2 research outputs found
A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares
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 = (lij ) of order n over a field K as the set of zeros
of the binomial ideal xi xj − xlij
: (i, j) is a non-empty cell inL ⊆ K[x1, . . . , xn].
Since 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