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 $\mathbb {F} _{2}^{m}$ are called $(n,m)$-functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.

Any $(n,m)$-function $F$ can be written as a vector $F=(f_{1},f_{2},\ldots f_{n})$ of $m$-dimensional Boolean functions $f_{1},f_{2},\ldots f_{n}$ 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 $F:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F} _{2}^{m}$ is the integer-valued function $W_{F}:\mathbb {F} _{2}^{n}\times \mathbb {F} _{2}^{m}$ defined by

$W_{F}(u,v)=\sum _{x\in \mathbb {F} _{2}^{n}}(-1)^{v\cdot F(x)+u\cdot x}$

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

$1_{G_{F}}(x,y)={\begin{cases}1&F(x)=y\\0&F(x)\neq y.\end{cases}}$

The Walsh spectrum of $F$ is the multi-set of all the values of its Walsh transform for all pairs $(u,v)\in \mathbb {F} _{2}^{n}\times {\mathbb {F} _{2}^{m}}^{*}$. 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 $(u,v)$ for which $W_{F}(u,v)\neq 0$.

Representations

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

Algebraic Normal Form

An $(n,m)$-function $F$ can be uniquely represented as a polynomial with coefficients in $\mathbb {F} _{2}^{m}$ of the form

$F(x)=\sum _{I\in {\cal {P}}(N)}a_{I}\,\left(\prod _{i\in I}x_{i}\right)=\sum _{I\in {\cal {P}}(N)}a_{I}\,x^{I},$

where ${\cal {P}}(N)$ is the power set of $N=\{1,\ldots ,n\}$ and the coefficients $a_{I}$ belong to $\mathbb {F} _{2}^{m}$. This representation is known as the algebraic normal form (ANF) of $F$. The algebraic degree of $F$, denoted $d^{\circ }(F)$ is then defined as the global degree of its ANF, i.e.

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 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} .

Univariate Representation

If we identify the vector space 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 \mathbb{F}_2^n} with the finite field 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 \mathbb{F}_{2^n}} with 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 2^n} elements, then any $(n,n)$-function can be uniquely represented as univariate polynomial over 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 \mathbb{F}_{2^n}} of degree at most 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 2^n-1} taking the form

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(x)=\sum_{j=0}^{2^n-1}\delta_jx^j~,~~\delta_j\in \Bbb{F}_{2^n}. }

This is the univariate representation 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} .

The algebraic degree 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} can be expressed using the 2-weight of the exponents of the univariate representation. Given a positive integer $j$, its 2-weight is the number of summands in its representation as a sum of powers of two; that is, if we write 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 j = \sum_{i = 0}^{n-1} j_s \cdot 2^s} for 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 j_s \in \{0,1\}} , then its 2-weigt is 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 w_2(j) = \sum_{i = 0}^{n-1} j_s} . The algebraic degree of $F$ can the be written as

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 \max_{j=0,\dots ,2^n-1/\, \delta_j\neq 0}w_2(j). }

Multi-dimensional Walsh Transform

TODO

Basic Properties of Vectorial Boolean Functions

Balanced Functions

An 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 (n,m)} -function 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} is called balanced if it takes every value 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 \mathbb{F}_2^m} precisely 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 2^{n-m}} times. In particular, an $(n,n)$-function is balanced if and only if it is a permutation.

An 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 (n,m)} -function 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} is balanced if and only if its component functions are balanced, i.e. if and only if 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 v \cdot F} is balanced for every 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 v \in {\mathbb{F}_2^m}^*} .

Covering Sequences

TODO

Algebraic Immunity

TODO