Algebraic immunity of vectorial boolean functions and boolean groebner bases

A.N Alekseychuk

doi:10.20535/tacs.2664-29132020.1.209431

Abstract

The basic concepts and results related to the Boolean Groebner bases and their application for computing the algebraic immunity of vectorial Boolean functions are considered. This parameter plays an important role for the security evaluation of block ciphers against algebraic attacks. Unlike the available works, the description is carried out at the elementary level using terms of Boolean functions theory. In addition, obtained proofs are shorter than the previous ones. This allows us to achieve significant progress in building the fundamentals of the theory (for the Boolean case) using only elementary methods.The paper can be useful for students and postgraduate students studying cryptology. It may also save time for professionals who want to get familiar with the mathematical techniques used in algebraic attacks on block ciphers.

Highlights

Introduction of the decision problem directly by the
Unlike the available works, the delem including the concepts of the (Boolean) Groebner scription is carried out with the help of elementary techbasis and algebraic immunity of vectorial Boolean func- niques and obtained proofs are shorter
At present there are several definitions of algebraic in contrast to the traditional approach to Groebner immunity of vectorial Boolean functions [1, 4, 5, 6, 7], bases of polynomial ideals, the deamong which the definition given by Ars-Faugère [4] is scription in the paper is based on the terms of Boolean the most appropriate from a practical point of view. functions theory

Summary

Introduction

Introduction of the decision problem (whether or not the algebraic immunity is above the specified threshold) directly by the. The main purpose is to prove the Ars-Faugère theorem [4], which makes possible to find algebraic immunity along with all equations of lowest degree. These equations result from the system of equations that describes a given vectorial Boolean function. At present there are several definitions of algebraic in contrast to the traditional approach to Groebner immunity of vectorial Boolean functions [1, 4, 5, 6, 7], bases of polynomial ideals (see [8], for example), the deamong which the definition given by Ars-Faugère [4] is scription in the paper is based on the terms of Boolean the most appropriate from a practical point of view.

Objectives

Conclusion