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* Parseval's relation: <center><math>\sum_{u\in\mathbb{F}_2^n}W_f^2(u)=2^{2n};</math></center>
* Parseval's relation: <center><math>\sum_{u\in\mathbb{F}_2^n}W_f^2(u)=2^{2n};</math></center>
* Poisson summation formula: for any vector subspace ๐ธ of ๐”ฝ<sub>2</sub><sup>๐‘›</sup> and for any elements ๐‘Ž,๐‘ in ๐”ฝ<sub>2</sub><sup>๐‘›</sup> <center><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></center>ย  for ๐ธ<sup>โŸ‚</sup> the orthogonal subspace of ๐ธ,{๐‘ขโˆˆ๐”ฝ<sub>2</sub><sup>๐‘›</sup> : ๐‘ขยท๐‘ฅ=0, for all ๐‘ฅโˆˆ๐ธ}.
* Poisson summation formula: for any vector subspace ๐ธ of ๐”ฝ<sub>2</sub><sup>๐‘›</sup> and for any elements ๐‘Ž,๐‘ in ๐”ฝ<sub>2</sub><sup>๐‘›</sup> <center><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></center>ย  for ๐ธ<sup>โŸ‚</sup> the orthogonal subspace of ๐ธ,{๐‘ขโˆˆ๐”ฝ<sub>2</sub><sup>๐‘›</sup> : ๐‘ขยท๐‘ฅ=0, for all ๐‘ฅโˆˆ๐ธ}.
=Equivalence of Boolean functions=
Two ๐‘›-variable Boolean functions ๐‘“,๐‘” are called <i>extended-affine equivalent</i> (shortly EA-equivalent) if there exists a linear automorphism ๐ฟ, an affine Boolean function ๐“ and a vecor ๐‘Ž such that ๐‘”(๐‘ฅ) = ๐‘“(๐ฟ(๐‘ฅ)+๐‘Ž)+๐“(๐‘ฅ).
A parameter that is preserved by EA-equivalence is called <i>EA-invariant</i>.

Revision as of 09:29, 27 September 2019

Introduction

Let ๐”ฝ2๐‘› be the vector space of dimension ๐‘› over the finite field with two elements. The vector space can also be endowed with the structure of the field, the finite field with 2๐‘› elements, ๐”ฝ2๐‘›. A function [math]\displaystyle{ f : \mathbb{F}_2^n\rightarrow\mathbb{F} }[/math] is called a Boolean function in dimenstion ๐‘› (or ๐‘›-variable Boolean function).

Given [math]\displaystyle{ x=(x_1,\ldots,x_n)\in\mathbb{F}_2^n }[/math], the support of x is the set [math]\displaystyle{ supp_x=\{i\in\{1,\ldots,n\} : x_i=1 \} }[/math]. The Hamming weight of ๐‘ฅ is the size of its support ([math]\displaystyle{ w_H(x)=|supp_x| }[/math]). Similarly the Hamming weight of a Boolean function ๐‘“ is the size of its support, i.e. the set [math]\displaystyle{ \{x\in\mathbb{F}_2^n : f(x)\ne0 \} }[/math]. The Hamming distance of two functions ๐‘“,๐‘” is the size of the set [math]\displaystyle{ \{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 ๐‘“.

๐‘ฅ ๐‘“(๐‘ฅ)
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 ๐‘›-variable Boolean function can be represented by a multivariate polynomial over ๐”ฝ2 of the form

[math]\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). }[/math]

Such representation is unique and it is the algebraic normal form of ๐‘“ (shortly ANF).

The degree of the ANF is called the algebraic degree of the function, ๐‘‘ยฐ๐‘“=max { |๐ผ| : ๐‘Ž๐ผ≠0 }.

Trace representation

We identify the vector space with the finite field and we consider ๐‘“ an ๐‘›-variable Boolean function of even weight (hence of algebraic degree at most ๐‘›-1). The map admits a uinque representation as a univariate polynomial of the form

[math]\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}, }[/math]

with ฮ“๐‘› set of integers obtained by choosing one element in each cyclotomic coset of 2 ( mod 2๐‘›-1), ๐˜ฐ(๐˜ซ) size of the cyclotomic coset containing ๐˜ซ, ๐˜ˆ๐˜ซ ∈ ๐”ฝ2๐˜ฐ(๐˜ซ), Tr๐”ฝ2๐˜ฐ(๐˜ซ)/๐”ฝ2 trace function from ๐”ฝ2๐˜ฐ(๐˜ซ) to ๐”ฝ2.

Such representation is also called the univariate representation .

๐‘“ can also be simply presented in the form [math]\displaystyle{ \mbox{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(P(x)) }[/math] where ๐˜— is a polynomial over the finite field F2๐‘› but such representation is not unique, unless ๐˜ฐ(๐˜ซ)=๐‘› for every ๐˜ซ such that ๐˜ˆ๐˜ซ≠0.

The Walsh transform

The Walsh transform ๐‘Š๐‘“ is the descrete Fourier transform of the sign function of ๐‘“, i.e. (-1)๐‘“(๐‘ฅ). With an innner product in ๐”ฝ2๐‘› ๐‘ฅยท๐‘ฆ, the value of ๐‘Š๐‘“ at ๐‘ขโˆˆ๐”ฝ2๐‘› is the following sum (over the integers)

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

The set [math]\displaystyle{ \{ u\in\mathbb{F}_2^n : W_f(u)\ne0 \} }[/math] is the Walsh support of ๐‘“.

Properties of the Walsh transform

For every ๐‘›-variable Boolean function ๐‘“ we have the following relations.

  • Inverse Walsh transform: for any element ๐‘ฅ of ๐”ฝ2๐‘› we have
    [math]\displaystyle{ \sum_{u\in\mathbb{F}_2^n}W_f(u)(-1)^{u\cdot x}= 2^n(-1)^{f(x)}; }[/math]
  • Parseval's relation:
    [math]\displaystyle{ \sum_{u\in\mathbb{F}_2^n}W_f^2(u)=2^{2n}; }[/math]
  • Poisson summation formula: for any vector subspace ๐ธ of ๐”ฝ2๐‘› and for any elements ๐‘Ž,๐‘ in ๐”ฝ2๐‘›
    [math]\displaystyle{ \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 ๐ธโŸ‚ the orthogonal subspace of ๐ธ,{๐‘ขโˆˆ๐”ฝ2๐‘› : ๐‘ขยท๐‘ฅ=0, for all ๐‘ฅโˆˆ๐ธ}.

Equivalence of Boolean functions

Two ๐‘›-variable Boolean functions ๐‘“,๐‘” are called extended-affine equivalent (shortly EA-equivalent) if there exists a linear automorphism ๐ฟ, an affine Boolean function ๐“ and a vecor ๐‘Ž such that ๐‘”(๐‘ฅ) = ๐‘“(๐ฟ(๐‘ฅ)+๐‘Ž)+๐“(๐‘ฅ). A parameter that is preserved by EA-equivalence is called EA-invariant.