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A proof of a conjecture on permutation trinomials

Daniele Bartoli, Mohit Pal, Pantelimon Stanica

Abstract

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding $f(X)=X^{q(p-1)+1} +αX^{pq}+X^{q+p-1}$, on finite fields $\mathbb{F}_{q^2}, q=p^k$, from [A. Rai, R. Gupta, {\it Further results on a class of permutation trinomials}, Cryptogr. Commun. 15 (2023), 811--820].

A proof of a conjecture on permutation trinomials

Abstract

In this paper we use algebraic curves and other algebraic number theory methods to show the validity of a permutation polynomial conjecture regarding , on finite fields , from [A. Rai, R. Gupta, {\it Further results on a class of permutation trinomials}, Cryptogr. Commun. 15 (2023), 811--820].

Paper Structure

This paper contains 6 sections, 9 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of the conjecture for
  4. Proof of the conjecture for
  5. Proof of the conjecture for
  6. The proof of the conjecture for

Key Result

Lemma 2.1

Let $\mathcal{A}$ and $\mathcal{B}$ be two plane curves. For any affine point $P$, the intersection number satisfies the inequality with equality if and only if the tangents at $P$ to $\mathcal{A}$ are all distinct from the tangents at $P$ to $\mathcal{B}$.

Theorems & Definitions (12)

  • Conjecture 1.1
  • Lemma 2.1
  • Theorem 2.2: Bézout Theorem
  • Theorem 2.3: Aubry-Perret bound AubryPerret
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • ...and 2 more