Difference between revisions of "Boolean Functions"

From Boolean Functions
Jump to: navigation, search
m (Equivalence of Boolean functions)
Line 77: Line 77:
  
 
=Equivalence of Boolean functions=
 
=Equivalence of Boolean functions=
Two 𝑛-variable Boolean functions 𝑓,𝑔 are called <i>extended-affine equivalent</i> (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).
+
Two 𝑛-variable Boolean functions 𝑓,𝑔 are called <i>extended-affine equivalent</i> (shortly EA-equivalent) if there exists a linear automorphism 𝐿, an affine Boolean function 𝓁 and a vecor 𝑎 such that <center>𝑔(𝑥) = 𝑓(𝐿(𝑥)+𝑎)+𝓁(𝑥).</center>
 
A parameter that is preserved by EA-equivalence is called <i>EA-invariant</i>.
 
A parameter that is preserved by EA-equivalence is called <i>EA-invariant</i>.

Revision as of 10:29, 27 September 2019

Introduction

Let 𝔽2𝑛 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 2𝑛 elements, 𝔽2𝑛. A function is called a Boolean function in dimenstion 𝑛 (or 𝑛-variable Boolean function).

Given , the support of x is the set . The Hamming weight of 𝑥 is the size of its support (). Similarly the Hamming weight of a Boolean function 𝑓 is the size of its support, i.e. the set . The Hamming distance of two functions 𝑓,𝑔 is the size of the set .

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

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| : 𝑎𝐼≠0 }.

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

with Γ𝑛 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, 𝘈𝘫 ∈ 𝔽2𝘰(𝘫), Tr𝔽2𝘰(𝘫)/𝔽2 trace function from 𝔽2𝘰(𝘫) to 𝔽2.

Such representation is also called the univariate representation .

𝑓 can also be simply presented in the form where 𝘗 is a polynomial over the finite field F2𝑛 but such representation is not unique, unless 𝘰(𝘫)=𝑛 for every 𝘫 such that 𝘈𝘫≠0.

The Walsh transform

The Walsh transform 𝑊𝑓 is the descrete Fourier transform of the sign function of 𝑓, i.e. (-1)𝑓(𝑥). With an innner product in 𝔽2𝑛 𝑥·𝑦, the value of 𝑊𝑓 at 𝑢∈𝔽2𝑛 is the following sum (over the integers)

The set 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 𝔽2𝑛 we have
  • Parseval's relation:
  • Poisson summation formula: for any vector subspace 𝐸 of 𝔽2𝑛 and for any elements 𝑎,𝑏 in 𝔽2𝑛
    for 𝐸 the orthogonal subspace of 𝐸,{𝑢∈𝔽2𝑛 : 𝑢·𝑥=0, for all 𝑥∈𝐸}.

Equivalence of Boolean functions

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 EA-equivalence is called EA-invariant.