[Paper Review] Reversed Dickson Polynomials of the Third Kind

Neranga Fernando|arXiv (Cornell University)|Feb 15, 2016

Coding theory and cryptography6 references3 citations

TL;DR

This paper investigates reversed Dickson polynomials of the third kind, denoted $ D_{n,2}(1,x) $, over finite fields $ \mathbb{F}_q $, where $ q = p^e $ and $ p $ is prime. It establishes necessary conditions for such polynomials to be permutation polynomials and provides an explicit evaluation of the sum $ \sum_{a \in \mathbb{F}_q} D_{n,2}(1,a) $, leveraging functional equations and generating functions to derive recursive formulas for the sum across various degrees.

ABSTRACT

Let $p$ be a prime and $q=p^e$. We discuss the properties of the reversed Dickson polynomial $D_{n,2}(1,x)$ of the third kind. We also give several necessary conditions for the reversed Dickson polynomial of the third kind $D_{n,2}(1,x)$ to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_q}D_{n,2}(1,a)$.

Motivation & Objective

• To analyze the permutation properties of reversed Dickson polynomials of the third kind $ D_{n,2}(1,x) $ over finite fields $ \mathbb{F}_q $.
• To establish necessary conditions for $ D_{n,2}(1,x) $ to be a permutation polynomial over $ \mathbb{F}_q $.
• To compute the sum $ \sum_{a \in \mathbb{F}_q} D_{n,2}(1,a) $ explicitly for all $ n $, using generating functions and functional identities.
• To connect the structure of these polynomials to known sequences and Jacobsthal polynomials, enhancing their algebraic interpretation.

Proposed method

• Derives a functional equation for $ F_n(1,x) = D_{n,2}(1,x) $ using the substitution $ x = y + y^{-1} $, linking it to roots in $ \mathbb{F}_{q^2} $.
• Uses the identity $ F_n(a,x) = a^n F_n(1, x/a^2) $ to reduce the study of $ F_n(a,x) $ to $ F_n(1,x) $, simplifying analysis.
• Applies generating functions and power series expansions to model the sum $ \sum_{a \in \mathbb{F}_q} F_n(1,a) $, expressing it via coefficients in a rational function expansion.
• Employs recursive coefficient comparison in power series to derive closed-form expressions for the sum at different degrees.
• Utilizes known results from [2] on reversed Dickson polynomials of the second kind to build recursive relations for the sum.
• Relies on the structure of $ h(z) $, a rational function derived from $ (z - z^q)^{q-1} $, to model the generating function of the sum.

Experimental results

Research questions

• RQ1 Under what conditions is the reversed Dickson polynomial of the third kind $ D_{n,2}(1,x) $ a permutation polynomial over $ \mathbb{F}_q $?
• RQ2 What is the explicit value of the sum $ \sum_{a \in \mathbb{F}_q} D_{n,2}(1,a) $ for all $ n $?
• RQ3 How do the functional equations of $ D_{n,2}(1,x) $ relate to those of Dickson and Chebyshev polynomials?
• RQ4 What connections exist between $ D_{n,2}(1,x) $ and Jacobsthal polynomials or integer sequences like A010892 and A128834?

Key findings

• The sum $ \sum_{a \in \mathbb{F}_q} D_{n,2}(1,a) $ is explicitly computed via recursive formulas based on coefficients of a generating function.
• For $ 1 \leq j \leq q-1 $, the sum satisfies $ \sum_{a \in \mathbb{F}_q} D_{j,2}(1,a) = -c_j + \frac{j}{2^{j-1}} $, where $ c_j $ are coefficients from a rational function expansion.
• For $ n = q $, the sum is $ \sum_{a \in \mathbb{F}_q} D_{q,2}(1,a) = c_1 - c_q $, with $ c_k $ derived from the generating function.
• For $ n = lq + j $ with $ 1 \leq l \leq q-2 $ and $ 1 \leq j \leq q-1 $, the sum satisfies a recursive relation involving $ c_{lq+j} $ and previous sum values.
• For $ n = q^2 - q + j $, the sum is given by $ \sum_{i=j}^{q-1} c_{q^2 + i} + \frac{j}{2^{q^2 - q + j - 1}} $, showing dependence on higher-order coefficients.
• The evaluation $ D_{n,2}(1, \frac{1}{4}) = \frac{n}{2^{n-1}} $ is confirmed, linking the polynomial to a known rational sequence.