GOST R 34.12-2015: Block Cipher "Kuznyechik"
draft-dolmatov-kuznyechik-02

Abstract

This document is intended to be a source of information about the Russian Federal standard block cipher with block length of n=128 bits, which is also referred as "Kuznyechik" [GOST3412-2015].This algorithm is one of the Russian cryptographic standard algorithms (called GOST algorithms).

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1. Scope
2. General Information
3. Definitions and Notations

3.1. Definitions
3.2. Notations

4. Parameter Values

4.1. Nonlinear Bijection
4.2. Linear Transformation
4.3. Transformations
4.4. Key schedule
4.5. Basic encryption algorithm

4.5.1. Encryption
4.5.2. Decryption

5. Examples (Informative)

5.1. Transformation S
5.2. Transformation R
5.3. Transformation L
5.4. Key schedule
5.5. Test encryption
5.6. Test decryption

6. Security Considerations
7. References

7.1. Normative References
7.2. Informative References

Author's Address

1. Scope

The Russian Federal standard [GOST3412-2015] specifies basic block ciphers used as cryptographic techniques for information processing and information protection including the provision of confidentiality, authenticity, and integrity of information during information transmission, processing and storage in computer-aided systems.

The cryptographic algorithms specified in this Standard are designed both for hardware and software implementation. They comply with modern cryptographic requirements, and put no restrictions on the confidentiality level of the protected information.

The Standard applies to developing, operation, and modernization of the information systems of various purposes.

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].

2. General Information

The block cipher "Kuznyechik" [GOST3412-2015] was developed by the Center for Information Protection and Special Communications of the Federal Security Service of the Russian Federation with participation of the Open Joint-Stock company "Information Technologies and Communication Systems" (InfoTeCS JSC). GOST R 34.12-2015 was approved and introduced by Decree #749 of the Federal Agency on Technical Regulating and Metrology on 19.06.2015.

Terms and concepts in the standard comply with the following international standards:

ISO/IEC 10116 [ISO-IEC10116],
series of standards ISO/IEC 18033 [ISO-IEC18033-1], [ISO-IEC18033-3].

3. Definitions and Notations

The following terms and their corresponding definitions are used in the standard.

3.1. Definitions

Definitions

encryption algorithm: process which transforms plaintext into ciphertext (Clause 2.19 of [ISO-IEC18033-1]),
decryption algorithm: process which transforms ciphertext into plaintext (Clause 2.14 of [ISO-IEC18033-1]),
basic block cipher: block cipher which for a given key provides a single invertible mapping of the set of fixed-length plaintext blocks into ciphertext blocks of the same length,
block: string of bits of a defined length (Clause 2.6 of [ISO-IEC18033-1]),
block cipher: symmetric encipherment system with the property that the encryption algorithm operates on a block of plaintext, i.e. a string of bits of a defined length, to yield a block of ciphertext (Clause 2.7 of [ISO-IEC18033-1]),
- Note: In GOST R 34.12-2015, it is established that the terms "block cipher" and "block encryption algorithm" are synonyms.
encryption: reversible transformation of data by a cryptographic algorithm to produce ciphertext, i.e., to hide the information content of the data (Clause 2.18 of [ISO-IEC18033-1]),
round key: sequence of symbols which is calculated from the key and controls a transformation for one round of a block cipher,
key: sequence of symbols that controls the operation of a cryptographic transformation (e.g., encipherment, decipherment) (Clause 2.21 of [ISO-IEC18033-1]),
- Note: In GOST R 34.12-2015, the key must be a binary sequence.
plaintext: unencrypted information (Clause 3.11 of [ISO-IEC10116]),
key schedule: calculation of round keys from the key,
decryption: reversal of a corresponding encipherment (Clause 2.13 of [ISO-IEC18033-1]),
symmetric cryptographic technique: cryptographic technique that uses the same secret key for both the originator`s and the recipient`s transformation (Clause 2.32 of [ISO-IEC18033-1]),
cipher: alternative term for encipherment system (Clause 2.20 of [ISO-IEC18033-1]),
ciphertext: data which has been transformed to hide its information content (Clause 3.3 of [ISO-IEC10116]).

3.2. Notations

The following notations are used in the standard:

V*: the set of all binary vector-strings of a finite length (hereinafter referred to as the strings) including empty string,
V_s: the set of all binary strings of length s, where s is a non-negative integer; substrings and string components are enumerated from right to left starting from zero,
U[*]W: direct (Cartesian) product of two set Us and W,
|A|: the number of components (the length) of a string A belonging to V* (if A is an empty string, then |A| = 0),
A||B: concatenation of strings A, B both belonging to V*, i.e., a string from V_(|A|+|B|), where the left substring from V_|A| is equal to A and the right subdtring from V_|B| is equal to B,
Z_(2^n): ring of residues modulo 2^n,
Q: finite field GF(2)[x]/p(x), where p(x)=x^8+x^7+x^6+x+1 belongs to GF(2)[x]; elements of field Q are represented by integers in such way that element z_0+z_1*theta+...+z_7*theta^7 belonging to Q corresponds to integer z_0+2*z_1+...+2^7*z_7, where z_i=0 or z_i=1, i=0,1,...,7 and theta denotes a residue class modulo p(x) containing x,
(xor): exclusive-or of the two binary strings of the same length,
Vec_s: Z_(2^s) -> V_s: bijective mapping which maps an element from ring Z_(2^s) into its binary representation, i.e., for an element z of the ring Z_(2^s), represented by the residue z_0 + (2*z_1) + ... + (2^(s-1)*z_(s-1)), where z_i in {0, 1}, i = 0, ..., n-1, the equality Vec_s(z) = z_(s-1)||...||z_1||z_0 holds,
Int_s: V_s -> Z_(2^s): the mapping inverse to the mapping Vec_s, i.e., Int_s = Vec_s^(-1),
delta: V_8 -> Q: bijective mapping which maps a binary string from V_8 into an element from field Q as follows: string z_7||...||z_1||z_0, where z_i in {0, 1}, i = 0, ..., 7, corresponds to the element z_0+(z_1*theta)+...+(z_7*theta^7) belonging to Z,
nabla: Q -> V8: the mapping inverse to the mapping nabla, i.e., delta = nabla^(-1),
PS: composition of mappings, where the mapping S applies first,
P^s: composition of mappings P^(s-1) and P, where P^1=P,

4. Parameter Values

4.1. Nonlinear Bijection

The bijective nonlinear mapping is a substitution: Pi = (Vec_8)Pi'(Int_8): V_8 -> V_8, where Pi': Z_(2^8) -> Z_(2^8). The values of the substitution Pi' are specified below as an array Pi' = (Pi'(0), Pi'(1), ... , Pi'(255)):

 Pi' =
(	252, 238, 221,  17, 207, 110,  49,  22, 251, 196, 250,
 	218,  35, 197,   4,  77, 233, 119, 240, 219, 147,  46,
 	153, 186,  23,  54, 241, 187,  20, 205,  95, 193, 249,
 	 24, 101,  90, 226,  92, 239,  33, 129,  28,  60,  66,
 	139,   1, 142,  79,   5, 132,   2, 174, 227, 106, 143,
 	160,   6,  11, 237, 152, 127, 212, 211,  31, 235,  52,
 	 44,  81, 234, 200,  72, 171, 242,  42, 104, 162, 253,
 	 58, 206, 204, 181, 112,  14,  86,   8,  12, 118,  18,
	191, 114,  19,  71, 156, 183,  93, 135,  21, 161, 150,
 	 41,  16, 123, 154, 199, 243, 145, 120, 111, 157, 158,
 	178, 177,  50, 117,  25,  61, 255,  53, 138, 126, 109,
 	 84, 198, 128, 195, 189,  13,  87, 223, 245,  36, 169,
 	 62, 168,  67, 201, 215, 121, 214, 246, 124,  34, 185,
 	  3, 224,  15, 236, 222, 122, 148, 176, 188, 220, 232,
 	 40,  80,  78,  51,  10,  74, 167, 151,  96, 115,  30,
 	  0,  98,  68,  26, 184,  56, 130, 100, 159,  38,  65,
 	173,  69,  70, 146,  39,  94,  85,  47, 140, 163, 165,
 	125, 105, 213, 149,  59,   7,  88, 179,  64, 134, 172,
 	 29, 247,  48,  55, 107, 228, 136, 217, 231, 137, 225,
 	 27, 131,  73,  76,  63, 248, 254, 141,  83, 170, 144,
 	202, 216, 133,  97,  32, 113, 103, 164,  45,  43,   9,
 	 91, 203, 155,  37, 208, 190, 229, 108,  82,  89, 166,
 	116, 210, 230, 244, 180, 192, 209, 102, 175, 194,  57,
 	 75,  99, 182).

4.2. Linear Transformation

The linear transformation is denoted by l: (V_8)^16 -> V_8, and defined as:

		l(a_15,...,a_0) = nabla(148*delta(a_15) + 32*delta(a_15) + 133*delta(a_13) + 
		16*delta(a_12) + 194*delta(a_11) + 192*delta(a_10) + 1*delta(a_9) + 251*delta(a_8) + 
		1*delta(a_7) + 192*delta(a_6) + 194*delta(a_5) + 16*delta(a_4) + 
		133*delta(a_3) + 32*delta(a_2) + 148*delta(a_1) +1*delta(a_0)),

for all a_i belonging to V_8, i = 0, 1, ..., 15, where the addition and multiplication operations are in the field Q, and constants are elements of the field as defined above.

4.3. Transformations

 pi^(-1)' =
(	 165,  45,  50, 143,  14,  48,  56, 192,  84, 230, 158,  57,  85, 126,  82, 145,
     100,   3,  87,  90,  28,  96,   7,  24,  33, 114, 168, 209,  41, 198, 164,  63,
     224,  39, 141,  12, 130, 234, 174, 180, 154,  99,  73, 229,  66, 228,  21, 183,
     200,   6, 112, 157,  65, 117,  25, 201, 170, 252,  77, 191,  42, 115, 132, 213,
     195, 175,  43, 134, 167, 177, 178,  91,  70, 211, 159, 253, 212,  15, 156,  47,
     155,  67, 239, 217, 121, 182,  83, 127, 193, 240,  35, 231,  37,  94, 181,  30,
     162, 223, 166, 254, 172,  34, 249, 226,  74, 188,  53, 202, 238, 120,   5, 107,
      81, 225,  89, 163, 242, 113,  86,  17, 106, 137, 148, 101, 140, 187, 119,  60,
     123,  40, 171, 210,  49, 222, 196,  95, 204, 207, 118,  44, 184, 216,  46,  54,
     219, 105, 179,  20, 149, 190,  98, 161,  59,  22, 102, 233,  92, 108, 109, 173,
      55,  97,  75, 185, 227, 186, 241, 160, 133, 131, 218,  71, 197, 176,  51, 250,
     150, 111, 110, 194, 246,  80, 255,  93, 169, 142,  23,  27, 151, 125, 236,  88,
     247,  31, 251, 124,   9,  13, 122, 103,  69, 135, 220, 232,  79,  29,  78,   4,
     235, 248, 243,  62,  61, 189, 138, 136, 221, 205,  11,  19, 152,   2, 147, 128,
     144, 208,  36,  52, 203, 237, 244, 206, 153,  16,  68,  64, 146,  58,   1,  38,
      18,  26,  72, 104, 245, 129, 139, 199, 214,  32,  10,   8,   0,  76, 215, 116 ).

The following transformations are applicable for encryption and decryption algorithms:

X[x]:V_128->V_128: X[k](a)=x(xor)a, where k, a belong to V_128,
S:V_128-> V_128: S(a)=(a_15||...||a_0)=pi(a_15)||...||pi(a_0), where a_15||...||a_0 belongs to V_128, a_i belongs to V_8, i=0,1,...,15,
S^(-1):V_128-> V_128: the inverse transformation of S, which may be calculated, for example, as follows: S^(-1)(a_15||...||a_0)=pi^(-1) (a_15)||...||pi^(-1)(a_0), where a_15||...||a_0 belongs to V_128, a_i belongs to V_8, i=0,1,...,15, pi^(-1) is the inverse of pi.,
R:V_128-> V_128: R(a_15||...||a_0)=l(a_15,...,a_0)||a_15||...||a_1, where a_15||...||a_0 belongs to V_128, a_i belongs to V_8, i=0,1,...,15,
L:V_128-> V_128: L(a)=R^(16)(a), where a belongs to V_128,
R^(-1):V_128-> V_128: the inverse transformation of R, which may be calculated, for example, as follows: R^(-1)(a_15||...||a_0)=a_14||a_13||...||a_0||l(a_14,a_13,...,a_0,a_15), where a_15||...||a_0 belongs to V_128, a_i belongs to V_8, i=0,1,...,15, pi^(-1) is the inverse of pi,
L^(-1):V_128-> V_128: L^(-1)(a)=(R^(-1))(16)(a), where a belongs to V_128,
F[k]:V_128[*]V_128 -> V_128[*]V_128: F[k](a_1,a_0)=(LSX[k](a_1)(xor)a_0,a_1), where k, a_0, a_1 belong to V_128.

4.4. Key schedule

Key schedule uses round constants C_i belonging to V_128, i=1, 2, ..., 32, defined as

		C_i=L(Vec_128(i)), i=1,2,...,32.

Round keys K_i, i=1, 2, ..., 10 are derived from key K=k_255||...||k_0 belonging to V_256, k_i belongs to V_1, i=0, 1, ..., 255, as follows:

		K_1=k_255||...||k_128;
		K_2=k_127||...||k_0;
		(K_(2i+1),K_(2i+2))=F[C_(8(i-1)+8)]... F[C_(8(i-1)+1)](K_(2i-1),K_(2i)), i=1,2,3,4.

4.5. Basic encryption algorithm

4.5.1. Encryption

Depending on the values of round keys K_1,...,K_10, the encryption algorithm is a substitution E_(K_1,...,K_10) defined as follows:

		E_(K_1,...,K_10)(a)=X[K_10]LSX[K_9]...LSX[K_2]LSX[K_1](a),

where a belongs to V_128.

4.5.2. Decryption

Depending on the values of round keys K_1,...,K_10, the decryption algorithm is a substitution D_(K_1,...,K_10) defined as follows:

		D_(K_1,...,K_10)(a)=X[K_1]L^(-1)S^(-1)X[K_2]...L^(-1)S^(-1)X[K_9] L^(-1)S^(-1)X[K_10](a),

where a belongs to V_128.

5. Examples (Informative)

This section is for information only and is not a normative part of the standard.

5.1. Transformation S

		S(ffeeddccbbaa99881122334455667700) = b66cd8887d38e8d77765aeea0c9a7efc, 
		S(b66cd8887d38e8d77765aeea0c9a7efc) = 559d8dd7bd06cbfe7e7b262523280d39, 
		S(559d8dd7bd06cbfe7e7b262523280d39) = 0c3322fed531e4630d80ef5c5a81c50b, 
		S(0c3322fed531e4630d80ef5c5a81c50b) = 23ae65633f842d29c5df529c13f5acda.

5.2. Transformation R

		R(00000000000000000000000000000100) = 94000000000000000000000000000001, 
		R(94000000000000000000000000000001) = a5940000000000000000000000000000, 
		R(a5940000000000000000000000000000) = 64a59400000000000000000000000000, 
		R(64a59400000000000000000000000000) = 0d64a594000000000000000000000000.

5.3. Transformation L

		L(64a59400000000000000000000000000) = d456584dd0e3e84cc3166e4b7fa2890d, 
		L(d456584dd0e3e84cc3166e4b7fa2890d) = 79d26221b87b584cd42fbc4ffea5de9a, 
		L(79d26221b87b584cd42fbc4ffea5de9a) = 0e93691a0cfc60408b7b68f66b513c13, 
		L(0e93691a0cfc60408b7b68f66b513c13) = e6a8094fee0aa204fd97bcb0b44b8580.

5.4. Key schedule

In this test example, the key is equal to:

		K = 8899aabbccddeeff0011223344556677fedcba98765432100123456789abcdef.

		K_1 = 8899aabbccddeeff0011223344556677, 
		K_2 = fedcba98765432100123456789abcdef.

		C_1 = 6ea276726c487ab85d27bd10dd849401, 
		X[C_1](K_1) = e63bdcc9a09594475d369f2399d1f276, 
		SX[C_1](K_1) = 0998ca37a7947aabb78f4a5ae81b748a, 
		LSX[C_1](K_1) = 3d0940999db75d6a9257071d5e6144a6,
		F[C_1](K_1, K_2) = = (c3d5fa01ebe36f7a9374427ad7ca8949, 8899aabbccddeeff0011223344556677).

		C_2 = dc87ece4d890f4b3ba4eb92079cbeb02, 
		F [C_2]F [C_1](K_1, K_2) = (37777748e56453377d5e262d90903f87, c3d5fa01ebe36f7a9374427ad7ca8949).

		C_3 = b2259a96b4d88e0be7690430a44f7f03,
		F[C_3]...F[C_1](K_1, K_2) = (f9eae5f29b2815e31f11ac5d9c29fb01, 37777748e56453377d5e262d90903f87).

		C_4 = 7bcd1b0b73e32ba5b79cb140f2551504,
		F[C_4]...F[C_1](K_1, K_2) = (e980089683d00d4be37dd3434699b98f, f9eae5f29b2815e31f11ac5d9c29fb01).

		C_5 = 156f6d791fab511deabb0c502fd18105,
		F[C_5]...F[C_1](K_1, K_2) = (b7bd70acea4460714f4ebe13835cf004, e980089683d00d4be37dd3434699b98f).

		C_6 = a74af7efab73df160dd208608b9efe06,
		F[C_6]...F[C_1](K_1, K_2) = (1a46ea1cf6ccd236467287df93fdf974, b7bd70acea4460714f4ebe13835cf004).

		C_7 = c9e8819dc73ba5ae50f5b570561a6a07,
		F[C_7]...F [C_1](K_1, K_2) = (3d4553d8e9cfec6815ebadc40a9ffd04, 1a46ea1cf6ccd236467287df93fdf974).

		C_8 = f6593616e6055689adfba18027aa2a08,
		(K_3, K_4) = F [C_8]...F [C_1](K_1, K_2) = (db31485315694343228d6aef8cc78c44, 3d4553d8e9cfec6815ebadc40a9ffd04).

The round keys K_i, i = 1, 2, ..., 10, take the following values:

		K_1 = 8899aabbccddeeff0011223344556677, 
		K_2 = fedcba98765432100123456789abcdef, 
		K_3 = db31485315694343228d6aef8cc78c44, 
		K_4 = 3d4553d8e9cfec6815ebadc40a9ffd04,
		K_5 = 57646468c44a5e28d3e59246f429f1ac, 
		K_6 = bd079435165c6432b532e82834da581b, 
		K_7 = 51e640757e8745de705727265a0098b1, 
		K_8 = 5a7925017b9fdd3ed72a91a22286f984, 
		K_9 = bb44e25378c73123a5f32f73cdb6e517, 
		K_10 = 72e9dd7416bcf45b755dbaa88e4a4043.

5.5. Test encryption

In this test example, encryption is performed on the round keys specified in clause 5.4. Let the plaintext be

		a = 1122334455667700ffeeddccbbaa9988,

then

		X[K_1](a) = 99bb99ff99bb99ffffffffffffffffff,
		SX[K_1](a) = e87de8b6e87de8b6b6b6b6b6b6b6b6b6, 
		LSX[K_1](a) = e297b686e355b0a1cf4a2f9249140830, 
		LSX[K_2]LSX[K_1](a) = 285e497a0862d596b36f4258a1c69072,
		LSX[K_3]...LSX[K_1](a) = 0187a3a429b567841ad50d29207cc34e, 
		LSX[K_4]...LSX[K_1](a) = ec9bdba057d4f4d77c5d70619dcad206, 
		LSX[K_5]...LSX[K_1](a) = 1357fd11de9257290c2a1473eb6bcde1, 
		LSX[K_6]...LSX[K_1](a) = 28ae31e7d4c2354261027ef0b32897df, 
		LSX[K_7]...LSX[K_1](a) = 07e223d56002c013d3f5e6f714b86d2d, 
		LSX[K_8]...LSX[K_1](a) = cd8ef6cd97e0e092a8e4cca61b38bf65, 
		LSX[K_9]...LSX[K_1](a) = 0d8e40e4a800d06b2f1b37ea379ead8e.

Then the ciphertext is

		b = X[K_10]LSX[K_9]...LSX[K_1](a) = 7f679d90bebc24305a468d42b9d4edcd.

5.6. Test decryption

In this test example, decryption is performed on the round keys specified in clause 5.4. Let the ciphertext be

		b = 7f679d90bebc24305a468d42b9d4edcd,