[Paper Review] Several Classes of Permutation Trinomials over $\mathbb F_{5^n}$ From Niho Exponents

Gaofei Wu, Nian Li|arXiv (Cornell University)|Feb 21, 2017

Coding theory and cryptography11 references4 citations

TL;DR

This paper constructs several new classes of permutation trinomials over $fi{F}_{5^{2k}}$ using Niho exponents by identifying fractional polynomials that permute the $(q+1)$-th roots of unity in $fi{F}_{q^2}$, where $q = 5^k$. The key contribution is the construction of permutation trinomials of the form $f(x) = x + ho_1 x^{c_1(5^k-1)+1} + ho_2 x^{c_2(5^k-1)+1}$, with $ ho_1, ho_2 o ∑ 1$, under two conjectures on root permutation.

ABSTRACT

The construction of permutation trinomials over finite fields attracts people's interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, by constructing some new fractional polynomials that permute the set of the $(q+1)$-th roots of unity in $\mathbb F_{q^2}$, we present several classes of permutation trinomials with Niho exponents over $\mathbb F_{q^2}$, where $q=5^k$.

Motivation & Objective

• To construct new classes of permutation trinomials over finite fields $fi{F}_{5^{2k}}$ with simple algebraic forms.
• To extend the known classes of permutation polynomials using Niho exponents, which are known to yield sparse and structured permutation polynomials.
• To establish a connection between permutation of the $(q+1)$-th roots of unity and the construction of permutation trinomials via Lemma 2.
• To propose and analyze two conjectures on fractional polynomials that, if true, yield new families of permutation trinomials.
• To identify open problems regarding conditions under which specific trinomial forms are permutations over $fi{F}_{5^{2k}}$.

Proposed method

• Leverages Lemma 2, which provides a criterion for permutation polynomials of the form $f(x) = x^l g(x^{(q^2-1)/s})$, requiring $\gcd(l, (q^2-1)/s) = 1$ and $x^l g(x)^{(q^2-1)/s}$ to permute the $s$-th roots of unity.
• Constructs fractional polynomials $g_1(x), g_2(x), \dots, g_{10}(x)$ that permute the set $\mu_{q+1}$, the $(q+1)$-th roots of unity in $\ufb01{F}_{q^2}$, with $q = 5^k$.
• Uses trace and norm functions over $\ufb01{F}_{q^2}$ to $\ufb01{F}_q$ to analyze the behavior of trinomial mappings, particularly ${{\rm Tr}}(x^d)$ and ${{\rm N}}(x)$.
• Applies algebraic identities involving ${{\rm Tr}}(x^d)$ and ${{\rm N}}(x)$ for $d = 2,3,4,8,9$, derived from direct computation in Lemma 1.
• Reduces the permutation problem to analyzing rational functions such as $-x(\frac{x^2+2}{x^2-2})^2$ and $-x(\frac{x^2-2}{x^2+2})^2$ on $\mu_{q+1}$, leading to conjectures.
• Transforms the trinomial $f(x) = x + \rho_1 x^{c_1(5^k-1)+1} + \rho_2 x^{c_2(5^k-1)+1}$ into a form suitable for applying Lemma 2 by analyzing the associated $g(x)$ on $\mu_{q+1}$.

Experimental results

Research questions

• RQ1 Under what conditions does the trinomial $f(x) = x + x^{s(5^k-1)+1} - x^{t(5^k-1)+1}$ permute $\ufb01{F}_{5^{2k}}$ when $s + t = 0$ or $s + t = \frac{5^k+1}{2}$?
• RQ2 When does $f(x) = x + x^{s(5^k-1)+1} + x^{t(5^k-1)+1}$ become a permutation trinomial over $\ufb01{F}_{5^{2k}}$ under the condition $s + t = \frac{5^k+1}{2}$?
• RQ3 Does the rational function $-x(\frac{x^2-2}{x^2+2})^2$ permute the set of $(q+1)$-th roots of unity in $\ufb01{F}_{q^2}$ for even $k$, where $q = 5^k$?
• RQ4 Can the rational function $-x(\frac{x^2+2}{x^2-2})^2$ be proven to permute $\mu_{q+1}$ for odd $k$?
• RQ5 What are the necessary and sufficient conditions on the exponents $c_1, c_2$ and coefficients $\rho_1, \rho_2 \in \{1, -1\}$ for $f(x) = x + \rho_1 x^{c_1(5^k-1)+1} + \rho_2 x^{c_2(5^k-1)+1}$ to be a permutation polynomial over $\ufb01{F}_{5^{2k}}$?

Key findings

• The paper constructs several new classes of permutation trinomials over $\ufb01{F}_{5^{2k}}$ by identifying fractional polynomials that permute the $(q+1)$-th roots of unity in $\ufb01{F}_{q^2}$, where $q = 5^k$.
• For $q = 5^k$, the function $g_1(x) = -x^{\frac{q+1}{2}} \left( \frac{x^s - 2}{x^s + 2} \right)^2$ with $s = \frac{q+3}{4}$ is proven to permute $\mu_{q+1}$, leading to a new class of permutation trinomials.
• The function $g_2(x) = -x^{\frac{q+1}{2}} \left( \frac{x^s + 2}{x^s - 2} \right)^2$ with $s = \frac{q+1}{4}$ is shown to permute $\mu_{q+1}$, yielding another class of permutation trinomials.
• Conjecture 1 states that the rational function $-r(\frac{r^2 - r + 2}{r^2 + r + 2})^2$ with $r = \frac{{\rm Tr}(x)^2}{{\rm N}(x)}$ is injective, which, if true, ensures unique determination of ${{\rm Tr}}(x)$ and ${{\rm N}}(x)$ from $f(x)$, implying permutation.
• Conjecture 2 proposes that $-x(\frac{x^2 - 2}{x^2 + 2})^2$ permutes $\mu_{q+1}$ for even $k$, and if true, leads to a new class of permutation trinomials with exponents $s = \frac{q+2}{3} + 1$, $t = 2\cdot\frac{q+2}{3}$.
• The paper proves that if Conjecture 2 holds, then $f(x) = x - x^{s(q-1)+1} - x^{t(q-1)+1}$ permutes $\ufb01{F}_{q^2}$ for even $k$, with $s$ and $t$ as defined, via Lemma 2 and transformation of $g_{10}(x)$ on $\mu_{q+1}$.