A hashing algorithm in the symmetric group

Implementations based on the MSc thesis "A hashing algorithm based on a one-way function in the symmetric group" :
https://lnu.diva-portal.org/smash/record.jsf?pid=diva2%3A1662325&dswid=5727

Two new operations $(\gg ,\ll )$ work as mirrored one way functions in $S_n$ , in the sense that:

• Given $b$ and $a\gg b$ or $b\ll a$ it is hard to retrieve $a$.
• $(a\gg b)\ll a=a\gg (b\ll a)=b$.

We develop a hashing algorithm based on these operations by encoding blocks into permutations and exploiting the algebraic incompatibility of the operations with the usual product of elements in ${\mathbb{Z}}_p^{\ast }$ .

C++ implementation

main.cpp
hash_sn.cpp
hash_sn.h

The Boost library is needed for the compilation (https://www.boost.org/).

Example on windows command line:

Compilation:

C:\path>g++ main.cpp hash_sn.cpp -I"include" -o hash_sn.exe  

Execution:

C:\path> hash_sn.exe digest_message.txt  
Reading file...  
Digesting message...  
Hash of digest_message.txt: 528a81a97baa7d4f4a08465852ce7369  

Python implementation + empirical tests.

hash_sn.py -> Hashing algorithm
hash_aux.py -> Auxiliary functions for hash_sn
hash_testing.py -> Empirical tests on the constructions $(\gg ,\ll )$
two_block_attack_hash_sn.py -> Simulate 2-block attack for small sizes (16-bit, 24-bit etc),
finite_field_arithmetic.py -> Compute inverses modulo a prime using Fermat's little theorem.

Obtain hashing value of string

• Parameters:
  m -> String to hash
  s -> Block size (128, 256, 512, 1024...)
  t -> Factorial for embedding in the symmetric group. If s=128, then t>34.
  p -> Prime between s and t! (https://bigprimes.org/).

• Small string digest example:

>>> p = 9272585787985760943894005456578885141087
>>> hash_sn.hash_sn('The quick brown fox jumps over the lazy dog',128 ,35, p).hash()
'f308af47709e72a537c1545eb91d0e67'

• File digest example:

>>> p = 1288079068764670493881163748072332651218703668359555082283935082653270651535749
>>> hash_sn.hash_sn(open('digest_message.txt').read(), 256, 58, p).hash()
'94d6f93bfc4b68cc941ad6ba2a6209c5f5479df07a64fc8c093674eb760dc363'

Empirical tests (see Section 2.4 of the thesis)

• Given a construction of ($\gg$), plot the density distribution of the classes of $(S_n/\sim )$:

>>> hash_testing.classes(t)

• Test avalanche effect of ($\gg$, $\ll$) by plotting $a\stackrel{k}{\gg }(b+i)$ for $i,k=0,1,2,3,4$:

>>> hash_testing.avalanche(a, b, s, t)

• Simulate the domain of hashing one single block (block size = 16 bits) and collisions encountered:

>>> hash_testing.one_block_domain(s, t)

• Plot a range $R$ of mappings $a\gg b$:

>>> hash_testing.randomness(t,R)

• Block cipher construction attempts:

>>> hash_testing.block_cipher_encrypt(block, key, s, t)