A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Abstract

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (l ij ) of order n over a field K as the set of zeros of the binomial idealSince isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic. KeywordsSymmetric-key algorithm • Image pattern • Partial Latin square • Affine algebraic set • Isomorphism Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

“…Notice that the second condition that we have just described was not explicitly indicated by Falcón, R.M. et al [36]. Nevertheless, as is illustrated in the following example, this condition is mandatory in order to obtain a symmetric relation.…”

“…Although the original definitions were established by Falcón, R.M. et al [36] as equivalence relations, it is not so. In what follows, we particularize both definitions in order to obtain two new equivalence relations among partial Latin squares of the same order and weight.…”

“…The subset P ⊆ Ent(L 1 ) ∩ Ent(L 2 ) was not established as an essential part of the original definition of being partial isotopic introduced by Falcón, R.M. et al [36]. Nevertheless, if it is not considered as such, then the resulting binary relation is intransitive.…”

“…(The case n = 3 was already analyzed by Falcón, R.M. et al [36].) A representative of each one of these classes is described in Figure 1.…”

“…In any case, the study of affine algebraic sets associated to Latin squares is still in the very initial stage. Thus, for instance, their distribution into isomorphism classes is only known for n ≤ 3 [36]. To deal with higher orders, it is necessary to delve into two new ways of classifying partial Latin squares.…”