Classification of quasigroups by image patterns
Date Issued
2007
Author(s)
Abstract
Given a finite quasigroup (Q, ∗), we define a quasigroup string
transformation e over the strings of elements from Q by e(a1a2 . . . an) =
b1b2 . . . bn if and only if bi = bi−1 ∗ ai for each i = 1, 2, . . . , n, where b0 is a
fixed element of Q, and ai are elements from Q. These kind of quasigroup
string transformations are used for designing several cryptographic primitives and error-correcting codes. Not all quasigroups are suitable for that
kind of designs. The set of quasigroups of given order can be separated in
two disjoint classes, the class of so called fractal quasigroups and the class of
non-fractal quasigroups. The classification is obtained by presenting several
consecutive sequences generated by e−transformations and their presentation in matrix form, used to produce suitable image pattern. We note that
the fractal quasigroups are usually not suitable for designing cryptographic
primitives.
transformation e over the strings of elements from Q by e(a1a2 . . . an) =
b1b2 . . . bn if and only if bi = bi−1 ∗ ai for each i = 1, 2, . . . , n, where b0 is a
fixed element of Q, and ai are elements from Q. These kind of quasigroup
string transformations are used for designing several cryptographic primitives and error-correcting codes. Not all quasigroups are suitable for that
kind of designs. The set of quasigroups of given order can be separated in
two disjoint classes, the class of so called fractal quasigroups and the class of
non-fractal quasigroups. The classification is obtained by presenting several
consecutive sequences generated by e−transformations and their presentation in matrix form, used to produce suitable image pattern. We note that
the fractal quasigroups are usually not suitable for designing cryptographic
primitives.
Subjects
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