Abstract
A b s t r a c t: Given a ¯nite alphabet A and a quasigroup operation ¤ on the set A, in earlier paper of ours we have de¯ned the quasigroup transformation E : A+ ! A+, where A+ is the set of all ¯nite strings with letters from A. Here we present several generalizations of the transformation E and we consider the conditions under which the transformed strings have uniform distributions of n-tuples of letters of A. The obtained results can be applied in cryptography, coding theory, de¯ning and improving pseudo random generators, and so on.
Highlights
Given a finite alphabet A and a quasigroup operation ∗ on the set A, in earlier paper of ours we have defined the quasigroup transformation E : A+ → A+, where A+ is the set of all finite strings with letters from A
The transformation E defined in (3) is not the only one that can be used as quasigroup string transformation
A Gd transformation was defined by using the quasigroup (6)
Summary
The quasigroup string transformations E and D, and their properties, were considered in several papers ([3], [4], [5], [6], [7]). Starting with an input message where the distribution of the letters is uniform, after one application of an E-transformation an output message with uniformly distributed pairs of letters will be obtained. That, after applying an E-transformation on arbitrary input message, an output message with uniformly distributed letters is obtained. It is shown that for the generalized transformations the following holds: if an input message has uniformly distributed n-tuples of letters, the output message has uniformly distributed n + d-tuples of letters for 1 ≤ d ≤ n.
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