GOST (hash function)

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:

• ${\displaystyle \mathcal{f}0{\mathcal{g}}^j}$ — j-bit block filled with zeroes.
• ${\displaystyle \mathcal{j}M\mathcal{j}}$ — length of the M block in bits modulo 2²⁵⁶.
• ${\displaystyle \mathcal{k}}$ — concatenation of two blocks.
• ${\displaystyle +}$ — arithmetic sum of two blocks modulo 2²⁵⁶.
• ${\displaystyle \oplus }$ — 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 ${\displaystyle M}$ is split into 256-bit blocks ${\displaystyle m_n,m_{n-1},m_{n-2},\dots ,m_1}$. In the case the last block ${\displaystyle m_n}$ 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 ${\displaystyle H_{\text{out}}=f(H_{\text{in}},m)}$, where ${\displaystyle H_{\text{out}}}$, ${\displaystyle H_{\text{in}}}$, ${\displaystyle m}$ are a 256-bit blocks.

Each message block, starting the first one, is processed by the step hash function ${\displaystyle f}$, to calculate intermediate hash value

${\displaystyle H_{i+1}=f(H_i,m_i)}$

The ${\displaystyle H_1}$ value can be arbitrary chosen, and usually is ${\displaystyle 0^{256}}$.

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

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

The ${\displaystyle h}$ is the desired value of the hash function of the message M.

So, the algorithm works as follows.

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

Step hash function

The step hash function ${\displaystyle f}$ maps two 256-bit blocks into one: ${\displaystyle H_{\text{out}}=f(H_{\text{in}},m)}$.

It consist of three parts:

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

Key generation

The keys generating algorithm uses:

• Two transformations of 256-bit blocks:
  • Transformation ${\displaystyle A(Y)=A(y_4\mathcal{k}{y}_3\mathcal{k}{y}_2\mathcal{k}{y}_1)=(y_1\oplus y_2)\mathcal{k}{y}_4\mathcal{k}{y}_3\mathcal{k}{y}_2}$, where ${\displaystyle y_1,y_2,y_3,y_4}$ are 64-bit sub-blocks of Y.
  • Transformation ${\displaystyle P(Y)=P(y_{32}\mathcal{k}{y}_{31}\mathcal{k}\dots \mathcal{k}{y}_1)=y_{\phi (32)}\mathcal{k}{y}_{\phi (31)}\mathcal{k}\dots \mathcal{k}{y}_{\phi (1)}}$, where ${\displaystyle \phi (i+1+4(k-1))=8i+k,i=0,\dots ,3,k=1,\dots ,8}$, and ${\displaystyle y_{32},y_{31},\dots ,y_1}$ are 8-bit sub-blocks of Y.
• Three constants:
  • C₂ = 0
  • C₃ = 0xff00ffff000000ffff0000ff00ffff0000ff00ff00ff00ffff00ff00ff00ff00
  • C₄ = 0

The algorithm:

1. ${\displaystyle U:=H_{\text{in}},V:=m,W:=U\oplus V,K_1=P(W)}$
2. For j = 2, 3, 4 do the following:

   ${\displaystyle U:=A(U)\oplus C_j,V:=A(A(V)),W:=U\oplus V,K_j=P(W)}$

Enciphering transformation

After the keys generation, the enciphering of ${\displaystyle H_{\text{in}}}$ is done using GOST 28147-89 in the mode of simple substitution on keys ${\displaystyle K_1,K_2,K_3,K_4}$. Let's denote the enciphering transformation as E (enciphering 64-bit data using 256-bit key). For enciphering, the ${\displaystyle H_{\text{in}}}$ is split into four 64-bit blocks: ${\displaystyle H_{\text{in}}=h_4\mathcal{k}{h}_3\mathcal{k}{h}_2\mathcal{k}{h}_1}$, and each of these blocks is enciphered as:

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

After this, the result blocks are concatenated into one 256-bit block: ${\displaystyle S=s_4\mathcal{k}{s}_3\mathcal{k}{s}_2\mathcal{k}{s}_1}$.

Shuffle transformation

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

First we define the ψ function, doing LFSR on a 256-bit block:

${\displaystyle \psi (Y)=\psi (y_{16}\mathcal{k}{y}_{15}\mathcal{k}\dots \mathcal{k}{y}_2\mathcal{k}{y}_1)=(y_1\oplus y_2\oplus y_3\oplus y_4\oplus y_{13}\oplus y_{16})\mathcal{k}{y}_{16}\mathcal{k}{y}_{15}\mathcal{k}\dots \mathcal{k}{y}_3\mathcal{k}{y}_2}$,

where ${\displaystyle y_{16},y_{15},\dots ,y_2,y_1}$ are 16-bit sub-blocks of the Y.

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

${\displaystyle H_1}$ = 0x00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000.

Although the GOST R 34.11 94 standard itself doesn't specify the algorithm initial value ${\displaystyle H_1}$ and S-box of the enciphering transformation ${\displaystyle E}$, 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.

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.