Almost Perfect Nonlinear (APN) Functions

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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 (π‘₯₁,π‘₯β‚‚) 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 β†’ 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 π‘Žβ‰ 0, express the number of input pairs with difference π‘Ž 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(π‘Ž,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 β‰₯ 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 π‘Ž 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 π‘₯, 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 π‘₯.

Characterizations

Walsh transform[1]

Any (𝑛,π‘š)-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 ∈ 𝔽2π‘š instead of just the nonzero ones. In this case, the inequality for (𝑛,π‘š)-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 β†’ 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 π‘Ž ∈ 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