GOST (hash function)

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 notations:

f 0 g j — j-bit block filled with zeroes.

j M j — length of the M block in bits modulo 2²⁵⁶.

k — concatenation of two blocks.

+ — arithmetic sum of two blocks modulo 2²⁵⁶

⊕ — 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 o u t   =   f ( H i n , m ) , where H o u t , H i n , 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   := i n i t i a l — Initial 256-bit value of the hash function, determined by user.
   2. Σ   :=   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. Σ   :=   Σ   +   m i - calculate control sum
3. Compression function of final iteration:
   1. L   :=   L   +   j   m n   j - calculate the full message length in bits
   2. m n   :=   0 256   −   j m n j k m n - pad the last message with zeroes
   3. Σ   :=   Σ   +   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 ,   Σ ) - 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 o u t   =   f ( H i n ,   m ) . It consist of three parts:

Generating of keys K 1 ,   K 2 ,   K 3 ,   K 4

Enciphering transformation   H i n 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   k   y 3   k   y 2   k   y 1 ) = ( y 1 ⊕ y 2 )   k   y 4   k   y 3   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 k y 31 k … k y 1 ) = y φ ( 32 ) k y φ ( 31 ) k … k y φ ( 1 ) , where φ ( i + 1 + 4 ( k − 1 ) ) = 8 i + k ,   i = 0 , … , 3 ,   k = 1 , … , 8 , and y 32 ,   y 31 ,   … ,   y 1 are 8-bit sub-blocks of Y.

Three constants:

C₂ = 0

C₃ = 0xff00ffff000000ffff0000ff00ffff0000ff00ff00ff00ffff00ff00ff00ff00

C₄ = 0

The algorithm:

1. U   :=   H i n , V   :=   m , W   :=   U   ⊕   V , K 1   =   P ( W )
2. For j = 2,3,4 do the following:

Enciphering transformation

After the keys generation, the enciphering of H i n 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 i n is split into four 64-bit blocks: H i n = h 4 k h 3 k h 2 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 k s 3 k s 2 k s 1 .

Shuffle transformation

On the last step, the shuffle transformation is applied to H i n , S and m using a Linear feedback shift register. In the result, the intermediate hash value H o u t is obtained.

First we define the ψ function, doing LFSR on a 256-bit block: ψ ( Y ) = ψ ( y 16 k y 15 k . . . k y 2 k y 1 ) = ( y 1 ⊕ y 2 ⊕ y 3 ⊕ y 4 ⊕ y 13 ⊕ y 16 ) k y 16 k y 15 k . . . k y 3 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 o u t = ψ 61 ( H i n ⊕ ψ ( m ⊕ ψ 12 ( S ) ) ) , where ψ i denotes an i-th power of the ψ 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.

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.

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 preimage attacks in 2¹⁹² time (2ⁿ time refers to the approximate number of times the algorithm was calculated in the attack).

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 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

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

GOST("This is message, length=32 bytes") = b1c466d37519b82e8319819ff32595e047a28cb6f83eff1c6916a815a637fffaGOST("Suppose the original message has length = 50 bytes") = 471aba57a60a770d3a76130635c1fbea4ef14de51f78b4ae57dd893b62f55208

More test vectors:

GOST("") = ce85b99cc46752fffee35cab9a7b0278abb4c2d2055cff685af4912c49490f8dGOST("a") = d42c539e367c66e9c88a801f6649349c21871b4344c6a573f849fdce62f314ddGOST("message digest") = ad4434ecb18f2c99b60cbe59ec3d2469582b65273f48de72db2fde16a4889a4dGOST( 128 characters of 'U' ) = 53a3a3ed25180cef0c1d85a074273e551c25660a87062a52d926a9e8fe5733a4GOST( 1000000 characters of 'a' ) = 5c00ccc2734cdd3332d3d4749576e3c1a7dbaf0e7ea74e9fa602413c90a129fa

Hashes for CryptoPro parameters

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

GOST("") = 981e5f3ca30c841487830f84fb433e13ac1101569b9c13584ac483234cd656c0GOST("a") = e74c52dd282183bf37af0079c9f78055715a103f17e3133ceff1aacf2f403011GOST("abc") = b285056dbf18d7392d7677369524dd14747459ed8143997e163b2986f92fd42cGOST("message digest") = bc6041dd2aa401ebfa6e9886734174febdb4729aa972d60f549ac39b29721ba0GOST("The quick brown fox jumps over the lazy dog") = 9004294a361a508c586fe53d1f1b02746765e71b765472786e4770d565830a76GOST("ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789") = 73b70a39497de53a6e08c67b6d4db853540f03e9389299d9b0156ef7e85d0f61GOST("12345678901234567890123456789012345678901234567890123456789012345678901234567890") = 6bc7b38989b28cf93ae8842bf9d752905910a7528a61e5bce0782de43e610c90GOST("This is message, length=32 bytes") = 2cefc2f7b7bdc514e18ea57fa74ff357e7fa17d652c75f69cb1be7893ede48ebGOST("Suppose the original message has length = 50 bytes") = c3730c5cbccacf915ac292676f21e8bd4ef75331d9405e5f1a61dc3130a65011GOST(128 of "U") = 1c4ac7614691bbf427fa2316216be8f10d92edfd37cd1027514c1008f649c4e8GOST(1000000 of "a") = 8693287aa62f9478f7cb312ec0866b6c4e4a0f11160441e8f4ffcd2715dd554f