# GOST (hash function)

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Template:Infobox cryptographic 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.

The hash function is based on the GOST block cipher.

## AlgorithmEdit

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 notations Edit

The algorithm descriptions uses the following notations:

• $\mathcal{f}0\mathcal{g}^j$ — j-bit block filled with zeroes.
• $\mathcal{j}M\mathcal{j}$ — length of the M block in bits modulo 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 Edit

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 lentgh 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 $h$.

### The step hash function Edit

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

#### Generating of keys Edit

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 = 0
C3 = 0xff00ffff000000ffff0000ff00ffff0000ff00ff00ff00ffff00ff00ff00ff00
C4 = 0


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 Edit

After the keys generation, the enciphering of $H_{in}$ 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 (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 Edit

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

First we define the ψ function, doing LSFR 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 Edit

The GOST R 34.11 94 standard itself doesn't specify the algorithm initial value $h$ and S-Box of the enciphering transformation $E$, but uses the following values in the samples sections [1]. It should be noted that these parameters are specified by RFC 4357 as test parameters and are not recommended for use in production applications. A "production ready" parameter set is also specified as part of RFC 4357 (see section 11.2).

The starting vector

h=0x00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000.


The S-Box for the $E$ transformation

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

## CryptanalysisEdit

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 preimage attacks in 2192 time.[2]

## GOST hashesEdit

The following are some examples of GOST hashes:

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


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:

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


Samples 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


Other samples:

GOST("") =
ce85b99cc46752fffee35cab9a7b0278abb4c2d2055cff685af4912c49490f8d

GOST("a") =
d42c539e367c66e9c88a801f6649349c21871b4344c6a573f849fdce62f314dd

GOST("message digest") =

GOST( 128 characters of 'U' ) =
53a3a3ed25180cef0c1d85a074273e551c25660a87062a52d926a9e8fe5733a4

GOST( 1000000 characters of 'a' ) =
5c00ccc2734cdd3332d3d4749576e3c1a7dbaf0e7ea74e9fa602413c90a129fa