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 $\mathbb {F} _{2}^{n}$ to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {F} _{2}^{m}} are called Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (n,m)} -functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.

Any Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (n,m)} -function $F$ can be written as a vector Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=(f_{1},f_{2},\ldots f_{n})} of $m$-dimensional Boolean functions Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{1},f_{2},\ldots f_{n}} 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.

Generalities on Boolean functions

Walsh transform

The Walsh transform of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F} _{2}^{m}} is the integer-valued function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{F}:\mathbb {F} _{2}^{n}\times \mathbb {F} _{2}^{m}} defined by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{F}(u,v)=\sum _{x\in \mathbb {F} _{2}^{n}}(-1)^{v\cdot F(x)+u\cdot x}}

It can be observed that the Walsh transform of some $F$ is in fact the Fourier transform of the indicator of its graph, i.e. the Fourier transform of the function $1_{G_{F}}$ defined as

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1_{G_{F}}(x,y)={\begin{cases}1&F(x)=y\\0&F(x)\neq y.\end{cases}}}

The Walsh spectrum of $F$ is the multi-set of all the values of its Walsh transform for all pairs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (u,v)\in \mathbb {F} _{2}^{n}\times {\mathbb {F} _{2}^{m}}^{*}} . The extended Walsh spectrum of $F$ is the multi-set of the absolute values of its Walsh transform, and the Walsh support of $F$ is the set of pairs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (u,v)} for which Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle W_{F}(u,v)\neq 0} .

Representations

Vectorial Boolean functions can be represented in a number of different ways.

Algebraic Normal Form

An Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (n,m)} -function $F$ can be uniquely represented as a polynomial with coefficients in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {F} _{2}^{m}} of the form

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x)=\sum _{I\in {\cal {P}}(N)}a_{I}\,\left(\prod _{i\in I}x_{i}\right)=\sum _{I\in {\cal {P}}(N)}a_{I}\,x^{I},}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\cal {P}}(N)} is the power set of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=\{1,\ldots ,n\}} and the coefficients Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{I}} belong to $\mathbb {F} _{2}^{m}$. This representation is known as the algebraic normal form (ANF) of $F$. The algebraic degree of $F$, denoted Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d^{\circ }(F)} is then defined as the global degree of its ANF, i.e.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d^{\circ }(F)=\ max\{|I|/\,a_{I}\neq (0,\dots ,0);I\in {\cal {P}}(N)\}}

and is equal to the maximal algebraic degree of the coordinate functions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} .