Testing the properties of large quasigroups

Abstract

With the growing importance of data security a growing effort to find new approaches to the cryptographic algorithms designs appears. One of the trends is to research the use of other algebraic structures than the traditional, such as a quasigroup. Quasigroups are equivalent to the more familiar Latin squares. There are many characteristics that must quasigroups have from the cryptography point of view. They have to be non-commutative, non-associative, non-idempotent and so on. If one want to work with quasigroups of a large order, effective methods of testing their properties are necessary. In this paper we present several experiments on various types of guasigroups and their results.

An acceleration of quasigroup operations by residue arithmetic

“…The properties of hash algorithms based on quasigroups were investigated intensively. 24,47 A statistical evaluation of hash functions utilizing selected quasigroups of different orders (5003, 2 16 , and 2 64 ) and one group of order 5003 confirmed that the statistical properties of hash values indeed depend on the quality of the underlying algebraic structures. The analysis of hashes obtained for a large text data set showed that the use of an unsuitable structure (eg, an associative group) results in a bad distribution of the number of changed bits and causes a fault in the avalanche property.…”

“…An example of multiplication table for the quasigroup of modular subtraction of order n, 24 defined by An example of multiplication table for the quasigroup of modular subtraction of order n, 24 defined by…”

“…The multiplication operation is in a quasigroup defined either by a multiplication table or by alternative methods such as functional equations, 46 vector valued Boolean functions, 5,14,57,58 multivariate quadratic polynomials, 4,5,14,28 and, eg, analytic formulas. 24,25,47 The multiplication tables of finite quasigroups of order n have in each row and column only unique elements from Q and correspond to latin squares. In another words, every row and every column of a multiplication table are permutations of all elements from Q.…”

“…7,34-37 The underlying mathematical principles are sound 11,13 and can be used for symbolic computations. 24,25,32,[42][43][44] Computational experiments demonstrated that the randomness of sequences obtained by quasigroup transformations is influenced by the used quasigroup, 32 and the computation speed of a quasigroup-based hash function depends on the order of the used quasigroup as well. 11,16,17,34,38 They can be easily implemented on highly parallel 11,14,[21][22][23]39 and resource constrained (embedded) 5,13,[17][18][19][20]34,37,38,40 platforms.…”

“…The use of large quasigroups can further improve the strength of cryptographic operations. 24,25,32,[42][43][44] Computational experiments demonstrated that the randomness of sequences obtained by quasigroup transformations is influenced by the used quasigroup, 32 and the computation speed of a quasigroup-based hash function depends on the order of the used quasigroup as well. Alternative quasigroup representations that do not store their multiplication tables in computer memory yield increased computational costs.…”