Vectorial Boolean Functions

Introduction

Let [math]\displaystyle{ \mathbb{F}_2^n }[/math] be the vector space of dimension [math]\displaystyle{ n }[/math] over the finite field [math]\displaystyle{ \mathbb{F}_2 }[/math] with two elements. Functions from [math]\displaystyle{ \mathbb{F}_2^m }[/math] to [math]\displaystyle{ \mathbb{F}_2^n }[/math] are called [math]\displaystyle{ (m,n) }[/math]-functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.

Any [math]\displaystyle{ (m,n) }[/math]-function [math]\displaystyle{ F }[/math] can be written as a vector [math]\displaystyle{ F = (f_1, f_2, \ldots f_m) }[/math] of [math]\displaystyle{ n }[/math]-dimensional Boolean functions [math]\displaystyle{ f_1, f_2, \ldots f_m }[/math] which are called the coordinate functions of [math]\displaystyle{ F }[/math].

Cryptanalytic attacks

Vectorial Boolean functions, also referred to as "S-boxes", or "Substitution boxes", in the context of cryptography, are a fundamental building block of block ciphers and are crucial to their security: more precisely, the resistance of the block cipher to cryptanalytic attacks directly depends on the properties of the S-boxes used in its construction.

The main types of cryptanalytic attacks that result in the definition of design criteria for S-boxes are the following:

• the differential attack introduced by Biham and Shamir; to resist it, an S-box must have low differential uniformity;
• the linear attack introduced by Matsui; to resist it, an S-box must have high nonlinearity;
• the higher order differential attack; to resist it, an S-box must have high algebraic degree;
• the interpolation attack; to resist it, the univariate representation of an S-box must have high degree, and its distance to the set of low univariate degree functions must be large;
• algebraic attacks.