Grasshopper (block cipher)

Grasshopper
GOST R 34.12-2015

General
Designers	InfoTeCS JSC
First published	2015
Certification	GOST, and FSS
Cipher detail
Key sizes	256 bits Feistel network
Block sizes	128 bits
Structure	Substitution-permutation network
Rounds	10
Best public cryptanalysis
A meet-in-the-middle attack on 5 rounds.

Kuznyechik (Russian: Кузнечик) is a symmetric block cipher. It has a block size of 128 bits and key length of 256 bits. It is defined in the National Standard of the Russian Federation GOST R 34.12-2015 and also in RFC 7801.

The name of the cipher can be translated from Russian as Grasshopper, however, the standard explicitly says that the English name for the cipher is Kuznyechik (/kʊznˈɛtʃɪk/). The designers claim that by naming the cipher Kuznyechik they follow the trend of difficult to pronounce algorithm names set up by Rijndael and Keccak.

The standard GOST R 34.12-2015 defines the new cipher in addition to the old GOST block cipher (now called Magma) one and does not declare the old cipher obsolete.

Kuznyechik is based on a substitution-permutation network, though the key schedule employs a Feistel network. The first block cipher with a mixed structure.

Designations

$\mathbb F$ — Finite field $GF(2^8)$ $x^8 + x^7 + x^6 + x + 1$.

$Bin_8 : \Z_p \rightarrow V_8$ — $\Z_p$ ($p=2^8$)

${Bin_8}^{-1}: V_8 \rightarrow \Z_p$ — $Bin_8$.

$\delta: V_8 \rightarrow \mathbb F$ — $\mathbb F$.

$\delta^{-1}: \mathbb F \rightarrow V_8$ — $\delta$

Description

For encryption, decryption and key generation, the following functions:

$Add_2[k](a)=k\oplus a$, где $k$, $a$ — двоичные строки вида $a=a_{15}||$…$||a_{0}$ ($||$ — symbol Concatenation strings).

$N(a)=S(a_{15})||$…$||S(a_{0}).~~N^{-1}(a)$ — обратное к $N(a)$ преобразование.

$G(a)=\delta(a_{15},$…$,a_0)||a_{15}||$…$||a_{1}.$

$G^{-1}(a)$ — обратное к $G(a)$ преобразование, причём $G^{-1}(a)=a_{14}||a_{13}||$…$||a_{0}||\delta(a_{14},a_{13},$…$,a_0,a_{15}).$

$H(a)=G^{16}(a)$, где $G^{16}$ — композиция преобразований $G^{15}$ и $G$ и т. д.

$F[k](a_1,a_0)=(HNAdd_2[k](a_1)\oplus a_0, a_1).$

The nonlinear transformation

Non-linear transformation is given by substituting S = Bin₈ S' Bin₈⁻¹.

Значения подстановки S' заданы в виде массива S' = (S'(0), S'(1), …, S'(255)):

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

Linear transformation

$\gamma$: $\gamma(a_{15},$…$, a_0) = \delta^{-1}~(148 * \delta(a_{15}) + 32 * \delta(a_{14}) + 133 * \delta(a_{13}) + 16 * \delta(a_{12}) +$ $194 * \delta(a_{11}) + 192 * \delta(a_{10}) + 1 * \delta(a_9) + 251 * \delta(a_8) + 1 * \delta(a_7) + 192 * \delta(a_6) +$ $194 * \delta(a_5) + 16 * \delta(a_4) + 133 * \delta(a_3) + 32 * \delta(a_2) + 148 * \delta(a_1) + 1 * \delta(a_0)),$

operations of addition and multiplication are carried out in the field $\mathbb F$.

Key generation

key generation algorithm uses iterative constant $C_i=H(Bin_{128}(i))$, i=1,2,…32. Sets the shared key $K=k_{255}||$…$||k_0$.

Iterated keys

$K_1=k_{255}||$…$||k_{128}$

$K_2=k_{127}||$…$||k_{0}$

$(K_{2i+1}, K_{2i+2})=F[C_{8(i-1)+8}]$…$F[C_{8(i-1)+1}](K_{2i+1}, K_{2i}), i=1, 2, 3, 4.$

Encryption algorithm

$E(a)=Add_2[K_{10}]HNAdd_2[K_9]$…$HNAdd_2[K_2]HNAdd_2[K_1](a),$ where a — 128-bit string.

Decryption algorithm

$D(a)=Add_2[K_{1}]H^{-1}N^{-1}Add_2[K_2]$…$H^{-1}N^{-1}Add_2[K_9]H^{-1}N^{-1}Add_2[K_{10}](a).$

Cryptanalysis

Riham AlTawy and Amr M. Youssef describe a meet-in-the-middle attack on the 5-round reduced Kuznyechik which allows to recover the key with time complexity of 2¹⁴⁰, memory complexity of 2¹⁵³, and data complexity of 2¹¹³.

Alex Biryukov, Leo Perrin, and Aleksei Udovenko published a paper in which they show that the S-Boxes of Kuznyechik and Streebog were not created pseudo-randomly but using a hidden algorithm which they were able to reverse engineer.

Riham AlTawy, Onur Duman, and Amr M. Youssef published two fault attacks on Kuznyechik which show the importance of protecting the implementations of the cipher.