Boolean Functions: Difference between revisions

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The Hamming weight of 𝑥 is the size of its support (<math>w_H(x)=|supp_x|</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>.
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 is the size of the set <math>\{x\in\mathbb{F}_2^n : f(x)\neq g(x) \}\ (w_H(f\oplus g))</math>.
The Hamming distance of two functions 𝑓,𝑔 (𝖽<sub>𝐻</sub>(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=
=Representation of a Boolean function=
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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 𝓌<sub>2</sub>(𝑗) is the Hamming weight of the binary expansion of 𝑗.
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 𝓌<sub>2</sub>(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<sub>𝐻</sub>(𝑓)=2<sup>𝑛-1</sup>.
* If 𝑑°𝑓=2 then 𝓌<sub>𝐻</sub>(𝑓)=2<sup>𝑛-1</sup> or 𝓌<sub>𝐻</sub>(𝑓)=2<sup>𝑛-1</sup>±2<sup>𝑛-1-ℎ</sup>, with 0≤ℎ≤𝑛/2.
* If 𝑑°𝑓≤𝑟 and 𝑓 nonzero then 𝓌<sub>𝐻</sub>(𝑓)≥2<sup>𝑛-𝑟</sup>.
* 𝓌<sub>𝐻</sub>(𝑓) is odd if and only if 𝑑°𝑓=𝑛.


=The Walsh transform=
=The Walsh transform=
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With an innner product in 𝔽<sub>2</sub><sup>𝑛</sup> 𝑥·𝑦, the value of 𝑊<sub>𝑓</sub> at 𝑢∈𝔽<sub>2</sub><sup>𝑛</sup> is the following sum (over the integers)
With an innner product in 𝔽<sub>2</sub><sup>𝑛</sup> 𝑥·𝑦, the value of 𝑊<sub>𝑓</sub> at 𝑢∈𝔽<sub>2</sub><sup>𝑛</sup> is the following sum (over the integers)
<center><math>W_f(u)=\sum_{x\in\mathbb{F}_2^n}(-1)^{f(x)+x\cdot u},</math></center>
<center><math>W_f(u)=\sum_{x\in\mathbb{F}_2^n}(-1)^{f(x)+x\cdot u},</math></center>
The set <math>\{ u\in\mathbb{F}_2^n : W_f(u)\ne0 \}</math> is the <i>Walsh support</i> of 𝑓.
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 <i>Walsh support</i> of 𝑓.


==Properties of the Walsh transform==
==Properties of the Walsh transform==
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* Poisson summation formula: for any vector subspace 𝐸 of 𝔽<sub>2</sub><sup>𝑛</sup> and for any elements 𝑎,𝑏 in 𝔽<sub>2</sub><sup>𝑛</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>  for 𝐸<sup>⟂</sup> the orthogonal subspace of 𝐸,{𝑢∈𝔽<sub>2</sub><sup>𝑛</sup> : 𝑢·𝑥=0, for all 𝑥∈𝐸}.
* Poisson summation formula: for any vector subspace 𝐸 of 𝔽<sub>2</sub><sup>𝑛</sup> and for any elements 𝑎,𝑏 in 𝔽<sub>2</sub><sup>𝑛</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>  for 𝐸<sup>⟂</sup> the orthogonal subspace of 𝐸,{𝑢∈𝔽<sub>2</sub><sup>𝑛</sup> : 𝑢·𝑥=0, for all 𝑥∈𝐸}.


=Equivalence of Boolean functions=
=Equivalences of Boolean functions=
Two 𝑛-variable Boolean functions 𝑓,𝑔 are called <i>affine equivalent</i> if there exists a linear automorphism 𝐿 and a vecor 𝑎 such that <center>𝑔(𝑥) = 𝑓(𝐿(𝑥)+𝑎).</center>
 
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>
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 an equivalence relation is called <i>invariant</i>.
 
* 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>.
 
=Properties important for cryptographic applications=
 
==Balanced functions==
An 𝑛-variable Boolean function 𝑓 is called <em>balanced</em> if 𝓌<sub>𝐻</sub>(𝑓)=2<sup>𝑛-1</sup>, so its output is uniformly distributed.
Such functions cannot have maximal degree.
Most cryptographic applications use balanced Boolean functions.
 
==The Nonlinearity==
The <em>nonlinearity</em> 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,
<center><math> \mathcal{NL}(f)=\min_{g\in\mathcal{A}}d_H(f,g)</math></center>
 
* 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 <em>covering radius bound</em> <math>\mathcal{NL}(f)\le2^{n-1}-2^{n/2-1}</math>.
* A function achieving the covering radius bound with equality is called [[Bent Boolean Functions| bent]] (𝑛 is an even integer and the function is not balanced).
* 𝑓 is bent if and only if 𝑊<sub>𝑓</sub>(𝑢)=±2<sup>𝑛/2</sup>, for every 𝑢∈𝔽<sub>2</sub><sup>𝑛</sup>.
* 𝑓 is bent if and only if for any nonzero element 𝑎 the Boolean function 𝐷<sub>𝑎</sub>𝑓(𝑥)=𝑓(𝑥+𝑎)+𝑓(𝑥) is balanced.
 
==Correlation-immunity order==
A Boolean function 𝑓 is <em>𝑚-th order correlation-immune</em> if the probability distribution of the output is unaltered when any 𝑚 input variables are fixed.
Balanced 𝑚-th order correlation-immune functions are called <em>𝑚-resilient</em>.
 
Given 𝑓 a 𝑛-variable function with correlation-immunity of order 𝑚 then <center>𝑚+𝑑°𝑓≤𝑛.</center>
If 𝑓 is also balanced, then <center>𝑚+𝑑°𝑓≤𝑛-1.</center>

Latest revision as of 14:39, 25 October 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 [math]\displaystyle{ f : \mathbb{F}_2^n\rightarrow\mathbb{F} }[/math] is called a Boolean function in dimenstion 𝑛 (or 𝑛-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 𝑥 is the size of its support ([math]\displaystyle{ 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]\displaystyle{ \{x\in\mathbb{F}_2^n : f(x)\ne0 \} }[/math]. The Hamming distance of two functions 𝑓,𝑔 (𝖽𝐻(𝑓,𝑔)) 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 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]\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 𝑓 (shortly ANF).

The degree of the ANF is called the algebraic degree of the function, 𝑑°𝑓=max { |𝐼| : 𝑎𝐼≠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∈𝔽2𝑛 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]\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 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 [math]\displaystyle{ \mbox{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(P(x)) }[/math] 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.

When we consider the trace representation of of a function, then the algebraic degree is given by [math]\displaystyle{ \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 𝓌𝐻(𝑓)=2𝑛-1.
  • If 𝑑°𝑓=2 then 𝓌𝐻(𝑓)=2𝑛-1 or 𝓌𝐻(𝑓)=2𝑛-1±2𝑛-1-ℎ, with 0≤ℎ≤𝑛/2.
  • If 𝑑°𝑓≤𝑟 and 𝑓 nonzero then 𝓌𝐻(𝑓)≥2𝑛-𝑟.
  • 𝓌𝐻(𝑓) is odd if and only if 𝑑°𝑓=𝑛.

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)

[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 \}=\{ 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 𝔽2𝑛 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 𝐸 of 𝔽2𝑛 and for any elements 𝑎,𝑏 in 𝔽2𝑛
    [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 𝐸 the orthogonal subspace of 𝐸,{𝑢∈𝔽2𝑛 : 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]\displaystyle{ W_g(u)=(-1)^{u\cdot L^{-1}(a)}W_f(L^{-1}(u)) }[/math].

Properties important for cryptographic applications

Balanced functions

An 𝑛-variable Boolean function 𝑓 is called balanced if 𝓌𝐻(𝑓)=2𝑛-1, so its output is uniformly distributed. Such functions cannot have maximal degree. Most cryptographic applications use balanced Boolean functions.

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]\displaystyle{ \mathcal{NL}(f)=\min_{g\in\mathcal{A}}d_H(f,g) }[/math]
  • For every 𝑓 we have [math]\displaystyle{ \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]\displaystyle{ \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 and the function is not balanced).
  • 𝑓 is bent if and only if 𝑊𝑓(𝑢)=±2𝑛/2, for every 𝑢∈𝔽2𝑛.
  • 𝑓 is bent if and only if for any nonzero element 𝑎 the Boolean function 𝐷𝑎𝑓(𝑥)=𝑓(𝑥+𝑎)+𝑓(𝑥) is balanced.

Correlation-immunity order

A Boolean function 𝑓 is 𝑚-th order correlation-immune if the probability distribution of the output is unaltered when any 𝑚 input variables are fixed. Balanced 𝑚-th order correlation-immune functions are called 𝑚-resilient.

Given 𝑓 a 𝑛-variable function with correlation-immunity of order 𝑚 then

𝑚+𝑑°𝑓≤𝑛.

If 𝑓 is also balanced, then

𝑚+𝑑°𝑓≤𝑛-1.