Difference between revisions of "Almost Perfect Nonlinear (APN) Functions"

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(Created page with "= Background and definition = Almost perfect nonlinear (APN) functions are the class of <math>(n,n)</math> Vectorial Boolean Functions that provide optimum resistance to...")
 
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= Characterizations =
 
= Characterizations =
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== Autocorrelation functions of the directional derivatives ==
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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
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<div><math>\mathcal{F}(f) = \sum_{x \in \mathbb{F}_{2^n}} (-1)^{f(x)} = 2^n - 2wt(f).</math></div>
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Any <math>(n,n)</math>-function <math>F</math> satisfies
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<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}} \mathcal{F}(D_af_\lambda) = 2^{2n+1}</math></div>
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for any <math>a \in \mathbb{F}_{2^n}^*</math>. Equality occurs if and only if <math>F</math> is APN.

Revision as of 11:00, 15 January 2019

Background and definition

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

The formal definition of an APN function is usually given through the values

which, for , express the number of input pairs with difference that map to a given . The existence of a pair with a high value of makes the function vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity as

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

Characterizations

Autocorrelation functions of the directional derivatives

Given a Boolean function , the autocorrelation function of is defined as

Any -function satisfies

for any . Equality occurs if and only if is APN.