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A PTR polynomial for the Hughes planes and a new class of permutation polynomials involving Catalan numbers

Stephen Brittain, Robert S. Coulter, Alice Man Wa Hui

TL;DR

This work constructs a reduced PTR polynomial $T$ over the finite field ${\mathbb F}_{Q}$ that coordinatizes Hughes planes built from regular nearfields. Central to the construction are the involutory permutation polynomial $\phi_k$, the trace map, and the polynomials $t_q$, together with Catalan-number identities that introduce Catalan and generalized Catalan coefficients into the PTR. The authors obtain three infinite classes of permutation polynomials arising from evaluations of $T$, with the notable feature that Catalan-type numbers appear as coefficients. They also analyze differential uniformity of the resulting PPs, showing maximal DU in several cases, which informs cryptographic suitability. The results connect combinatorial Catalan structures with finite geometric objects and yield new infinite PP families linked to Hughes planes.

Abstract

Hughes introduced the projective planes that bear his name in 1957 and they have since been studied extensively. However, until now, no polynomial representation of a planar ternary ring that represents them has been determined. In this paper, we rectify this omission by determining a reduced PTR polynomial for any Hughes plane defined over a regular nearfield. The polynomials obtained provide a new surprising connection: both the Catalan numbers and generalized Catalan numbers occur among the coefficients, depending on the representation. Since every PTR polynomial has connections with several classes of permutation polynomials, we obtain three new infinite classes of permutation polynomials as a consequence of our main result, and these, too, involve the Catalan numbers. The differential uniformity of new permutation polynomials is also determined.

A PTR polynomial for the Hughes planes and a new class of permutation polynomials involving Catalan numbers

TL;DR

This work constructs a reduced PTR polynomial over the finite field that coordinatizes Hughes planes built from regular nearfields. Central to the construction are the involutory permutation polynomial , the trace map, and the polynomials , together with Catalan-number identities that introduce Catalan and generalized Catalan coefficients into the PTR. The authors obtain three infinite classes of permutation polynomials arising from evaluations of , with the notable feature that Catalan-type numbers appear as coefficients. They also analyze differential uniformity of the resulting PPs, showing maximal DU in several cases, which informs cryptographic suitability. The results connect combinatorial Catalan structures with finite geometric objects and yield new infinite PP families linked to Hughes planes.

Abstract

Hughes introduced the projective planes that bear his name in 1957 and they have since been studied extensively. However, until now, no polynomial representation of a planar ternary ring that represents them has been determined. In this paper, we rectify this omission by determining a reduced PTR polynomial for any Hughes plane defined over a regular nearfield. The polynomials obtained provide a new surprising connection: both the Catalan numbers and generalized Catalan numbers occur among the coefficients, depending on the representation. Since every PTR polynomial has connections with several classes of permutation polynomials, we obtain three new infinite classes of permutation polynomials as a consequence of our main result, and these, too, involve the Catalan numbers. The differential uniformity of new permutation polynomials is also determined.

Paper Structure

This paper contains 10 sections, 18 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The Hughes planes
  3. The regular nearfields
  4. The PTR function for the Hughes planes
  5. An involutory permutation polynomial
  6. Some identities involving binomial coefficients and the Catalan numbers
  7. Determining the PTR polynomial
  8. Proof of \ref{['hughesptr']}
  9. A version of \ref{['hughesptr']} involving generalized Catalan numbers
  10. Differential uniformity considerations

Key Result

Theorem 2

Let $T\in{\mathbb F}_{Q}[X,Y,Z]$ be the polynomial defined by where and Here, $C[n]$ is the $n$-th Catalan number and $t_n(X)=X^n-X$. Then $T$ is a reduced PTR polynomial for the Hughes plane of order $Q$ defined over the regular nearfield.

Theorems & Definitions (35)

  • Definition 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4: Lucas' Theorem
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 25 more