# Difference between revisions of "Vectorial Boolean Functions"

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= Introduction = | = Introduction = | ||

− | + | Let <math>\mathbb{F}_2^n</math> be the vector space of dimension <math>n</math> over the finite field <math>\mathbb{F}_2</math> with two elements. Functions from <math>\mathbb{F}_2^m</math> to <math>\mathbb{F}_2^n</math> are called <span class="definition"><math>(m,n)</math>-functions</span> or simply <span class="definition">vectorial Boolean functions</span> when the dimensions of the vector spaces are implicit or irrelevant. | |

+ | |||

+ | Any <math>(m,n)</math>-function <math>F</math> can be written as a vector <math>F = (f_1, f_2, \ldots f_m)</math> of <math>n</math>-dimensional [[Boolean functions]] <math>f_1, f_2, \ldots f_m</math> which are called the <span class="definition">coordinate functions</span> of <math>F</math>. | ||

== Cryptanalytic attacks == | == 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. | ||

= Generalities on Boolean functions = | = Generalities on Boolean functions = |

## Revision as of 21:26, 30 December 2018

## Contents

# Introduction

Let be the vector space of dimension over the finite field with two elements. Functions from to are called -functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.

Any -function can be written as a vector of -dimensional Boolean functions which are called the coordinate functions of .

## 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.