Boolean Functions

Introduction

Let $\mathbb {F} _{2}^{n}$ be the vector space of dimension n over the finite field with two elements. The vector space can also be endowed with the structure of the field, the finite field with $2^{n}{\mbox{ elements, }}\mathbb {F} _{2^{n}}$. A function $f:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F}$ is called a Boolean function in dimenstion n (or n-variable Boolean function).

Given $x=(x_{1},\ldots ,x_{n})\in \mathbb {F} _{2}^{n}$, the support of x is the set $supp_{x}=\{i\in \{1,\ldots ,n\}:x_{i}=1\}$. The Hamming weight of x is the size of its support ($w_{H}(x)=|supp_{x}|$). Similarly the Hamming weight of a Boolean function f is the size of its support, i.e. the set $\{x\in \mathbb {F} _{2}^{n}:f(x)\neq 0\}$. The Hamming distance of two functions f,g is the size of the set $\{x\in \mathbb {F} _{2}^{n}:f(x)\neq g(x)\}\ (w_{H}(f\oplus g))$.

Representation of a Boolean function

There exist different ways to represent a Boolean function. A simple, but often not efficient, one is by its truth-table. For example consider the following truth-table for a 3-variable Boolean function f.

x			f(x)
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	1
1	1	0	0
1	1	1	1

Algebraic normal form

An n-variable Boolean function can be represented by a multivariate polynomial over $\mathbb {F}$ of 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)=\bigoplus_{I\subseteq\{1,\ldots,n\}}a_i\Big(\prod_{i\in I}x_i\Big)\in\mathbb{F}_2[x_1,\ldots,x_n]/(x_1^2\oplus x_1,\ldots,x_n^2\oplus x_n). }

Such representation is unique and it is the algebraic normal form of f (shortly ANF).

The degree of the ANF is called the algebraic degree of the 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 d^0f=\max \{ |I| : a_I\ne0 \}} .

Trace representation

We identify the vector space with the finite field and we consider f an n-variable Boolean function of even weight (hence of algebraic degree at most n-1). The map admits a uinque representation as a univariate polynomial of 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\in\Gamma_n}\mbox{Tr}_{\mathbb{F}_{2^{o(j)}}/\mathbb{F}_2}(A_jx^j), \quad x\in\mathbb{F}_{2^n}, }

with Γ_n set of integers obtained by choosing one element in each cyclotomic coset of 2 ( mod 2ⁿ-1), o(j) size of the cyclotomic coset containing j, A_j ∈ 𝔽_2^o(j), Tr_{𝔽2^o(j)/𝔽2} trace function from 𝔽_{2^o(j) to 𝔽2.}

Such representation is also called the univariate representation .

f can also be simply presented in 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 \mbox{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(P(x))} where P is a polynomial over the finite field F_2ⁿ but such representation is not unique, unless o(j)=n for every j such that A_j≠0.