23 89 01 62 DE 31 C0
E_K(Y_2):
00000: 94 33 00 06 10 F7 F2 AE
Y_3:
00000: 56 23 89 01 62 DE 31 C1
E_K(Y_3):
00000: 97 B7 AA 6D 73 C5 87 57
Y_4:
00000: 56 23 89 01 62 DE 31 C2
E_K(Y_4):
00000: 94 15 52 8B FF C9 E8 0A
Y_5:
00000: 56 23 89 01 62 DE 31 C3
E_K(Y_5):
00000: 03 F7 68 BF F1 82 D6 70
Y_6:
00000: 56 23 89 01 62 DE 31 C4
E_K(Y_6):
00000: FD 05 F8 4E 9B 09 D2 FE
Y_7:
00000: 56 23 89 01 62 DE 31 C5
E_K(Y_7):
00000: DA 4D 90 8A 95 B1 75 C4
Y_8:
00000: 56 23 89 01 62 DE 31 C6
E_K(Y_8):
00000: 65 99 73 96 DA C2 4B D7
Y_9:
00000: 56 23 89 01 62 DE 31 C7
E_K(Y_9):
00000: A9 00 50 4A 14 8D EE 26
C:
00000: C7 95 06 6C 5F 9E A0 3B 85 11 33 42 45 91 85 AE
00010: 1F 2E 00 D6 BF 2B 78 5D 94 04 70 B8 BB 9C 8E 7D
00020: 9A 5D D3 73 1F 7D DC 70 EC 27 CB 0A CE 6F A5 76
00030: 70 F6 5C 64 6A BB 75 D5 47 AA 37 C3 BC B5 C3 4E
00040: 03 BB 9C
2. Padding step:
A_1 || ... || A_h:
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 00 00 00 00 00 00 00
C_1 || ... || C_q:
00000: C7 95 06 6C 5F 9E A0 3B 85 11 33 42 45 91 85 AE
00010: 1F 2E 00 D6 BF 2B 78 5D 94 04 70 B8 BB 9C 8E 7D
00020: 9A 5D D3 73 1F 7D DC 70 EC 27 CB 0A CE 6F A5 76
00030: 70 F6 5C 64 6A BB 75 D5 47 AA 37 C3 BC B5 C3 4E
00040: 03 BB 9C 00 00 00 00 00
3. Authentication tag T generation step:
1^1 || ICN:
00000: 92 DE F0 6B 3C 13 0A 59
Z_1:
00000: 2B 07 3F 04 94 F3 72 A0
H_1:
00000: 70 8A 78 19 1C DD 22 AA
current sum:
00000: D6 BB 5B EA 81 93 12 62
Z_2:
00000: 2B 07 3F 05 94 F3 72 A0
H_2:
00000: 6F 02 CC 46 4B 2F A0 A3
current sum:
00000: DD 1C 82 4E 91 78 49 A5
Z_3:
00000: 2B 07 3F 06 94 F3 72 A0
H_3:
00000: 9F 81 F2 26 FD 19 6F 05
current sum:
00000: 05 89 22 17 F6 5A DA C7
Z_4:
00000: 2B 07 3F 07 94 F3 72 A0
H_4:
00000: B9 C2 AC 9B E5 B5 DF F9
current sum:
00000: D1 DB 9B 7F C4 9E 7C 97
Z_5:
00000: 2B 07 3F 08 94 F3 72 A0
H_5:
00000: 74 B5 EC 96 55 1B F8 88
current sum:
00000: 56 45 F6 B5 18 5C B7 1A
Z_6:
00000: 2B 07 3F 09 94 F3 72 A0
H_6:
00000: 7E B0 21 A4 03 5B 04 C3
current sum:
00000: 3F C2 C2 E6 FB EE D0 4D
Z_7:
00000: 2B 07 3F 0A 94 F3 72 A0
H_7:
00000: C2 A9 C3 A8 70 4D 9B B0
current sum:
00000: 15 47 1F B5 CD 8E 6C 02
Z_8:
00000: 2B 07 3F 0B 94 F3 72 A0
H_8:
00000: F5 D5 05 A8 7B 83 83 B5
current sum:
00000: 12 56 78 96 1D 40 E0 93
Z_9:
00000: 2B 07 3F 0C 94 F3 72 A0
H_9:
00000: F7 95 E7 5F DE B8 93 3C
current sum:
00000: 6E F4 0A B0 C1 5F 20 48
Z_10:
00000: 2B 07 3F 0D 94 F3 72 A0
H_10:
00000: 65 A1 A3 E6 80 F0 81 45
current sum:
00000: A4 64 A7 08 FF 45 14 22
Z_11:
00000: 2B 07 3F 0E 94 F3 72 A0
H_11:
00000: 1C 74 A5 76 4C B0 D5 95
current sum:
00000: 60 94 4E 05 D0 85 75 14
Z_12:
00000: 2B 07 3F 0F 94 F3 72 A0
H_12:
00000: DC 84 47 A5 14 E7 83 E7
current sum:
00000: EE 98 B9 B5 0F F7 83 E8
Z_13:
00000: 2B 07 3F 10 94 F3 72 A0
H_13:
00000: A7 E3 AF E0 04 EE 16 E3
current sum:
00000: C0 39 0F A2 28 AF 6D CB
Z_14:
00000: 2B 07 3F 11 94 F3 72 A0
H_14:
00000: A5 AA BB 0B 79 80 D0 71
current sum:
00000: 73 E0 6E 07 EF 37 CD CC
Z_15:
00000: 2B 07 3F 12 94 F3 72 A0
H_15:
00000: 6E 10 4C C9 33 52 5C 5D
current sum:
00000: 2F 40 69 0A EB 53 F5 39
Z_16:
00000: 2B 07 3F 13 94 F3 72 A0
H_16:
00000: 83 11 B6 02 4A A9 66 C1
len(A) || len(C):
00000: 00 00 01 48 00 00 02 18
sum (xor) ( H_16 (x) ( len(A) || len(C) ) ):
00000: 73 CE F4 4B AE 6B DB 61
Tag T:
00000: A7 92 80 69 AA 10 FD 10
A.2.2. Example 2
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
1. Encryption step:
0^1 || ICN:
00000: 00 77 66 55 44 33 22 11
Y_1:
00000: 5B 2A 7E 60 4F 9F BB 95
E_K(Y_1):
00000: 48 A6 A5 17 0D 52 9D B1
C:
00000: 6A 95 E1 42 6B 25 9D 4E
2. Padding step:
A_1 || ... || A_h:
00000:
C_1 || ... || C_q:
00000: 6A 95 E1 42 6B 25 9D 4E
3. Authentication tag T generation step:
1^1 || ICN:
00000: 80 77 66 55 44 33 22 11
Z_1:
00000: 59 73 54 78 7E 52 E6 EB
H_1:
00000: EC E3 F9 DA 11 8C 7D 95
current sum:
00000: 25 D0 E4 20 7B 6B F6 3D
Z_2:
00000: 59 73 54 79 7E 52 E6 EB
H_2:
00000: 31 0C 0D AC C9 D0 4D 93
len(A) || len(C):
00000: 00 00 00 00 00 00 00 40
sum (xor) ( H_2 (x) ( len(A) || len(C) ) ):
00000: 66 D3 8F 12 0F 78 92 49
Tag T:
00000: 33 4E E2 70 45 0B EC 9E
Contributors
Evgeny Alekseev
CryptoPro
Email: alekseev@cryptopro.ru
Alexandra Babueva
CryptoPro
Email: babueva@cryptopro.ru
Lilia Akhmetzyanova
CryptoPro
Email: lah@cryptopro.ru
Grigory Marshalko
TC 26
Email: marshalko_gb@tc26.ru
Vladimir Rudskoy
TC 26
Email: rudskoy_vi@tc26.ru
Alexey Nesterenko
National Research University Higher School of Economics
Email: anesterenko@hse.ru
Lidia Nikiforova
CryptoPro
Email: nikiforova@cryptopro.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
Ekaterina Griboedova
CryptoPro
Email: griboedovaekaterina@gmail.com