Boolean Functions

Introduction

Let 𝔽₂^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 elements, 𝔽_2^𝑛. A function <math>f : \mathbb{F}_2^n\rightarrow\mathbb{F}</math> is called a Boolean function in dimenstion 𝑛 (or 𝑛-variable Boolean function).

Given <math>x=(x_1,\ldots,x_n)\in\mathbb{F}_2^n</math>, the support of x is the set <math>supp_x=\{i\in\{1,\ldots,n\} : x_i=1 \}</math>. The Hamming weight of 𝑥 is the size of its support (<math>w_H(x)=|supp_x|</math>). Similarly the Hamming weight of a Boolean function 𝑓 is the size of its support, i.e. the set <math>\{x\in\mathbb{F}_2^n : f(x)\ne0 \}</math>. The Hamming distance of two functions 𝑓,𝑔 (𝖽_𝐻(𝑓,𝑔)) is the size of the set <math>\{x\in\mathbb{F}_2^n : f(x)\neq g(x) \}\ (w_H(f\oplus g))</math>.

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 𝔽₂ of the form

<math> 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). </math>

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, 𝑑°𝑓=max { |𝐼| : 𝑎_𝐼≠0 }.

Based on the algebraic degree we called 𝑓

• affine if 𝑑°𝑓=1, linear if 𝑑°𝑓=1 and 𝑓(𝟎)=0;
• quadratic if 𝑑°𝑓=2.

Affine functions are of the form 𝑓(𝑥)= 𝑢⋅𝑥+𝑒, for 𝑢∈𝔽₂^𝑛 and 𝑒∈𝔽₂

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

<math> 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}, </math>

with Γ_𝑛 set of integers obtained by choosing one element in each cyclotomic coset of 2 ( mod 2^𝑛-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 <math> \mbox{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(P(x))</math> where 𝘗 is a polynomial over the finite field F_2^𝑛 but such representation is not unique, unless 𝘰(𝘫)=𝑛 for every 𝘫 such that 𝘈_𝘫≠0.

When we consider the trace representation of of a function, then the algebraic degree is given by <math>\max_{j\in\Gamma_n | A_j\ne0}w_2(j)</math>, where 𝓌₂(𝑗) is the Hamming weight of the binary expansion of 𝑗.

On the weight of a Boolean function

For 𝑓 a 𝑛-variable Booleand function the following relations about its weight are satisfied.

• If 𝑑°𝑓=1 then 𝓌_𝐻(𝑓)=2^𝑛-1.
• If 𝑑°𝑓=2 then 𝓌_𝐻(𝑓)=2^𝑛-1 or 𝓌_𝐻(𝑓)=2^𝑛-1±2^𝑛-1-ℎ, with 0≤ℎ≤𝑛/2.
• If 𝑑°𝑓≤𝑟 and 𝑓 nonzero then 𝓌_𝐻(𝑓)≥2^𝑛-𝑟.
• 𝓌_𝐻(𝑓) is odd if and only if 𝑑°𝑓=𝑛.

The Walsh transform

The Walsh transform 𝑊_𝑓 is the descrete Fourier transform of the sign function of 𝑓, i.e. (-1)^𝑓(𝑥). With an innner product in 𝔽₂^𝑛 𝑥·𝑦, the value of 𝑊_𝑓 at 𝑢∈𝔽₂^𝑛 is the following sum (over the integers)

<math>W_f(u)=\sum_{x\in\mathbb{F}_2^n}(-1)^{f(x)+x\cdot u},</math>

The set <math>\{ u\in\mathbb{F}_2^n : W_f(u)\ne0 \}=\{ u\in\mathbb{F}_2^n : W_f(u)=1 \}</math> is the Walsh support of 𝑓.

Properties of the Walsh transform

For every 𝑛-variable Boolean function 𝑓 we have the following relations.

• Inverse Walsh transform: for any element 𝑥 of 𝔽₂^𝑛 we have <math> \sum_{u\in\mathbb{F}_2^n}W_f(u)(-1)^{u\cdot x}= 2^n(-1)^{f(x)};</math>
• Parseval's relation: <math>\sum_{u\in\mathbb{F}_2^n}W_f^2(u)=2^{2n};</math>
• Poisson summation formula: for any vector subspace 𝐸 of 𝔽₂^𝑛 and for any elements 𝑎,𝑏 in 𝔽₂^𝑛 <math> \sum_{u\in a+E^\perp}(-1)^{b\cdot u}W_f(u) = |E^\perp|(-1)^{a\cdot b}\sum_{x\in b+E}(-1)^{f(x)+a\cdot x},</math> for 𝐸^⟂ the orthogonal subspace of 𝐸,{𝑢∈𝔽₂^𝑛 : 𝑢·𝑥=0, for all 𝑥∈𝐸}.

Equivalences of Boolean functions

Two 𝑛-variable Boolean functions 𝑓,𝑔 are called affine equivalent if there exists a linear automorphism 𝐿 and a vecor 𝑎 such that

𝑔(𝑥) = 𝑓(𝐿(𝑥)+𝑎).

Two 𝑛-variable Boolean functions 𝑓,𝑔 are called extended-affine equivalent (shortly EA-equivalent) if there exists a linear automorphism 𝐿, an affine Boolean function 𝓁 and a vecor 𝑎 such that

𝑔(𝑥) = 𝑓(𝐿(𝑥)+𝑎)+𝓁(𝑥).

A parameter that is preserved by an equivalence relation is called invariant.

• The degree is invariant under affine equivalence and, for not affine functions, also under EA-equivalence.
• If 𝑓,𝑔 are affine equivalent, then <math>W_g(u)=(-1)^{u\cdot L^{-1}(a)}W_f(L^{-1}(u))</math>.

The Nonlinearity

The nonlinearity of a function 𝑓 is defined as its minimal distance to affine functions, i.e. called 𝒜 the set of all affine 𝑛-variable functions,

<math> \mathcal{NL}(f)=\min_{g\in\mathcal{A}}d_H(f,g)</math>

• For every 𝑓 we have <math>\mathcal{NL}(f)=2^{n-1}-\frac{1}{2}\max_{u\in\mathbb{F}_2^n}|W_f(u)|</math>.
• From Parseval relation we obtain the covering radius bound <math>\mathcal{NL}(f)\le2^{n-1}-2^{n/2-1}</math>.
• A function achieving the covering radius bound with equality is called bent (𝑛 is an even integer).
• 𝑓 is bent if and only if 𝑊_𝑓(𝑢)=±2^𝑛/2, for every 𝑢∈𝔽₂^𝑛.