Almost Perfect Nonlinear (APN) Functions: Difference between revisions

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= Background and definition =
= Background and definition =


Almost perfect nonlinear (APN) functions are the class of <math>(n,n)</math> [[Vectorial Boolean Functions]] that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function <math>F</math> is efficient when fixing some difference <math>\delta</math> and computing the output of <math>F</math> for all pairs of inputs <math>(x_1,x_2)</math> whose difference is <math>\delta</math> produces output pairs with a difference distribution that is far away from uniform.
Almost perfect nonlinear (APN) functions are the class of (𝑛,𝑛) [[Vectorial Boolean Functions]] that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function 𝐹 is efficient when fixing some difference 𝛿 and computing the output of 𝐹 for all pairs of inputs (𝑥₁,𝑥₂) whose difference is 𝛿 produces output pairs with a difference distribution that is far away from uniform.


The formal definition of an APN function <math>F : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}</math> is usually given through the values
The formal definition of an APN function 𝐹 : 𝔽<sub>2<sup>𝑛</sup></sub> → 𝔽<sub>2<sup>𝑛</sup></sub> is usually given through the values
<div><math>\Delta_F(a,b) = | \{ x \in \mathbb{F}_{2^n} : F(x) + F(a+x) = b \}|</math></div>
<div><math>\Delta_F(a,b) = | \{ x \in \mathbb{F}_{2^n} : F(x) + F(a+x) = b \}|</math></div>
which, for <math>a \ne 0</math>, express the number of input pairs with difference <math>a</math> that map to a given <math>b</math>. The existence of a pair <math>(a,b) \in \mathbb{F}_{2^n}^* \times \mathbb{F}_{2^n}</math> with a high value of <math>\Delta_F(a,b)</math> makes the function <math>F</math> vulnerable to differential cryptanalysis. This motivates the definition of ''differential uniformity'' as
which, for 𝑎≠0, express the number of input pairs with difference 𝑎 that map to a given 𝑏. The existence of a pair <math>(a,b) \in \mathbb{F}_{2^n}^* \times \mathbb{F}_{2^n}</math> with a high value of Δ<sub>𝐹(𝑎,𝑏)</sub> makes the function 𝐹 vulnerable to differential cryptanalysis. This motivates the definition of ''differential uniformity'' as
<div><math>\Delta_F = \max \{ \Delta_F(a,b) : a \in \mathbb{F}_{2^n}^*, b \in \mathbb{F}_{2^n} \}</math></div>
<div><math>\Delta_F = \max \{ \Delta_F(a,b) : a \in \mathbb{F}_{2^n}^*, b \in \mathbb{F}_{2^n} \}</math></div>
which clearly satisfies <math>\Delta_F \ge 2</math> for any function <math>F</math>. The functions meeting this lower bound are called ''almost perfect nonlinear (APN)''.
which clearly satisfies Δ<sub>𝐹</sub> ≥ 2 for any function 𝐹. The functions meeting this lower bound are called ''almost perfect nonlinear (APN)''.


The characterization by means of the derivatives below suggests the following definition: a v.B.f. <math>F</math> is said to be ''strongly-plateuaed'' if, for every <math>a</math> and every <math>v</math>, the size of the set <math>\{ b \in \mathbb{F}_2^n : D_aD_bF(x) = v \}</math> does not depend on <math>x</math>, or, equivalently, the size of the set <math>\{ b \in \mathbb{F}_2^n : D_aF(b) = D_aF(x) + v \}</math> does not depend on <math>x</math>.
The characterization by means of the derivatives below suggests the following definition: a v.B.f. 𝐹 is said to be ''strongly-plateuaed'' if, for every 𝑎 and every 𝑣, the size of the set <math>\{ b \in \mathbb{F}_2^n : D_aD_bF(x) = v \}</math> does not depend on 𝑥, or, equivalently, the size of the set <math>\{ b \in \mathbb{F}_2^n : D_aF(b) = D_aF(x) + v \}</math> does not depend on 𝑥.


= Characterizations =
= Characterizations =
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== Walsh transform<ref name="chavau1994">Florent Chabaud, Serge Vaudenay, ''Links between differential and linear cryptanalysis'', Workshop on the Theory and Application of Cryptographic Techniques, 1994 May 9, pp. 356-365, Springer, Berlin, Heidelberg</ref> ==
== Walsh transform<ref name="chavau1994">Florent Chabaud, Serge Vaudenay, ''Links between differential and linear cryptanalysis'', Workshop on the Theory and Application of Cryptographic Techniques, 1994 May 9, pp. 356-365, Springer, Berlin, Heidelberg</ref> ==


Any <math>(n,m)</math>-function <math>F</math> satisfies
Any (𝑛,𝑚)-function 𝐹 satisfies


<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}^*} W_F^4(a,b) \ge 2^{2n}(3 \cdot 2^{n+m} - 2^{m+1} - 2^{2n})</math></div>
<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}^*} W_F^4(a,b) \ge 2^{2n}(3 \cdot 2^{n+m} - 2^{m+1} - 2^{2n})</math></div>
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with equality characterizing APN functions.
with equality characterizing APN functions.


In particular, for <math>(n,n)</math>-functions we have
In particular, for (𝑛,𝑛)-functions we have


<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^n}^*} W_F^4(a,b) \ge 2^{3n+1}(2^n-1)</math></div>
<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^n}^*} W_F^4(a,b) \ge 2^{3n+1}(2^n-1)</math></div>
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with equality characterizing APN functions.
with equality characterizing APN functions.


Sometimes, it is more convenient to sum through all <math>b \in \mathbb{F}_{2^m}</math> instead of just the nonzero ones. In this case, the inequality for <math>(n,m)</math>-functions becomes
Sometimes, it is more convenient to sum through all 𝑏 ∈ 𝔽<sub>2<sup>𝑚</sup></sub> instead of just the nonzero ones. In this case, the inequality for (𝑛,𝑚)-functions becomes


<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}} W_F^4(a,b) \ge 2^{2n + m}(3 \cdot 2^n - 2)</math></div>
<div><math>\sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}} W_F^4(a,b) \ge 2^{2n + m}(3 \cdot 2^n - 2)</math></div>


and the particular case for <math>(n,n)</math>-functions becomes
and the particular case for (𝑛,𝑛)-functions becomes


<div><math>\sum_{a,b \in \mathbb{F}_{2^n}} W_F^4(a,b) \ge 2^{3n+1}(3 \cdot 2^{n-1} - 1)</math></div>
<div><math>\sum_{a,b \in \mathbb{F}_{2^n}} W_F^4(a,b) \ge 2^{3n+1}(3 \cdot 2^{n-1} - 1)</math></div>
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== Autocorrelation functions of the directional derivatives <ref name="bercanchalai2006"> Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, ''On Almost Perfect Nonlinear Functions Over GF(2^n)'', IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70</ref> ==
== Autocorrelation functions of the directional derivatives <ref name="bercanchalai2006"> Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, ''On Almost Perfect Nonlinear Functions Over GF(2^n)'', IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70</ref> ==


Given a Boolean function <math>f : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_2</math>, the ''autocorrelation function'' of <math>f</math> is defined as
Given a Boolean function 𝑓 : 𝔽<sub>2<sup>𝑛</sup></sub> → 𝔽<sub>2</sub>, the ''autocorrelation function'' of 𝑓 is defined as
<div><math>\mathcal{F}(f) = \sum_{x \in \mathbb{F}_{2^n}} (-1)^{f(x)} = 2^n - 2wt(f).</math></div>
<div><math>\mathcal{F}(f) = \sum_{x \in \mathbb{F}_{2^n}} (-1)^{f(x)} = 2^n - 2wt(f).</math></div>


Any <math>(n,n)</math>-function <math>F</math> satisfies
Any (𝑛,𝑛)-function 𝐹 satisfies
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}} \mathcal{F}(D_af_\lambda) = 2^{2n+1}</math></div>
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}} \mathcal{F}(D_af_\lambda) = 2^{2n+1}</math></div>
for any <math>a \in \mathbb{F}_{2^n}^*</math>. Equality occurs if and only if <math>F</math> is APN.
for any 𝑎 ∈ 𝔽*<sub>2<sup>𝑛</sup></sub> . Equality occurs if and only if <math>F</math> is APN.


This allows APN functions to be characterized in terms of the ''sum-of-square-indicator'' <math>\nu(f)</math> defined as
This allows APN functions to be characterized in terms of the ''sum-of-square-indicator'' <math>\nu(f)</math> defined as
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for <math>\varphi_a(x) = {\rm Tr}(ax)</math>.
for <math>\varphi_a(x) = {\rm Tr}(ax)</math>.


Then any <math>(n,n)</math> function <math>F</math> satisfies
Then any (𝑛,𝑛) function 𝐹 satisfies
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \nu(f_\lambda) \ge (2^n-1)2^{2n+1}</math></div>
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \nu(f_\lambda) \ge (2^n-1)2^{2n+1}</math></div>
and equality occurs if and only if <math>F</math> is APN.
and equality occurs if and only if 𝐹 is APN.


Similar techniques can be used to characterize permutations and APN functions with plateaued components.
Similar techniques can be used to characterize permutations and APN functions with plateaued components.
= Characterization of Plateaued Functions =
== Characterization by the Derivatives <ref name="carletPlateaued">Carlet C. Boolean and vectorial plateaued functions and APN functions. IEEE Transactions on Information Theory. 2015 Nov;61(11):6272-89.</ref> ==
=== First characterization ===
Using the fact that two integer-valued functions over <math>\mathbb{F}_2^m</math> are equal precisely when their Fourier transforms are equal, one can obtain the following characterization.
Let <math>F</math> be an <math>(n,m)</math>-function. Then:
- <math>F</math> is plateaued if and only if, for every <math>v \in \mathbb{F}_2^m</math>, the size of the set
<div><math> \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \}</math></div>
does not depend on <math>x</math>;
- <math>F</math> is plateaued with single amplitude if and only if the size of the above set depends neither on <math>x</math> nor on <math>v</math> when <math>v \ne 0</math>.
Moreover:
- for any <math>(n,m)</math>-function <math>F</math>, the value distribution of <math>D_aD_bF(x)</math> equals the value distribution of <math>D_aF(b) + D_aF(x)</math> as <math>(a,b)</math> ranges over <math>(\mathbb{F}_2^n)^2</math>;
- if two plateuaed functions have the same distribution, then for every <math>u</math>, their component functions at <math>u</math> have the same amplitude.
=== Characterization in the Case of Unbalanced Components ===
Let <math>F</math> be an <math>(n,m)</math>-function. Then <math>F</math> is plateuaed with all components unbalanced if and only if, for every <math>v,x \in \mathbb{F}_2^n</math>, we have
<div><math> | \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \} | = | \{ (a,b) \in (\mathbb{F}_2^n)^2 : F(a) + F(b) = v \} |.</math></div>
Moreover, <math>F</math> is plateuaed with single amplitude if and only if, in addition, this value does not depend on <math>v</math> for <math>v \ne 0</math>.
=== Properties of Strongly-Plateaued Functions ===
A Boolean function is strongly-plateaued if and only if its partially-bent. A v.B.f. is strongly-plateaued if and only if all of its component functions are partially-bent. In particular, bent and quadratic Boolean and vectorial functions are strongly-plateaued.
The image set <math>{\rm Im}(D_aF)</math> of any derivative of a strongly-plateaued function <math>F</math> is an affine space.
=== Characterization by the Auto-Correlation Functions ===

Latest revision as of 12:54, 11 October 2019

Background and definition

Almost perfect nonlinear (APN) functions are the class of (𝑛,𝑛) Vectorial Boolean Functions that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function 𝐹 is efficient when fixing some difference 𝛿 and computing the output of 𝐹 for all pairs of inputs (𝑥₁,𝑥₂) whose difference is 𝛿 produces output pairs with a difference distribution that is far away from uniform.

The formal definition of an APN function 𝐹 : 𝔽2𝑛 → 𝔽2𝑛 is usually given through the values

[math]\displaystyle{ \Delta_F(a,b) = | \{ x \in \mathbb{F}_{2^n} : F(x) + F(a+x) = b \}| }[/math]

which, for 𝑎≠0, express the number of input pairs with difference 𝑎 that map to a given 𝑏. The existence of a pair [math]\displaystyle{ (a,b) \in \mathbb{F}_{2^n}^* \times \mathbb{F}_{2^n} }[/math] with a high value of Δ𝐹(𝑎,𝑏) makes the function 𝐹 vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity as

[math]\displaystyle{ \Delta_F = \max \{ \Delta_F(a,b) : a \in \mathbb{F}_{2^n}^*, b \in \mathbb{F}_{2^n} \} }[/math]

which clearly satisfies Δ𝐹 ≥ 2 for any function 𝐹. The functions meeting this lower bound are called almost perfect nonlinear (APN).

The characterization by means of the derivatives below suggests the following definition: a v.B.f. 𝐹 is said to be strongly-plateuaed if, for every 𝑎 and every 𝑣, the size of the set [math]\displaystyle{ \{ b \in \mathbb{F}_2^n : D_aD_bF(x) = v \} }[/math] does not depend on 𝑥, or, equivalently, the size of the set [math]\displaystyle{ \{ b \in \mathbb{F}_2^n : D_aF(b) = D_aF(x) + v \} }[/math] does not depend on 𝑥.

Characterizations

Walsh transform[1]

Any (𝑛,𝑚)-function 𝐹 satisfies

[math]\displaystyle{ \sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}^*} W_F^4(a,b) \ge 2^{2n}(3 \cdot 2^{n+m} - 2^{m+1} - 2^{2n}) }[/math]

with equality characterizing APN functions.

In particular, for (𝑛,𝑛)-functions we have

[math]\displaystyle{ \sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^n}^*} W_F^4(a,b) \ge 2^{3n+1}(2^n-1) }[/math]

with equality characterizing APN functions.

Sometimes, it is more convenient to sum through all 𝑏 ∈ 𝔽2𝑚 instead of just the nonzero ones. In this case, the inequality for (𝑛,𝑚)-functions becomes

[math]\displaystyle{ \sum_{a \in \mathbb{F}_{2^n}, b \in \mathbb{F}_{2^m}} W_F^4(a,b) \ge 2^{2n + m}(3 \cdot 2^n - 2) }[/math]

and the particular case for (𝑛,𝑛)-functions becomes

[math]\displaystyle{ \sum_{a,b \in \mathbb{F}_{2^n}} W_F^4(a,b) \ge 2^{3n+1}(3 \cdot 2^{n-1} - 1) }[/math]

with equality characterizing APN functions in both cases.

Autocorrelation functions of the directional derivatives [2]

Given a Boolean function 𝑓 : 𝔽2𝑛 → 𝔽2, the autocorrelation function of 𝑓 is defined as

[math]\displaystyle{ \mathcal{F}(f) = \sum_{x \in \mathbb{F}_{2^n}} (-1)^{f(x)} = 2^n - 2wt(f). }[/math]

Any (𝑛,𝑛)-function 𝐹 satisfies

[math]\displaystyle{ \sum_{\lambda \in \mathbb{F}_{2^n}} \mathcal{F}(D_af_\lambda) = 2^{2n+1} }[/math]

for any 𝑎 ∈ 𝔽*2𝑛 . Equality occurs if and only if [math]\displaystyle{ F }[/math] is APN.

This allows APN functions to be characterized in terms of the sum-of-square-indicator [math]\displaystyle{ \nu(f) }[/math] defined as

[math]\displaystyle{ \nu(f) = \sum_{a \in \mathbb{F}_{2^n}} \mathcal{F}^2(D_aF) = 2^{-n} \sum_{a \in \mathbb{F}_{2^n}} \mathcal{F}^4(f + \varphi_a) }[/math]

for [math]\displaystyle{ \varphi_a(x) = {\rm Tr}(ax) }[/math].

Then any (𝑛,𝑛) function 𝐹 satisfies

[math]\displaystyle{ \sum_{\lambda \in \mathbb{F}_{2^n}^*} \nu(f_\lambda) \ge (2^n-1)2^{2n+1} }[/math]

and equality occurs if and only if 𝐹 is APN.

Similar techniques can be used to characterize permutations and APN functions with plateaued components.

  1. Florent Chabaud, Serge Vaudenay, Links between differential and linear cryptanalysis, Workshop on the Theory and Application of Cryptographic Techniques, 1994 May 9, pp. 356-365, Springer, Berlin, Heidelberg
  2. Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, On Almost Perfect Nonlinear Functions Over GF(2^n), IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70