# GOST (hash function)

The GOST hash function, defined in the standards GOST R 34.11-94 and GOST 34.311-95 is a 256-bit cryptographic hash function. It was initially defined in the Russian national standard R 34.11-94 Information Technology – Cryptographic Information Security – Hash Function. The equivalent standard used by other member-states of the is GOST 34.311-95.

This function must not be confused with a different Streebog hash function, which is defined in the new revision of the standard GOST R 34.11-2012.

The GOST hash function is based on the .

## Algorithm

GOST processes a variable-length message into a fixed-length output of 256 bits. The input message is broken up into chunks of 256-bit blocks (eight 32-bit integers); the message is by appending as many zeros to it as are required to bring the length of the message up to 256 bits. The remaining bits are filled up with a 256-bit integer arithmetic sum of all previously hashed blocks and then a 256-bit integer representing the length of the original message, in bits.

### Basic notation

The algorithm descriptions uses the following notations:

• $mathcal{f}0mathcal{g}^j$ — j-bit block filled with zeroes.
• $mathcal{j}Mmathcal{j}$ — length of the M block in bits modulo 2256.
• $mathcal{k}$ — concatenation of two blocks.
• $+$ — arithmetic sum of two blocks modulo 2256
• $oplus$ — logical xor of two blocks

Further we consider that the little-order bit is located at the left of a block, and the high-order bit at the right.

### Description

The input message $M$ is split into 256-bit blocks $m_{n}, m_{n-1}, m_{n-2}, ... , m_{1}$. In the case the last block $m_{n}$ contains less than 256 bits, it is prepended left by zero bits to achieve the desired length.

Each block is processed by the step hash function $H_{out} = f(H_{in}, m)$, where $H_{out}$, $H_{in}$, $m$ are a 256-bit blocks.

Each message block, starting the first one, is processed by the step hash function $f$, to calculate intermediate hash value
$!H_{i+1}=f(H_{i}, m_{i})$
The $H_1$ value can be arbitrary chosen, and usually is $0^{256}$.

After $H_{n+1}$ is calculated, the final hash value is obtained in the following way

• $H_{n+2} = f(H_{n+1}, L)$, where L — is the length of the message M in bits modulo $2^{256}$
• $h = f(H_{n+2}, K)$, where K — is 256-bit control sum of M: $m_1 + m_2 + m_3 + ... + m_n$

The $h$ is the desired value of the hash function of the message M.

So, the algorithm works as follows.

1. Initialization:
1. $h := initial$ — Initial 256-bit value of the hash function, determined by user.
2. $Sigma := 0$ — Control sum
3. $L := 0$ — Message length
2. Compression function of internal iterarions: for i = 1 … n — 1 do the following (while $|M|>256$):
1. $h := f(h, m_i)$ – apply step hash function
2. $L := L + 256$ – recalculate message length
3. $Sigma := Sigma + m_i$ – calculate control sum
3. Compression function of final iteration:
1. $L := L + mathcal{j} m_n mathcal{j}$ – calculate the full message length in bits
2. $m_n := {0}^{256 - mathcal{j} m_n mathcal{j}} mathcal{k} m_n$ – pad the last message with zeroes
3. $Sigma := Sigma + m_n$ – update control sum
4. $h := f(h, m_n)$ – process the last message block
5. $h := f(h, L)$ – MD – strengthen up by hashing message length
6. $h := f(h, Sigma)$ – hash control sum
4. The output value is $h$.

### Step hash function

The step hash function $f$ maps two 256-bit blocks into one: $H_{out} = f(H_{in}, m)$. It consist of three parts:

• Generating of keys $K_1, K_2, K_3, K_4$
• Enciphering transformation $H_{in}$ using keys $K_1, K_2, K_3, K_4$
• Shuffle transformation

#### Key generation

The keys generating algorithm uses:

• Two transformations of 256-bit blocks:
• Transformation $A(Y)=A(y_4 mathcal{k} y_3 mathcal{k} y_2 mathcal{k} y_1) = (y_1 oplus y_2) mathcal{k} y_4 mathcal{k} y_3 mathcal{k} y_2$, where $y_1, y_2, y_3, y_4$ are 64-bit sub-blocks of Y.
• Transformation $P(Y) = P(y_{32} mathcal{k} y_{31} mathcal{k} dots mathcal{k} y_1) = y_{varphi(32)} mathcal{k} y_{varphi(31)} mathcal{k} dots mathcal{k} y_{varphi(1)}$, where $varphi (i + 1 + 4(k - 1))= 8i + k, i = 0, dots, 3, k = 1, dots, 8$, and $y_{32}, y_{31}, dots, y_{1}$ are 8-bit sub-blocks of Y.
• Three constants:
• C2 =
• C3 =
• C4 =

The algorithm:

1. $U := H_{in},quad V := m,quad W := U oplus V,quad K_1 = P(W)$
2. For j = 2,3,4 do the following:
$U := A(U) oplus C_j,quad V := A(A(V)),quad W := U oplus V,quad K_j = P(W)$

#### Enciphering transformation

After the keys generation, the enciphering of $H_{in}$ is done using in the mode of simple substitution on keys $K_1, K_2, K_3, K_4$. Let’s denote the enciphering transformation as E (Note: the E transformation enciphers 64-bit data using 256-bit key). For enciphering, the $H_{in}$ is split into four 64-bit blocks: $H_{in} = h_4 mathcal{k} h_3 mathcal{k} h_2 mathcal{k} h_1$, and each of these blocks is enciphered as:

• $s_1 = E(h_1, K_1)$
• $s_2 = E(h_2, K_2)$
• $s_3 = E(h_3, K_3)$
• $s_4 = E(h_4, K_4)$

After this, the result blocks are concatenated into one 256-bit block: $S = s_4 mathcal{k} s_3 mathcal{k} s_2 mathcal{k} s_1$.

#### Shuffle transformation

On the last step, the shuffle transformation is applied to $H_{in}$, S and m using a . In the result, the intermediate hash value $H_{out}$ is obtained.

First we define the ψ function, doing LFSR on a 256-bit block: $psi(Y) = psi(y_{16} mathcal{k} y_{15} mathcal{k} ... mathcal{k} y_2 mathcal{k} y_1) = (y_1 oplus y_2 oplus y_3 oplus y_4 oplus y_{13} oplus y_{16}) mathcal{k} y_{16} mathcal{k} y_{15} mathcal{k} ... mathcal{k} y_3 mathcal{k} y_2$, where $y_{16}, y_{15}, ... , y_{2}, y_{1}$ are 16-bit sub-blocks of the Y.

The shuffle transformation is $H_{out} = {psi}^{61}(H_{in} oplus psi(m oplus {psi}^{12}(S)))$, where ${psi}^i$ denotes an i-th power of the $psi$ function.

### Initial values

There are two commonly used sets of initial parameters for GOST R 34.11 94. The starting vector for the both sets is

`$H_1$=0x00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000. `

Although the GOST R 34.11 94 standard itself doesn’t specify the algorithm initial value $H_1$ and S-box of the enciphering transformation $E$, but uses the following “test parameters” in the samples sections.

#### “Test parameters” S-box

RFC 5831 specifies only these parameters, but RFC 4357 names them as “test parameters” and does not recommend them for use in production applications.

S-box number Value
1 4 10 9 2 13 8 0 14 6 11 1 12 7 15 5 3
2 14 11 4 12 6 13 15 10 2 3 8 1 0 7 5 9
3 5 8 1 13 10 3 4 2 14 15 12 7 6 0 9 11
4 7 13 10 1 0 8 9 15 14 4 6 12 11 2 5 3
5 6 12 7 1 5 15 13 8 4 10 9 14 0 3 11 2
6 4 11 10 0 7 2 1 13 3 6 8 5 9 12 15 14
7 13 11 4 1 3 15 5 9 0 10 14 7 6 8 2 12
8 1 15 13 0 5 7 10 4 9 2 3 14 6 11 8 12

#### CryptoPro S-box

The CryptoPro S-box comes from “production ready” parameter set developed by CryptoPro company, it is also specified as part of RFC 4357, section 11.2.

S-box number Value
1 10 4 5 6 8 1 3 7 13 12 14 0 9 2 11 15
2 5 15 4 0 2 13 11 9 1 7 6 3 12 14 10 8
3 7 15 12 14 9 4 1 0 3 11 5 2 6 10 8 13
4 4 10 7 12 0 15 2 8 14 1 6 5 13 11 9 3
5 7 6 4 11 9 12 2 10 1 8 0 14 15 13 3 5
6 7 6 2 4 13 9 15 0 10 1 5 11 8 14 12 3
7 13 14 4 1 7 0 5 10 3 12 8 15 6 2 9 11
8 1 3 10 9 5 11 4 15 8 6 7 14 13 0 2 12

## Cryptanalysis

In 2008, an attack was published that breaks the full-round GOST hash function. The paper presents a collision attack in 2105 time, and first and second in 2192 time (2n time refers to the approximate number of times the algorithm was calculated in the attack).

## GOST hash test vectors

### Hashes for “test parameters”

The 256-bit (32-byte) GOST hashes are typically represented as 64-digit hexadecimal numbers. Here are test vectors for the GOST hash with “test parameters”

`GOST("The quick brown fox jumps over the lazy og") = 77b7fa410c9ac58a25f49bca7d0468c9296529315eaca76bd1a10f376d1f4294 `

Even a small change in the message will, with overwhelming probability, result in a completely different hash due to the . For example, changing d to c:

`GOST("The quick brown fox jumps over the lazy og") = a3ebc4daaab78b0be131dab5737a7f67e602670d543521319150d2e14eeec445 `

Two samples coming from the GOST R 34.11-94 standard:

`GOST("This is message, length=32 bytes") = b1c466d37519b82e8319819ff32595e047a28cb6f83eff1c6916a815a637fffa  GOST("Suppose the original message has length = 50 bytes") = 471aba57a60a770d3a76130635c1fbea4ef14de51f78b4ae57dd893b62f55208 `

More test vectors:

`GOST("") = ce85b99cc46752fffee35cab9a7b0278abb4c2d2055cff685af4912c49490f8d  GOST("a") = d42c539e367c66e9c88a801f6649349c21871b4344c6a573f849fdce62f314dd  GOST("message digest") = ad4434ecb18f2c99b60cbe59ec3d2469582b65273f48de72db2fde16a4889a4d  GOST( 128 characters of 'U' ) = 53a3a3ed25180cef0c1d85a074273e551c25660a87062a52d926a9e8fe5733a4  GOST( 1000000 characters of 'a' ) = 5c00ccc2734cdd3332d3d4749576e3c1a7dbaf0e7ea74e9fa602413c90a129fa `

### Hashes for CryptoPro parameters

GOST algorithm with CryptoPro S-box generates different set of hash values.

`GOST("") = 981e5f3ca30c841487830f84fb433e13ac1101569b9c13584ac483234cd656c0  GOST("a") = e74c52dd282183bf37af0079c9f78055715a103f17e3133ceff1aacf2f403011  GOST("abc") = b285056dbf18d7392d7677369524dd14747459ed8143997e163b2986f92fd42c  GOST("message digest") = bc6041dd2aa401ebfa6e9886734174febdb4729aa972d60f549ac39b29721ba0  GOST("The quick brown fox jumps over the lazy dog") = 9004294a361a508c586fe53d1f1b02746765e71b765472786e4770d565830a76  GOST("ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789") = 73b70a39497de53a6e08c67b6d4db853540f03e9389299d9b0156ef7e85d0f61  GOST("12345678901234567890123456789012345678901234567890123456789012345678901234567890") = 6bc7b38989b28cf93ae8842bf9d752905910a7528a61e5bce0782de43e610c90  GOST("This is message, length=32 bytes") = 2cefc2f7b7bdc514e18ea57fa74ff357e7fa17d652c75f69cb1be7893ede48eb  GOST("Suppose the original message has length = 50 bytes") = c3730c5cbccacf915ac292676f21e8bd4ef75331d9405e5f1a61dc3130a65011  GOST(128 of "U") = 1c4ac7614691bbf427fa2316216be8f10d92edfd37cd1027514c1008f649c4e8  GOST(1000000 of "a") = 8693287aa62f9478f7cb312ec0866b6c4e4a0f11160441e8f4ffcd2715dd554f `

## Source

http://wikipedia.org/