Boolean Functions

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Introduction

Let 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 . 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 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, .

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.