Non-Boolean almost perfect nonlinear functions on non-Abelian groups

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.

💡 Research Summary

The paper “Non‑Boolean almost perfect nonlinear functions on non‑Abelian groups” extends the classical notions of almost perfect nonlinear (APN) functions and maximum nonlinearity from Boolean mappings on vector spaces over (\mathbb{F}_2) to arbitrary mappings between finite groups, allowing either the domain, the codomain, or both to be non‑Abelian. The authors begin by recalling that in the Boolean setting an APN function is one whose differential uniformity equals two; equivalently, for every non‑zero input difference (a) the equation (F(x+a)-F(x)=b) has at most two solutions. This property guarantees optimal resistance against differential cryptanalysis and is a cornerstone of modern S‑box design.

To generalize, the paper defines a difference operator for a map (F:K\to N) between finite groups (K) and (N) as (\Delta_aF(x)=F(xa)-F(x)), where the group operation replaces the vector addition. Because non‑Abelian groups lack commutativity, left‑ and right‑differences must be considered separately, and the authors adopt the left‑difference convention throughout. The differential distribution table (DDT) is then (\delta_F(a,b)=|{x\in K\mid\Delta_aF(x)=b}|). A function is called APN if for every non‑identity (a\in K) and every (b\in N) we have (\delta_F(a,b)\le 2). This definition reduces to the classical one when (K=N=\mathbb{F}_2^n).

The second major contribution is a Fourier‑analytic characterization that works for any finite groups. Let (\widehat{K}) and (\widehat{N}) denote the full sets of complex irreducible characters of (K) and (N), respectively. For (\psi\in\widehat{K}) and (\chi\in\widehat{N}) the generalized Walsh transform of (F) is defined as
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