Almost Perfect Nonlinear (APN) Functions: Difference between revisions
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= Characterizations = | = Characterizations = | ||
== 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>(p,q)</math>-function <math>F</math> satisfies | |||
<div><math>\sum_{a \in \mathbb{F}_{2^p}, b \in \mathbb{F}_{2^q}^*} W_F^4(a,b) \ge 2^{2p}(3 \cdot 2^{p+q} - 2^{q+1} - 2^{2p})</math></div> | |||
with equality characterizing APN functions. | |||
== 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> == |
Revision as of 08:42, 18 January 2019
Background and definition
Almost perfect nonlinear (APN) functions are the class of [math]\displaystyle{ (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]\displaystyle{ F }[/math] is efficient when fixing some difference [math]\displaystyle{ \delta }[/math] and computing the output of [math]\displaystyle{ F }[/math] for all pairs of inputs [math]\displaystyle{ (x_1,x_2) }[/math] whose difference is [math]\displaystyle{ \delta }[/math] produces output pairs with a difference distribution that is far away from uniform.
The formal definition of an APN function [math]\displaystyle{ F : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n} }[/math] is usually given through the values
which, for [math]\displaystyle{ a \ne 0 }[/math], express the number of input pairs with difference [math]\displaystyle{ a }[/math] that map to a given [math]\displaystyle{ b }[/math]. 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 [math]\displaystyle{ \Delta_F(a,b) }[/math] makes the function [math]\displaystyle{ F }[/math] vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity as
which clearly satisfies [math]\displaystyle{ \Delta_F \ge 2 }[/math] for any function [math]\displaystyle{ F }[/math]. The functions meeting this lower bound are called almost perfect nonlinear (APN).
Characterizations
Walsh transform[1]
Any [math]\displaystyle{ (p,q) }[/math]-function [math]\displaystyle{ F }[/math] satisfies
with equality characterizing APN functions.
Autocorrelation functions of the directional derivatives [2]
Given a Boolean function [math]\displaystyle{ f : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_2 }[/math], the autocorrelation function of [math]\displaystyle{ f }[/math] is defined as
Any [math]\displaystyle{ (n,n) }[/math]-function [math]\displaystyle{ F }[/math] satisfies
for any [math]\displaystyle{ a \in \mathbb{F}_{2^n}^* }[/math]. 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
for [math]\displaystyle{ \varphi_a(x) = {\rm Tr}(ax) }[/math].
Then any [math]\displaystyle{ (n,n) }[/math] function [math]\displaystyle{ F }[/math] satisfies
and equality occurs if and only if [math]\displaystyle{ F }[/math] is APN.
Similar techniques can be used to characterize permutations and APN functions with plateaued components.
- ↑ 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
- ↑ 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