There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order.

The Golomb-Welch conjecture deals with the existence of perfect eerror correcting Lee codes of word length n, P L(n, e) codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear P L(n, 2) codes for n ≤ 12. Using this new approach we also construct the first quasiperfect Lee codes for dimension n = 3, and show that, for fixed n, there are only finitely many such codes over Z.

No abstract

We present a new algorithm for a decision problem if two Latin squares are isotopic. Our modification has the same complexity as Miller’s algorithm, but in many practical situations is much faster. Based on our results we study also a zero-knowledge protocol suggested in [3]. From our results it follows that there are some problems in practical application of this protocol.

No abstract

Classical ciphers are used to encrypt plaintext messages written in a natural language in such a way that they are readable for sender or intended recipient only. Many classical ciphers can be broken by brute-force search through the key-space. Methods of artificial intelligence, such as optimization heuristics, can be used to narrow the search space, to speed-up text processing and text recognition in the cryptanalytic process. Here we present a broad overview of different AI techniques usable in cryptanalysis of classical ciphers. Specific methods to effectively recognize the correctly decrypted text among many possible decrypts are discussed in the next part Automated cryptanalysis – Language processing.

No abstract

It is well known that a congruence ax Á b .mod n/ has a solution if and only if gcd.a; n/ j b, and, if the condition is satisfied, the number of incongruent solutions equals gcd.a; n/. In 2010, Alomair, Clark and Poovendran proved that the congruence ax Á b .mod n/ has a solution coprime to n if and only if gcd.a; n/ D gcd.b; n/, as an auxiliary result playing a key role in a problem related to an electronic signature. In this paper we provide a concise proof of this result, together with a closed formula for the number of incongruent solutions coprime to n as well. Moreover, a bound is presented for the probability that, for randomly chosen a; b 2 Z, this congruence possesses at least one solution coprime to n.