Splitting quaternion algebras over quadratic number fields

We propose a randomized polynomial time algorithm for computing non-trivial zeros of quadratic forms in 4 or more variables over F q (t), where F q is a finite field of odd characteristic. The algorithm is based on a suitable splitting of the form into two forms and finding a common value they both represent. We make use of an effective formula for the number of fixed degree irreducible polynomials in a given residue class. We apply our algorithms for computing a Witt decomposition of a quadratic form, for computing an explicit isometry between quadratic forms and finding zero divisors in quaternion algebras over quadratic extensions of F q (t).

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“…Here we solve this problem using Algorithms 1 and 2. The method is a straightforward analogue of the algorithm from [12].…”

Section: An Applicationmentioning

confidence: 99%

“…In the final section we apply our results to find zero divisors in quaternion algebras over quadratic extensions of F q (t) or, equivalently, to find zeros of ternary quadratic forms over quadratic extensions of F q (t). The material of this part is the natural analogue of that presented in [12] over quadratic number fields.…”

Section: Introductionmentioning

confidence: 99%

Explicit equivalence of quadratic forms over Fq(t)

Ivanyos

Kutas²,

Rónyai

2019

Finite Fields and Their Applications

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“…We will refer to this problem as the explicit isomorphism problem. This is a well studied problem in computational algebra [5,19,20,22,24]. It has connections to arithmetic geometry , norm equations [20], parametrization of algebraic varieties [14] and error-correcting codes [13].…”

Section: Introductionmentioning

confidence: 99%

An identification system based on the explicit isomorphism problem

Kiss

Kutas

2021

AAECC

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We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

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“…They also show that finding explicit isomorphisms between central simple K-algebras of dimension n 2 over K can be reduced to finding an explicit isomorphism between an algebra A and M n 2 (K). Then in [30] (and independently in [15]) an algorithm was provided when A is isomorphic to…”

Section: Introductionmentioning

confidence: 99%

Explicit isomorphisms of quaternion algebras over quadratic global fields

Csahók¹,

Kutas²,

Mickaël³

et al. 2020

Preprint

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Let L be a separable quadratic extension of either Q or F q (t). We propose efficient algorithms for finding isomorphisms between quaternion algebras over L. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L-algebra. In order to obtain efficient algorithms in the characteristic 2 case, we propose an algorithm for finding nontrivial zeros of a regular quadratic form in four variables over F 2 k (t).

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Splitting quaternion algebras over quadratic number fields

Cited by 5 publications

References 17 publications

Explicit equivalence of quadratic forms over Fq(t)

Explicit equivalence of quadratic forms over Fq(t)

An identification system based on the explicit isomorphism problem

Explicit isomorphisms of quaternion algebras over quadratic global fields

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