Quasigroup String Processing - theory and applications (in cryptography, PRNG, fractal designs, ...)
by
Smile Markovski
Institut of Informatics, Faculty of Sciences, St. Cyril and Methodius University, Skopje, Macedonia

Quasigroup string processings can be defined in several ways and here we define two simplest (and more important) ones. Given a finite set A we denote by A⁺ the set of all finite strings on A. Let * be a quasigroup operation on A. Then another quasigroup operation \ on A is induced by * as follows:

x*y=z <===> x\z = y

and the following equations

x\(xy)=y,     x*(x\y)=y

are identities in the algebra (A, *, \). Fix an element l=a₀=b₀ in A and define two transformations E_l, D_l:A⁺ --> A⁺ as follows. Let a₁... a_n in A⁺. Then

El(a1... an)=b1... bn <===> bi+1=bi*ai+1,  i=0, 1, ..., n-1

Dl(a1... an)=c1... cn <===> ci+1=ai\ai+1,  i=0, 1, ..., n-1

We have that ED=1=DE, i.e E and D are permutations, and this allows us to use E and D for defining suitable encription and decription functions.

The most important properties of E and D are the following. Let E_{l1... lk} = E_l1... E_lk,  D_{l1... lk} = D_l1... D_lk. Then we have:

The distributions of s-tuple in E_{l1... lk}(a₁... a_n)  (1 <= s <= k) for enough large arbitrary string

Date received: April 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-55.