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Resolving a conjecture on permutation polynomials over 𝔽2n

  • Yi Li
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Abstract

Let δ∈𝔽2n satisfy Tr𝔽2n/𝔽2(δ)=1. We study the permutation behavior of f(x) = (1/(x2+x+δ))2k+x over 𝔽2n. Helleseth and Zinoviev proved that f(x) is a permutation for k=0,1, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for 0≤ k<n, f(x) is a permutation of 𝔽2n if and only if k=0 or k=1.

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Cite

BibTeXRIS
BibTeX
@misc{li2026resolving,
  title = {{Resolving a conjecture on permutation polynomials over $\mathbbF_2^n$}},
  author = {Yi Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03353},
  eprint = {2609.03353},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03353},
}
RIS
TY  - UNPB
TI  - Resolving a conjecture on permutation polynomials over 𝔽_2^n
AU  - Yi Li
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03353
UR  - https://arxiv.org/abs/2609.03353
AB  - Let δ∈𝔽_2^n satisfy Tr_𝔽_2^n/𝔽_2(δ)=1. We study the permutation behavior of f(x) = (1/(x^2+x+δ))^2^k+x over 𝔽_2^n. Helleseth and Zinoviev proved that f(x) is a permutation for k=0,1, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for 0≤ k<n, f(x) is a permutation of 𝔽_2^n if and only if k=0 or k=1.
ER  -