On the Comparison of Cryptographical Properties of Two Different Families of Graphs with Large Cycle Indicator

Michał Klisowski,Vasyl Ustimenko

doi:10.1007/s11786-012-0121-x

Michał Klisowski, Vasyl Ustimenko

Open Access

https://doi.org/10.1007/s11786-012-0121-x

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Abstract

The paper is devoted to the implementations of the public key algorithms based on simple algebraic graphs A(n, K) and D(n, K) defined over the same finite commutative ring K. If K is a finite field both families are families of graphs with large cycle indicator. In fact, the family D(n, F q ) is a family of graphs of large girth (f.g.l.g.) with c = 1, their connected components CD(n, F q ) form the f.g.l.g. with the speed of growth 4/3. Family A(n, q), char F q ≠ 2 is a family of connected graphs with large cycle indicator with the largest possible speed of growth. The computer simulation demonstrates the advantage (better density which is the number of monomial expressions) of public rules derived from A(n, q) in comparison with symbolic algorithm based on graphs D(n, q).

Highlights

Multivariate cryptography in the narrow sense is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over finite fields
If the polynomials have the degree two, we talk about multivariate quadratics
In publications [15] classes of stream ciphers and public key algorithms based on explicit construction of families of algebraic graphs of large girth D(n, q) and their generalisations D(n, K ), where K is general commutative ring (D(n, Fq ) = D(n, q)) were proposed

Summary

Introduction

Multivariate cryptography in the narrow sense (see Wikipedia) is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over finite fields. In publications [15] classes of stream ciphers and public key algorithms based on explicit construction of families of algebraic graphs of large girth D(n, q) and their generalisations D(n, K ), where K is general commutative ring (D(n, Fq ) = D(n, q)) were proposed It was shown later [42] that for each finite commutative ring K we can create a cubical polynomial map f of K n onto K n depending on string of regular elements 4 we consider an explicit construction of a family of affine algebraic digraphs of large girth over each finite commutative ring containing at least 3 regular elements Different properties of this family are investigated in [23,24,33,34,36,37].

On the Families of Directed Graphs of Large Girth

We can construct the public key map in the following manner

On the Family of Affine Digraph of Large Girth over Commutative Rings

On the Family of Graph of Large Cycle Indicator

On the Time Evaluation for the Public Rule

Conclusion