A parametrization of nonassociative Petit algebras K[t;σ]/K[t:σ](t^m−a) up to equivalence

Abstract

Let K/F be a cyclic Galois field extension of degree n with Gal(K/F)=⟨σ⟩. The nonassociative Petit algebras K[t;σ]/K[t;σ](t^m−a) have recently emerged as important tools in coding theory and cryptography, where the applications only require distinct algebras up to isomorphism. Algebras that are isomorphic via a monomial isomorphism of degree one are called equivalent. We present a novel and precise parametrization of the F-equivalence classes of these algebras in the key case that F contains a primitive nth root of unity and n| m. We evaluate the parametrization explicitly when F is a nonarchimedean local field, which has applications to the number field case recently used to construct a nonassociative variant of Learning with Errors.

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BibTeX RIS

BibTeX

@misc{nevins2026parametrization,
  title = {{A parametrization of nonassociative Petit algebras $K[t;σ]/K[t:σ](t^m-a)$ up to equivalence}},
  author = {Monica Nevins and Susanne Pumpluen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30921},
  eprint = {2610.30921},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30921},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A parametrization of nonassociative Petit algebras K[t;σ]/K[t:σ](t^m−a) up to equivalence
AU  - Monica Nevins
AU  - Susanne Pumpluen
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30921
UR  - https://arxiv.org/abs/2610.30921
AB  - Let K/F be a cyclic Galois field extension of degree n with Gal(K/F)=⟨σ⟩. The nonassociative Petit algebras K[t;σ]/K[t;σ](t^m−a) have recently emerged as important tools in coding theory and cryptography, where the applications only require distinct algebras up to isomorphism. Algebras that are isomorphic via a monomial isomorphism of degree one are called equivalent. We present a novel and precise parametrization of the F-equivalence classes of these algebras in the key case that F contains a primitive nth root of unity and n| m. We evaluate the parametrization explicitly when F is a nonarchimedean local field, which has applications to the number field case recently used to construct a nonassociative variant of Learning with Errors.
ER  -

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