[Paper Review] Determination of a Type of Permutation Trinomials over Finite Fields

Xiang‐dong Hou|arXiv (Cornell University)|Sep 13, 2013

Coding theory and cryptography28 references3 citations

TL;DR

This paper determines explicit necessary and sufficient conditions for trinomials of the form $ f = a x + b x^q + x^{2q-1} $ over finite fields $ \mathbb{F}_q $ to be permutation polynomials over $ \mathbb{F}_{q^2} $, distinguishing cases for odd and even $ q $. The key contribution is a complete classification of such permutation trinomials, which resolves a related problem on the permutation property of polynomials $ g_{n,q} $ defined by functional equations involving sums over $ \mathbb{F}_q $.

ABSTRACT

Let $f=a{ t x} +b{ t x}^q+{ t x}^{2q-1}\in\Bbb F_q[{ t x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\in\Bbb F_p[{ t x}]$ ($n\ge 0$, $p= ext{char}\,\Bbb F_q$) be the polynomial defined by the functional equation $\sum_{c\in\Bbb F_q}({ t x}+c)^n=g_{n,q}({ t x}^q-{ t x})$. We determine all $n$ of the form $n=q^α-q^β-1$, $α>β\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.

Motivation & Objective

• To determine necessary and sufficient conditions for trinomials $ f = a x + b x^q + x^{2q-1} $ to be permutation polynomials over $ \mathbb{F}_{q^2} $, where $ q $ is a prime power.
• To resolve a related open problem concerning the permutation property of polynomials $ g_{n,q} $ defined by the functional equation $ \sum_{c \in \mathbb{F}_q} (x + c)^n = g_{n,q}(x^q - x) $.
• To characterize all integers $ n = q^\alpha - q^\beta - 1 $, with $ \alpha > \beta \geq 0 $, for which $ g_{n,q} $ is a permutation polynomial over $ \mathbb{F}_{q^2} $.
• To provide a new method for proving uniqueness of solutions to the equation $ a x + b x^q + x^{2q-1} = y $ in $ \mathbb{F}_{q^2} $, leveraging algebraic identities and trace conditions.
• To establish a connection between the permutation property of $ f $ and the structure of $ g_{n,q} $, particularly when $ n = q^\alpha - q^\beta - 1 $.

Proposed method

• For odd $ q $, derive conditions on $ a, b \in \mathbb{F}_q^* $ such that $ f = a x + b x^q + x^{2q-1} $ permutes $ \mathbb{F}_{q^2} $, using discriminant analysis and substitution techniques.
• For even $ q $, employ trace conditions $ \operatorname{Tr}_{q/2}(1/(a+1)) = 0 $ and $ \operatorname{Tr}_{q/2}(1/b) = 0 $, and express solutions via rational functions in $ d \in \mathbb{F}_q \setminus \mathbb{F}_2 $.
• Use generating functions and coefficient extraction via constant term extraction in Laurent series to evaluate complex sums involving binomial coefficients modulo $ p $.
• Apply identities involving $ \sum_{l} \binom{l+2}{2} \binom{-l}{l} z^l $, $ \sum_{l} (l+1) \binom{-l}{l} z^l $, and similar, to simplify the functional equation for $ g_{n,q} $.
• Transform $ g_{n,q} $ modulo $ x^{q^2} - x $ via invertible variable substitution into the form $ A x + B x^q + C x^{2q-1} $, reducing the problem to Theorem A.
• Leverage known results on permutation polynomials and projective geometry to describe the set $ \mathcal{X} $ of projective triples $ [a:b:c] $ for which $ a x + b x^q + c x^{2q-1} $ is a PP over $ \mathbb{F}_{q^2} $.

Experimental results

Research questions

• RQ1 For which $ a, b \in \mathbb{F}_q $ is the trinomial $ f = a x + b x^q + x^{2q-1} $ a permutation polynomial over $ \mathbb{F}_{q^2} $ when $ q $ is odd?
• RQ2 What are the necessary and sufficient conditions on $ a, b \in \mathbb{F}_q $ for $ f = a x + b x^q + x^{2q-1} $ to permute $ \mathbb{F}_{q^2} $ when $ q $ is even?
• RQ3 For which integers $ n = q^\alpha - q^\beta - 1 $, $ \alpha > \beta \geq 0 $, is the polynomial $ g_{n,q} $ a permutation polynomial over $ \mathbb{F}_{q^2} $?
• RQ4 How can the functional equation $ \sum_{c \in \mathbb{F}_q} (x + c)^n = g_{n,q}(x^q - x) $ be used to characterize permutation polynomials $ g_{n,q} $?
• RQ5 What is the geometric structure of the set $ \mathcal{X} \subset \text{PG}(2, \mathbb{F}_q) $ parametrizing permutation trinomials of the form $ a x + b x^q + c x^{2q-1} $?

Key findings

• For odd $ q $, $ f = a x + b x^q + x^{2q-1} $ is a permutation polynomial over $ \mathbb{F}_{q^2} $ if and only if one of four conditions holds: (i) $ a(a-1) $ is a square and $ b^2 = a^2 + 3a $; (ii) $ a = 1 $ and $ b^2 - 4 $ is a square; (iii) $ a = 3, b = 0, q \equiv -1 \pmod{6} $; (iv) $ a = b = 0, q \equiv 1,3 \pmod{6} $.
• For even $ q > 2 $, $ f $ is a permutation polynomial over $ \mathbb{F}_{q^2} $ if and only if: (i) $ a \neq 1 $, $ \operatorname{Tr}_{q/2}(1/(a+1)) = 0 $, and $ b^2 = a^2 + a $; or (ii) $ a = 1 $, $ b \neq 0 $, and $ \operatorname{Tr}_{q/2}(1/b) = 0 $.
• All $ n = q^\alpha - q^\beta - 1 $ for which $ g_{n,q} $ is a permutation polynomial over $ \mathbb{F}_{q^2} $ are completely characterized: for even $ q $, the conditions are given in Theorem C; for odd $ q $, they follow from Theorem A via variable substitution.
• The set $ \mathcal{X} $ of projective triples $ [a:b:c] $ for which $ a x + b x^q + c x^{2q-1} $ is a PP over $ \mathbb{F}_{q^2} $ is explicitly described for even $ q $, as a union of four families parameterized by $ d \in \mathbb{F}_q \setminus \mathbb{F}_2 $.
• The computation of the constant term in Laurent series expansions leads to a surprising factorization of a degree-4 polynomial into rational functions, enabling the derivation of closed-form expressions for key sums.
• The identity $ \sum_{l} \binom{l+2}{2} \binom{-l}{l+1} z^l = \frac{3z(1+2z)}{(1+4z)^2} $ for $ q > 2 $ is established using residue calculus and trace-based coefficient extraction, crucial for proving the main theorems.