Almost Perfect Nonlinear (APN) Functions

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.

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

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

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

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

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).

Characterizations

Walsh transform

Any <math>(p,q)</math>-function <math>F</math> satisfies

<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>

with equality characterizing APN functions.

Autocorrelation functions of the directional derivatives

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

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

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

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

for any <math>a \in \mathbb{F}_{2^n}^*</math>. 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

<math>\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>\varphi_a(x) = {\rm Tr}(ax)</math>.

Then any <math>(n,n)</math> function <math>F</math> satisfies

<math>\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 <math>F</math> is APN.

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