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

The construction of permutation trinomials of the form $X^r(X^{α(2^m-1)}+X^{β(2^m-1)} + 1)$ over $\F_{2^{2m}}$, where $m,~r\text{ and }α> β$ are positive integers, is an active area of research. Several classes of permutation trinomials with fixed values of $α$, $β$ and $r$ have been studied. Here, we construct three new classes of permutation trinomials with $(α,β,r)\in{(7,5,7),(8,6,9),(10,4,11)}$ over $\F_{2^{2m}}$. We also analyze the quasi-multiplicative equivalence of the newly obtained classes of permutation trinomials to both the existing ones and to each other. Additionally, we prove the nonexistence of a class of permutation trinomials over $\F_{2^{2m}}$ of the same type for $r=9$, $α=7$, and $β=3$ when $m > 3$. Furthermore, we provide a proof for a conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (2024).

💡 Research Summary

The paper investigates permutation trinomials of the form
  f(X)=X^r·(X^{α(2^m−1)}+X^{β(2^m−1)}+1)
over the quadratic extension field 𝔽_{2^{2m}} (i.e., fields of even characteristic). While many families with small exponents have been studied, the authors introduce three completely new families corresponding to the exponent triples (α,β,r) = (7,5,7), (8,6,9) and (10,4,11).

The authors first recall the standard criterion (Lemma 2.2) that reduces the permutation property of f(X) to two conditions: (i) gcd(r,2^m−1)=1 and (ii) the auxiliary polynomial G(X)=X^r·(X^α+X^β+1)^{2^m−1} must permute the multiplicative subgroup µ_{2^m+1} (the “unit circle”). Condition (i) is a simple congruence check; the bulk of the work lies in establishing (ii).

For each new family, the authors construct the equation G(X)=G(Y) with X,Y∈µ_{2^m+1} and rewrite it as a bivariate polynomial f(X,Y). Using computer algebra (SageMath), they factor f(X,Y) over suitable extensions of 𝔽_{2^m}. The factorization always contains a linear factor (X+Y) and several quadratic factors of the shape
  X^2Y^2 + b·(X^2+Y^2) + c·XY + 1,
where b and c are powers of a primitive element b∈𝔽_{2^k} (k=10,4,6 respectively). The authors then apply trace maps Tr_{2^{k}m/2^m} to these coefficients, exploiting Lemma 3.1 (or analogous trace identities) to show that each quadratic factor cannot vanish for any pair (X,Y)∈µ_{2^m+1} unless X=Y. Consequently, G is injective on µ_{2^m+1}, and by Lemma 2.2 the original trinomial is a permutation of 𝔽_{2^{2m}}. The necessary congruence restrictions on m (e.g., m≢0 (mod 5) for the (10,4,11) family, m odd for (8,6,9), and m even with m≢0 (mod 3) for (7,5,7)) arise from ensuring the trace conditions and the coprimality gcd(r,2^m−1)=1.

In addition to the positive results, the paper proves a non‑existence theorem for the family X^9·(X^{7(2^m−1)}+X^{3(2^m−1)}+1). The authors associate the inner trinomial h(X)=X^7+X^3+1 with an algebraic curve C over 𝔽_{2^{2m}}. They demonstrate that C is absolutely irreducible and compute its genus g=3. Applying the Hasse–Weil bound |#C(𝔽_{2^{2m}})−(2^{2m}+1)| ≤ 2g·2^m, they show that for m>3 the number of rational points on C is too small to accommodate a permutation of the unit circle, thereby establishing that the trinomial cannot be a permutation in this range.

The paper also addresses the concept of quasi‑multiplicative (QM) equivalence, introduced by Wu et al. (2022), which considers transformations f(X) ↦ a·f(bX+c)+d (a,b∈𝔽_q^*, c,d∈𝔽_q) that preserve the number of monomials. The authors systematically examine whether any of the newly constructed families are QM‑equivalent to previously known families (including those from Gupta–Sharma 2016 and Yadav et al. 2024) or to each other. By exhaustive analysis of possible parameters (a,b,c,d) and exploiting the specific exponent patterns, they prove that all new families are QM‑inequivalent both to the existing literature and among themselves.

Finally, the authors settle a conjecture posed by Yadav, Gupta, Singh, and Yadav (2024) concerning the QM‑equivalence of two previously known families (with parameters (6,1,7) and (6,4,7)). Using the same trace‑based factorization technique, they show that no admissible QM transformation can map one family onto the other, thereby providing a complete proof of the conjecture.

Overall, the paper contributes three genuinely new permutation trinomials, establishes a non‑existence result via algebraic‑geometric methods, and clarifies the landscape of QM equivalence for this class of polynomials. The blend of explicit computational factorization, trace arguments, and Hasse–Weil theory makes the work technically robust, though the reliance on computer‑generated factorizations without detailed supplemental data slightly limits reproducibility. Future work could extend these techniques to higher‑degree trinomials, odd characteristic fields, or a systematic classification of QM equivalence classes.