Almost Perfect Nonlinear (APN) Functions

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 $\delta$ and computing the output of $F$ for all pairs of inputs $(x_{1},x_{2})$ whose difference is $\delta$ produces output pairs with a difference distribution that is far away from uniform.

The formal definition of an APN function $F:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{n}}$ is usually given through the values

$\Delta _{F}(a,b)=|\{x\in \mathbb {F} _{2^{n}}:F(x)+F(a+x)=b\}|$

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

$\Delta _{F}=\max\{\Delta _{F}(a,b):a\in \mathbb {F} _{2^{n}}^{*},b\in \mathbb {F} _{2^{n}}\}$

which clearly satisfies $\Delta _{F}\geq 2$ 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 $\{b\in \mathbb {F} _{2}^{n}:D_{a}D_{b}F(x)=v\}$ does not depend on $x$, or, equivalently, the size of the set $\{b\in \mathbb {F} _{2}^{n}:D_{a}F(b)=D_{a}F(x)+v\}$ does not depend on $x$.

Characterizations

Walsh transform

Any $(n,m)$-function $F$ satisfies

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{m}}^{*}}W_{F}^{4}(a,b)\geq 2^{2n}(3\cdot 2^{n+m}-2^{m+1}-2^{2n})$

with equality characterizing APN functions.

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

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{n}}^{*}}W_{F}^{4}(a,b)\geq 2^{3n+1}(2^{n}-1)$

with equality characterizing APN functions.

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

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{m}}}W_{F}^{4}(a,b)\geq 2^{2n+m}(3\cdot 2^{n}-2)$

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

$\sum _{a,b\in \mathbb {F} _{2^{n}}}W_{F}^{4}(a,b)\geq 2^{3n+1}(3\cdot 2^{n-1}-1)$

with equality characterizing APN functions in both cases.

Autocorrelation functions of the directional derivatives

Given a Boolean function $f:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2}$, the autocorrelation function of $f$ is defined as

${\mathcal {F}}(f)=\sum _{x\in \mathbb {F} _{2^{n}}}(-1)^{f(x)}=2^{n}-2wt(f).$

Any $(n,n)$-function $F$ satisfies

$\sum _{\lambda \in \mathbb {F} _{2^{n}}}{\mathcal {F}}(D_{a}f_{\lambda })=2^{2n+1}$

for any $a\in \mathbb {F} _{2^{n}}^{*}$. Equality occurs if and only if $F$ is APN.

This allows APN functions to be characterized in terms of the sum-of-square-indicator $\nu (f)$ defined as

$\nu (f)=\sum _{a\in \mathbb {F} _{2^{n}}}{\mathcal {F}}^{2}(D_{a}F)=2^{-n}\sum _{a\in \mathbb {F} _{2^{n}}}{\mathcal {F}}^{4}(f+\varphi _{a})$

for $\varphi _{a}(x)={\rm {Tr}}(ax)$.

Then any $(n,n)$ function $F$ satisfies

$\sum _{\lambda \in \mathbb {F} _{2^{n}}^{*}}\nu (f_{\lambda })\geq (2^{n}-1)2^{2n+1}$

and equality occurs if and only if $F$ is APN.

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