# 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 GOST R 34.11-94 Information Technology – Cryptographic Information Security – Hash Function. The equivalent standard used by other member-states of the CIS 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 GOST block cipher.

## 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 little endian integers); the message is padded 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 notation:
* ${\mathcal {f}}0{\mathcal {g}}^{j}$ — j-bit block filled with zeroes.
* ${\mathcal {j}}M{\mathcal {j}}$ — length of the M block in bits modulo 2$^{256}$.
* ${\mathcal {k}}$ — concatenation of two blocks.
* $+$ — arithmetic sum of two blocks modulo 2$^{256}$.
* $\oplus$ — 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 $m_{n},\,m_{n-1},\,m_{n-2},\,\ldots ,\,m_{1}$. 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 $H_{\text{out}}=f(H_{\text{in}},\,m)$, where $H_{\text{out}}$, $H_{\text{in}}$, $m$ 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: $m_{1}+m_{2}+m_{3}+\ldots +m_{n}$
The $h$ is the desired value of the hash function of the message M.
So, the algorithm works as follows.
* Initialization:
** $h:={\text{initial}}$ — Initial 256-bit value of the hash function, determined by user.
** $\Sigma :=0$ — Control sum
** $L:=0$ — Message length
* Compression function of internal iterations: for i = 1 … n — 1 do the following (while $|M|>256$):
** $h:=f(h,\,m_{i})$ – apply step hash function
** $L:=L+256$ – recalculate message length
** $\Sigma :=\Sigma +m_{i}$ – calculate control sum
* Compression function of final iteration:
** $L:=L+{\mathcal {j}}\,m_{n}\,{\mathcal {j}}$ – calculate the full message length in bits
** $m_{n}:={0}^{256-{\mathcal {j}}m_{n}{\mathcal {j}}}{\mathcal {k}}m_{n}$ – pad the last message with zeroes
** $\Sigma :=\Sigma +m_{n}$ – update control sum
** $h:=f(h,\,m_{n})$ – process the last message block
** $h:=f(h,\,L)$ – MD – strengthen up by hashing message length
** $h:=f(h,\,\Sigma )$ – hash control sum
* The output value is $h$.

### Step hash function

The step hash function $f$ maps two 256-bit blocks into one: $H_{\text{out}}=f(H_{\text{in}},\,m)$.
It consist of three parts:
* Generating of keys $K_{1},\,K_{2},\,K_{3},\,K_{4}$
* Enciphering transformation $H_{\text{in}}$ 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 $\varphi (i+1+4(k-1))=8i+k,\quad i=0,\,\dots ,\,3,\quad k=1,\,\dots ,\,8$, and $y_{32},\,y_{31},\,\dots ,\,y_{1}$ are 8-bit sub-blocks of Y.
* Three constants:
** C$_2$ = 0
** C$_3$ = 0xff00ffff000000ffff0000ff00ffff0000ff00ff00ff00ffff00ff00ff00ff00
** C$_4$ = 0
The algorithm:
* $U:=H_{\text{in}},\quad V:=m,\quad W:=U\ \oplus \ V,\quad K_{1}=P(W)$
* 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 $H_{\text{in}}$ is done using GOST 28147-89 in the mode of simple substitution on keys $K_{1},\,K_{2},\,K_{3},\,K_{4}$. Let's denote the enciphering transformation as E (enciphering 64-bit data using 256-bit key). For enciphering, the $H_{\text{in}}$ is split into four 64-bit blocks: $H_{\text{in}}=h_{4}{\mathcal {k}}h_{3}{\mathcal {k}}h_{2}{\mathcal {k}}h_{1}$, 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 $H_{\text{in}}$, S and m using a linear-feedback shift register. In the result, the intermediate hash value $H_{\text{out}}$ is obtained.
First we define the ψ function, doing LFSR on a 256-bit block:
    $\psi (Y)=\psi (y_{16}{\mathcal {k}}y_{15}{\mathcal {k}}\ldots {\mathcal {k}}y_{2}{\mathcal {k}}y_{1})=(y_{1}\oplus y_{2}\oplus y_{3}\oplus y_{4}\oplus y_{13}\oplus y_{16}){\mathcal {k}}y_{16}{\mathcal {k}}y_{15}{\mathcal {k}}\ldots {\mathcal {k}}y_{3}{\mathcal {k}}y_{2}$,
where $y_{16},y_{15},\ldots ,y_{2},y_{1}$ are 16-bit sub-blocks of the Y.
The shuffle transformation is $H_{\text{out}}=\psi ^{61}{\mathord {\left(H_{\text{in}}\oplus \psi \left(m\oplus \psi ^{12}(S)\right)\right)}}$, 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 both the 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 | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | 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 | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | Value | 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 2$^{105}$ time, and first and second preimage attacks in 2$^{192}$ time (2$^n$ 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"
Even a small change in the message will, with overwhelming probability, result in a completely different hash due to the avalanche effect. For example, changing d to c:
Two samples coming from the GOST R 34.11-94 standard:
More test vectors:

### Hashes for CryptoPro parameters

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