[Paper Review] Reversed Dickson polynomials of the fourth kind over finite fields

Kaimin Cheng, Shaofang Hong+1 more|arXiv (Cornell University)|Apr 15, 2016

Coding theory and cryptography1 references3 citations

TL;DR

This paper investigates the permutational behavior of reversed Dickson polynomials of the fourth kind, $ D_{n,3}(1,x) $, over finite fields $ bF_q $. Using generating functions and coefficient analysis, it derives an explicit formula for the first moment $ extstyleigsum_{a F_q} D_{n,3}(1,a) $, providing a key computational tool for studying permutation properties in characteristic $ p > 3 $.

ABSTRACT

In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$.

Motivation & Objective

• To study the permutational behavior of reversed Dickson polynomials of the fourth kind, $ D_{n,3}(1,x) $, over finite fields $ bF_q $.
• To derive an explicit formula for the first moment $ extstyleigsum_{a F_q} D_{n,3}(1,a) $, a critical invariant for permutation polynomial criteria.
• To establish a characterization of $ D_{n,3}(1,x) $ as a permutation polynomial using auxiliary polynomials and generating functions.
• To extend the Hermite criterion framework to this new class of reversed Dickson polynomials by computing the first moment explicitly.

Proposed method

• Define the reversed Dickson polynomial of the fourth kind via the formula $ D_{n,3}(a,x) = extstyleigsum_{i=0}^{l{n/2}} rac{n-3i}{n-i} inom{n-i}{i} (-x)^i a^{n-2i} $ for $ n o 1 $, with $ D_{0,3}(a,x) = -1 $.
• Use the identity $ D_{n,3}(a,x) = rac{(2a-y)y^n - (y+a)(a-y)^n}{2y-a} $ with $ x = y(a-y) $ to express the polynomial in terms of a rational function in $ y $.
• Construct a generating function $ G(t) = extstyleigsum_{n=1}^{q^2-1} ig( a_n - rac{3n-1}{2^n} ig) t^n $, where $ a_n = extstyleigsum_{a F_q} D_{n,3}(1,a) $, to analyze the first moment.
• Derive an expression for $ h(t) $, a rational function involving $ t^{q^2-1} $, $ t^q $, and binomial coefficients, to model the generating function's structure.
• Apply the binomial theorem to $ (t - t^q)^{q-1} $ to compute coefficients $ b_i $ of the polynomial $ extstyleigsum_{i=0}^{q^2-q} b_i t^i = -1 - (t - t^q)^{q-1} $.
• Use coefficient comparison in the equation $ (t^q - t^{q-1} - 1) extstyleigsum_{n=1}^{q^2-1} d_n t^n = ext{RHS} $, where $ d_n = a_n - rac{3n-1}{2^n} $, to recursively compute $ a_n $.

Experimental results

Research questions

• RQ1 What is the explicit value of the first moment $ extstyleigsum_{a F_q} D_{n,3}(1,a) $ for reversed Dickson polynomials of the fourth kind?
• RQ2 Under what conditions is $ D_{n,3}(1,x) $ a permutation polynomial over $ bF_q $, and how can this be characterized algebraically?
• RQ3 How does the generating function approach enable the computation of the first moment when direct evaluation is infeasible?
• RQ4 What role do the coefficients $ c_i $ of the rational function expansion play in reconstructing the moment sum?
• RQ5 Can the recursive structure of the moment sum be fully captured by the derived recurrence relations in terms of $ c_i $ and $ d_n $?

Key findings

• The first moment $ extstyleigsum_{a F_q} D_{n,3}(1,a) $ is explicitly computed as $ -c_j + rac{3j-1}{2^j} $ for $ 1 o j o q-1 $, where $ c_j $ is the coefficient of $ t^j $ in a rational function expansion.
• For $ n = q $, the moment is $ c_1 - c_q - rac{1}{2} $, showing a deviation from the simple $ rac{3n-1}{2^n} $ pattern due to field characteristic effects.
• For $ n = ext{multiple of } q $, the moment satisfies a recursive relation: $ extstyleigsum D_{ u q,3}(1,a) = extstyleigsum D_{( u-1)q,3}(1,a) - extstyleigsum D_{( u-1)q+1,3}(1,a) - c_{ u q} + rac{3}{2^ u} $, valid for $ 2 o u o q-2 $.
• For $ n = q^2 - q + j $ with $ 0 o j o q-1 $, the moment is $ extstyleigsum_{i=j}^{q-1} c_{q^2+i} + rac{3j-1}{2^j} $, indicating a shift in behavior near the field size.
• The coefficient $ b_i $ in the expansion of $ -1 - (t - t^q)^{q-1} $ is explicitly given as $ (-1)^{eta+1} inom{q-1}{eta} $ if $ eta + eta = q-1 $, $ -1 $ if $ eta = eta = 0 $, and 0 otherwise.
• The full moment sum $ extstyleigsum_{a F_q} D_{n,3}(1,a) $ is reconstructed from the recurrence relations involving $ c_i $, $ d_n $, and $ a_n = d_n + rac{3n-1}{2^n} $, providing a complete algorithmic evaluation.