Some new results on permutation trinomials over finite fields with even characteristic

Technical Summary: New Results on Permutation Trinomials over Finite Fields with Even Characteristic

Problem Statement
The paper addresses the construction and classification of permutation trinomials over finite fields of even characteristic, specifically $\mathbb{F}_{q^2}$ where $q = 2^m$. The focus is on polynomials of the form $f(X) = X^r(X^{\alpha(q-1)} + X^{\beta(q-1)} + 1)$, where $m, r, \alpha, \beta$ are positive integers and $\alpha > \beta$. While monomials are easily classified and permutation binomials with identity coefficients do not exist in even characteristic, the classification of permutation trinomials remains a challenging open area. The authors aim to identify new classes of such trinomials, verify their novelty against existing literature, and resolve specific conjectures regarding their equivalence.

Methodology
The authors employ a combination of algebraic criteria, computational factorization, and algebraic geometry:

1. Permutation Criteria: The primary tool is the Wan-Lidl criterion (Lemma 2.1) and its specialization for $\mathbb{F}_{q^2}$ (Lemma 2.2). This reduces the problem of determining if $f(X)$ permutes $\mathbb{F}_{q^2}$ to checking two conditions: $\gcd(r, q-1)=1$ and the permutation of the unit circle $\mu_{q+1}$ by the associated polynomial $g(X) = X^r(h(X))^{q-1}$.
2. Factorization and Injectivity: To prove that the associated polynomial $g(X)$ permutes $\mu_{q+1}$, the authors analyze the equation $g(X) = g(Y)$ for $X, Y \in \mu_{q+1}$. They transform this into a polynomial equation $f(X, Y) = 0$ and utilize SageMath to factorize $f(X, Y)$ over specific field extensions. They demonstrate that the only solution in the relevant domain is $X=Y$, thereby proving injectivity.
3. Quasi-Multiplicative (QM) Equivalence: To establish the novelty of the constructed classes, the authors apply the definition of QM equivalence (Definition 4.1). They utilize a two-step strategy: first, checking if the sets of exponents of two polynomials can be mapped to each other via a multiplier $d$ modulo $q^2-1$; second, comparing coefficients if the first step holds.
4. Nonexistence Provia Algebraic Geometry: To prove the nonexistence of a specific class of trinomials, the authors leverage the connection between permutation polynomials and algebraic curves. They construct a specific polynomial $H(X, Y)$ and prove its absolute irreducibility (Lemma 5.2). They then apply the Hasse-Weil bound (Aubry-Perret version) to show that the curve defined by $H(X, Y)$ possesses rational points that violate the permutation property for sufficiently large $m$.
5. Conjecture Resolution: The proof of the conjecture regarding QM equivalence involves solving systems of congruences derived from exponent sets, utilizing the Chinese Remainder Theorem (CRT) to determine the existence of a valid multiplier $d$.

Key Contributions and Results

• Three New Classes of Permutation Trinomials: The authors construct and prove the existence of three new classes of permutation trinomials over $\mathbb{F}_{2^{2m}}$ with the following parameter sets $(\alpha, \beta, r)$:

  1. $(7, 5, 7)$: Permutation occurs if and only if $m$ is even and $m \not\equiv 0 \pmod 3$.
  2. $(8, 6, 9)$: Permutation occurs if and only if $m$ is odd.
  3. $(10, 4, 11)$: Permutation occurs if and only if $m \not\equiv 0 \pmod 5$.
     The proofs involve detailed factorization of the associated bivariate polynomials over extensions of $\mathbb{F}_{q^2}$ and verification of injectivity on the unit circle.

• QM Inequivalence: The paper demonstrates that these three new classes are Quasi-Multiplicative (QM) inequivalent to each other and to all previously known classes of permutation trinomials listed in the literature (specifically those referenced in Table 1 of the paper). This confirms that the constructed classes represent genuinely new mathematical objects rather than transformations of existing ones.

• Nonexistence Result: The authors prove that the trinomial $X^9(X^{7(q-1)} + X^{3(q-1)} + 1)$ does not permute $\mathbb{F}_{2^{2m}}$ for $m > 3$. This is achieved by showing that the associated algebraic curve is absolutely irreducible and applying the Hasse-Weil bound to guarantee the existence of non-permuting rational points for $m \ge 18$, with smaller cases verified computationally.

• Resolution of a Conjecture: The paper provides an elementary proof for a conjecture proposed by Yadav, Gupta, Singh, and Yadav (2024). The conjecture states that if $F(X) = X^r(X^{\alpha(q-1)} + X^{\beta(q-1)} + 1)$ and $G(X) = X^{2\alpha-r}(X^{\alpha(q-1)} + X^{(\alpha-\beta)(q-1)} + 1)$ both permute $\mathbb{F}_{q^2}$, then they are QM equivalent. The authors explicitly compute the exponent $d$ that establishes this equivalence, offering a proof distinct from previous work by Yadav, Singh, and Gupta (2024).

Significance
The paper claims significance in extending the known landscape of permutation trinomials over finite fields with even characteristic. By identifying three new parameter sets, the authors expand the catalog of explicit constructions available for applications in cryptography, coding theory, and combinatorial design. Furthermore, the rigorous verification of QM inequivalence ensures that these new classes are not merely redundant representations of known polynomials. The nonexistence proof for the $(9, 7, 3)$ case narrows the search space for potential permutation trinomials, while the resolution of the conjecture clarifies the structural relationships between specific families of these polynomials. The work relies on established algebraic techniques but applies them to resolve specific open questions and conjectures in the field.