[Paper Review] New P$c$N and AP$c$N functions over finite fields

Yanan Wu, Nian Li+1 more|arXiv (Cornell University)|Oct 12, 2020

Coding theory and cryptography20 references4 citations

TL;DR

This paper presents new constructions of perfect $c$-nonlinear (P$c$N) and almost perfect $c$-nonlinear (AP$c$N) functions over finite fields using cyclotomic techniques, the switch method, and generalized AGW criteria. It introduces one class of P$c$N functions and four classes of AP$c$N functions, along with four additional classes via the AGW criterion, all achieving low $c$-differential uniformity, with several functions proven to be new under affine equivalence.

ABSTRACT

Functions with low $c$-differential uniformity were proposed in $2020$ and attracted lots of attention, especially the P$c$N and AP$c$N functions, due to their applications in cryptography. The objective of this paper is to study P$c$N and AP$c$N functions. As a consequence, we propose a class of P$c$N functions and four classes of AP$c$N functions by using the cyclotomic technique and the switch method. In addition, four classes of P$c$N or AP$c$N functions are presented by virtue of (generalized) AGW criterion.

Motivation & Objective

• To construct new classes of P$c$N and AP$c$N functions over finite fields with low $c$-differential uniformity for improved resistance against differential cryptanalysis.
• To extend existing constructions beyond monomials by developing multinomial-based P$c$N and AP$c%N functions.
• To employ advanced algebraic tools—specifically the cyclotomic technique, switch method, and generalized AGW criterion—to systematically generate new functions with provable $c$-differential properties.
• To verify the novelty of the constructed functions by comparing algebraic degrees, $c$-values, and field characteristics with known results, and to explore their invariance under affine equivalence.

Proposed method

• Utilizes the cyclotomic technique and switch method to analyze and solve the $c$-differential equation ${}_{c}D_{a}F(x) = b$ for $c \in \mathbb{F}_q \setminus \{1\}$, ensuring uniformity properties.
• Applies the generalized AGW criterion to construct new P$c$N and AP$c$N functions by verifying that $\phi(x)$ permutes a specific set $J$ and $\ker(\phi) \cap \mathbb{F}_q = \{0\}$, or is 2-to-1 over $\mathbb{F}_q$.
• Employs the structure $F(x) = u\phi(x) + \sum_{i=1}^{t} g\left(x^q - x\right)^{(q^n - 1)/d_i}$ to generate functions with $c$-differential uniformity 1 or 2.
• Uses trace functions $\operatorname{Tr}^{q^n}_q(x)$ and polynomial compositions to define piecewise functions that maintain low $c$-differential uniformity.
• Validates constructions via explicit examples over $\mathbb{F}_{2^4}$ and $\mathbb{F}_{q^3}$, demonstrating AP$c$N behavior for specific $c$ and field parameters.
• Compares new functions with known ones using algebraic degree, $c$-value, and characteristic $p$, confirming non-equivalence under affine transformation.

Experimental results

Research questions

• RQ1 Can new P$c$N and AP$c$N functions be constructed beyond monomials using algebraic techniques over finite fields?
• RQ2 How do cyclotomic techniques and the switch method enable the construction of functions with low $c$-differential uniformity?
• RQ3 To what extent do the generalized AGW criterion and its variants allow for systematic generation of P$c$N and AP$c$N functions?
• RQ4 Are the newly constructed functions equivalent to known ones under affine or CCZ-equivalence, particularly under $F \circ L(x)$ for affine permutations $L$?
• RQ5 What is the role of the kernel and image structure of $\phi(x)$ in ensuring $c$-differential uniformity in the AGW-based constructions?

Key findings

• One new class of P$c$N functions is constructed using the cyclotomic technique and switch method, with $c$-differential uniformity exactly 1.
• Four new classes of AP$c$N functions are proposed via the same method, achieving $c$-differential uniformity 2.
• Four additional classes of P$c$N or AP$c$N functions are derived using the generalized AGW criterion, with uniformity 1 or 2 depending on the properties of $\phi(x)$ and $g(x)$.
• An example over $\mathbb{F}_{2^4}$ confirms that $F(x) = x^2 + x + (x^q - x)^{(q^3 - 1)/3} + (x^q - x)^{(q^3 - 1)/5}$ is AP$c$N for all $c \in \mathbb{F}_q \setminus \{1\}$.
• The constructed functions are not equivalent to known ones under affine transformation $F \circ L(x)$, as confirmed by differences in algebraic degree, $c$-value, and field characteristic.
• The paper establishes that $c$-differential uniformity is not preserved under EA- or CCZ-equivalence, but is preserved under composition with affine permutations $L(x)$, which helps in verifying novelty.