Vectorial Boolean Functions

Introduction

Let $\mathbb {F} _{2}^{n}$ be the vector space of dimension $n$ over the finite field $\mathbb {F} _{2}$ with two elements. Functions from $\mathbb {F} _{2}^{m}$ to $\mathbb {F} _{2}^{n}$ are called $(m,n)$-functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.

Any $(m,n)$-function $F$ can be written as a vector $F=(f_{1},f_{2},\ldots f_{m})$ of $n$-dimensional Boolean functions $f_{1},f_{2},\ldots f_{m}$ which are called the coordinate functions of $F$.

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.