Fast arithmetic for the algebraic closure of finite fields

Feo, Luca De; Doliskani, Javad; Schost, Éric

doi:10.1145/2608628.2608672

Cited by 6 publications

(6 citation statements)

References 35 publications

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“…Then, for small values of s and q, one may use a table of irreducible polynomials. For larger values, the constructions [14,18,19] are reasonably efficient, and yield an irreducible polynomial in time less than quadratic in s. However negligible from an asymptotic point of view, the construction of the polynomial h and of the field k take a serious toll on the practical performances of Rains' algorithm. 3 This concludes the presentation of Rains' algorithm.…”

Section: Algorithm 5 Rains' Cyclotomic Algorithmmentioning

confidence: 99%

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Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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Let F q be a finite field. Given two irreducible polynomials f, g over F q , with deg f dividing deg g, the finite field embedding problem asks to compute an explicit description of a field embedding of. When deg f = deg g, this is also known as the isomorphism problem.This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better.We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

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Section: Algorithm 5 Rains' Cyclotomic Algorithmmentioning

confidence: 99%

“…This technique is not new [56,57,58,7], however it is seldom found in the literature. We briefly recall it, following the presentation of [19].…”

Section: Inverse Maps and Dualitymentioning

confidence: 99%

Computing isomorphisms and embeddings of finite fields

Brieulle

Feo

Doliskani

et al. 2019

ACM Commun. Comput. Algebra

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“…Of course, it is possible to use the extension of Kedlaya and Umans' algorithm mentioned before; this leads to algorithms with almost linear running time in a boolean model, but as we said above, they are difficult to put into practice. In an algebraic model, the paper [17] give two different algorithms, but none of them is quasi-linear.…”

Section: Compositamentioning

confidence: 99%

Algorithms for Finite Field Arithmetic

Schost

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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We review several algorithms to construct finite fields and perform operations such as field embedding. Following previous work by notably Shoup, de Smit and Lenstra or Couveignes and Lercier, as well as results obtained with De Feo and Doliskani, we distinguish between algorithms that build "towers" of finite fields, with degrees of the form , 2 , 3 , . . . and algorithms for composita. We show in particular how techniques that originate from algorithms for computing with triangular sets can be useful in such a context.

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“…All solutions presented so far have superquadratic complexity, i.e., d > 2. Recent work on embedding algorithms [11,12,14] yields subquadratic (more precisely, d ≤ 1.5) solutions for specially constructed (non-unique, non-general) families of irreducible polynomials, and even quasi-optimal ones (i.e., d = 1 + ε) if a quasi-linear modular composition algorithm is available. However these constructions involve counting points of random elliptic curves over finite fields, and have thus a rather high polynomial dependency in log p; for this reason, they are usually considered practical only for relatively small characteristic.…”

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confidence: 99%

Standard Lattices of Compatibly Embedded Finite Fields

Feo

Randriam

Rousseau

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field F p at once. They can also be used to represent the algebraic closureF p , and to represent all finite fields in a standard manner.The most well known constructions are Conway polynomials, and the Bosma-Cannon-Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time.Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representingF p .Compared to Bosma-Cannon-Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees.Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical.

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Fast arithmetic for the algebraic closure of finite fields

Cited by 6 publications

References 35 publications

Computing isomorphisms and embeddings of finite fields

Computing isomorphisms and embeddings of finite fields

Algorithms for Finite Field Arithmetic

Standard Lattices of Compatibly Embedded Finite Fields

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