Bilinear complexity of algebras and the Chudnovsky–Chudnovsky interpolation method

Randriambololona, Hugues

doi:10.1016/j.jco.2012.02.005

Cited by 34 publications

(74 citation statements)

References 20 publications

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“…At the very end of [20, section 5], it is discussed how our optimal solution to Riemann-Roch systems could be combined with extensions of the Chudnovsky-Chudnovsky method such as [9], that use evaluation at points of higher degree and with multiplicities. This discussion was not reproduced here, because these results are now superseded by [21,Th. 5.2(c)], which uses an even finer notion of generalized evaluation.…”

Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 96%

“…Thus, it becomes useful, for instance, if one works over a field F p of prime order. However, our method, in Proposition 2 as well as in [21,Th. 5.2(c)], still requires curves with sufficiently many points of degree 1, as asked by condition (13).…”

Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 98%

“…5.2(c)], which uses an even finer notion of generalized evaluation. Namely, the bounds from [21] involve the quantities…”

Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 99%

“…Gaps between prime numbers in a given residue class. (This is a translation of the paragraph at the bottom of [20, p. 31], and is also discussed in [21,Rem. 5.5]. )…”

Section: More Recent Developments and Questions That Remain Openmentioning

confidence: 99%

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Gaps between prime numbers and tensor rank of multiplication in finite fields

Randriam

2018

Des. Codes Cryptogr.

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We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field F p 2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time.Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over Fp, and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.

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Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 96%

Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 98%

“…5.2(c)], which uses an even finer notion of generalized evaluation. Namely, the bounds from [21] involve the quantities…”

Section: Generalizations Of the Basic Chudnovsky-chudnovsky Methodmentioning

confidence: 99%

“…Gaps between prime numbers in a given residue class. (This is a translation of the paragraph at the bottom of [20, p. 31], and is also discussed in [21,Rem. 5.5]. )…”

Section: More Recent Developments and Questions That Remain Openmentioning

confidence: 99%

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Gaps between prime numbers and tensor rank of multiplication in finite fields

Randriam

2018

Des. Codes Cryptogr.

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“…The main theorem of this work is Theorem 27 and it states that Algorithm 4 is able to find all decompositions of a bilinear map over a finite field. It can be used for proving lower bounds on the rank of a bilinear map and it has applications for improving upper bounds on the Chudnovsky-Chudnovsky algorithms [8,23,22]. Specifically, we compute all the decompositions for the short product of polynomials P and Q modulo X 5 and the product of 3 × 2 by 2 × 3 matrices.…”

Section: Introductionmentioning

confidence: 99%

Improved method for finding optimal formulas for bilinear maps in a finite field

Covanov

2019

Theoretical Computer Science

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In 2012, Barbulescu, Detrey, Estibals and Zimmermann proposed a new framework to exhaustively search for optimal formulae for evaluating bilinear maps over finite fields, such as Strassen or Karatsuba formulae. The main contribution of this work is a new criterion to aggressively prune useless branches in the exhaustive search, thus leading to the computation of new optimal formulae. We apply in particular our approach to the short product modulo X 5 and the circulant product modulo (X 5 − 1). Moreover, we are able to prove that there is essentially only one optimal decomposition of the product of 3 × 2 by 2 × 3 matrices up to the action of some group of automorphisms.

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