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On APN functions in odd characteristic, the disproof of a conjecture and related problems

Daniele Bartoli, Pantelimon Stanica

TL;DR

The paper tackles the existence of APN functions in odd characteristic by disproving the Pal–Budaghyan conjecture, showing that the proposed family cannot be APN for large $q$ (specifically beyond $9587$) and that no infinite APN family exists beyond the cases listed in BP24. The authors deploy a fusion of function-field theory (Kummer extensions, constants, ramification) and Weil-sum analysis to bound differential behavior and derive not-APN results, including a stronger non-APN statement in the $j=k$ specialization. They also explore related candidate families, providing computational data and theoretical arguments that APN-ness is not sustained as the field size grows, thereby narrowing the landscape of APN permutations in odd characteristic. The work has cryptographic significance by eliminating a broad proposed source of APN functions and clarifying which constructions can yield robust differential properties in large finite fields.

Abstract

In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality $q$ is larger than~$9587$, then those functions will never be APN. Moreover, we discuss other connected families of functions, for potential APN functions, but we show that they are not good candidates for APNess if the underlying field is large, in spite of the fact that they though they are APN for small environments.

On APN functions in odd characteristic, the disproof of a conjecture and related problems

TL;DR

The paper tackles the existence of APN functions in odd characteristic by disproving the Pal–Budaghyan conjecture, showing that the proposed family cannot be APN for large (specifically beyond ) and that no infinite APN family exists beyond the cases listed in BP24. The authors deploy a fusion of function-field theory (Kummer extensions, constants, ramification) and Weil-sum analysis to bound differential behavior and derive not-APN results, including a stronger non-APN statement in the specialization. They also explore related candidate families, providing computational data and theoretical arguments that APN-ness is not sustained as the field size grows, thereby narrowing the landscape of APN permutations in odd characteristic. The work has cryptographic significance by eliminating a broad proposed source of APN functions and clarifying which constructions can yield robust differential properties in large finite fields.

Abstract

In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality is larger than~, then those functions will never be APN. Moreover, we discuss other connected families of functions, for potential APN functions, but we show that they are not good candidates for APNess if the underlying field is large, in spite of the fact that they though they are APN for small environments.
Paper Structure (6 sections, 10 theorems, 64 equations, 3 tables)

This paper contains 6 sections, 10 theorems, 64 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries from Function Field Theory
  3. A "potential" infinite class of APN functions
  4. A related class of potential APNs
  5. The Difference Distribution Table of a general class of functions in terms of Weil sums
  6. The case j=k

Key Result

Theorem 2.1

Sti Consider an algebraic function field $\mathbb F$ with constant field $\mathbb L$ containing a primitive $n$-th root of unity ($n>1$ and $n$ relatively prime to the characteristic of $\mathbb L$). Let $u\in \mathbb F$ be such that there is a place $Q$ of $\mathbb F$ with $\gcd(v_Q(u),n)=1$ (see S

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 8 more