Sequencings and Directed Graphs with Applications to Cryptography

Abstract

A motivation for this paper is to find a practical mechanism for generating permutations of a finite set of consecutive positive integers so that the resultant spacings between originally consecutive numbers i and i + i, are now different for each i. This is equivalent to finding complete Latin squares. The ordered set of spacings in such a permutation is called a sequencing. The set of partial sums of the terms in a sequencing is called a directed terrace. For lack of standard terminology, the associated permutations here are called quick trickles.

This paper concerns methods of finding such sequencings, in part by finding constraints on their existence, so that search time can be substantially reduced. A second approach is to represent a quick trickle permutation as a directed graph. The sequencings then are represented by chords of different lengths. Various methods can be used to rearrange the chords and obtain additional sequencings and groups of quick trickle permutations.

It is well known that complete Latin squares of size n ×n can be found if n is even but not if n is odd. This is equivalent to saying that the group of integers (1, 2, 3,..., n) under the operation of multiplication mod n is sequenceable if and only if n is even. A more general concept is introduced which is termed quasi-sequenceable. This applies to both even and odd sizes.

The application to block encryption, or the so-called substitution permutation system is briefly described. The net result is interround mixing, deterministically generated under key control, and quickly replaceable with a new pattern of equal merit.

Currently, typical block encryption systems use algorithms which are fixed and publicly known. The motivation here is to develop block encryption systems using algorithms which are variable, secret, generated and periodically changed by the key.