Efficient computations in central simple algebras using Amitsur cohomology

2023/07/01 by Péter Kutas, Kutas, Péter, Mickaël Montessinos +1
Mathematics · #11Y40 (Primary) 16H05 #68Q12 (Secondary) #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2307.00261

openalex publication_date 2023/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an efficient computational representation of central simple algebras using Brauer factor sets. Using this representation and polynomial quantum algorithms for number theoretical tasks such as factoring and S-unit group computation, we give a polynomial quantum algorithm for the explicit isomorphism problem over number field, which relies on a heuristic concerning the irreducibility of the characteristic polynomial of a random matrix with algebraic integer coefficients. We present another version of the algorithm which does not need any heuristic but which is only polynomial if the degree of the input algebra is bounded.

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