Geometric characterization of p-exceptional monomial GAPN functions

Abstract

On finite fields of characteristic p, PN (perfect nonlinear) functions for odd p and APN (almost perfect nonlinear) functions for even p are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even p to all p. One of the main targets of studies on such highly nonlinear functions is their classification. While p-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over 𝔽_p is called a p-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on 𝔽_pⁿ for infinitely many positive integers n. In this paper, we give a geometric characterization of p-exceptional monomial GAPN functions. To this end, for a prime power q, we establish a geometric characterization of non-zero polynomials in 𝔽_q[x,y] having no absolutely irreducible factor defined over 𝔽_q.

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BibTeX RIS

BibTeX

@misc{kuroda2026geometric,
  title = {{Geometric characterization of $p$-exceptional monomial GAPN functions}},
  author = {Masamichi Kuroda and Kentaro Mitsui},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.08969},
  eprint = {2608.08969},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08969},
}

RIS

TY  - UNPB
TI  - Geometric characterization of p-exceptional monomial GAPN functions
AU  - Masamichi Kuroda
AU  - Kentaro Mitsui
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.08969
UR  - https://arxiv.org/abs/2608.08969
AB  - On finite fields of characteristic p, PN (perfect nonlinear) functions for odd p and APN (almost perfect nonlinear) functions for even p are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even p to all p. One of the main targets of studies on such highly nonlinear functions is their classification. While p-exceptional monomial PN and $2$-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over 𝔽_p is called a p-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on 𝔽_p^n for infinitely many positive integers n. In this paper, we give a geometric characterization of p-exceptional monomial GAPN functions. To this end, for a prime power q, we establish a geometric characterization of non-zero polynomials in 𝔽_q[x,y] having no absolutely irreducible factor defined over 𝔽_q.
ER  -

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