Curves with many points and multiplication in finite fileds

Shparlinski, Igor E.; Tsfasman, M. A.; Vlăduţ, Serge

doi:10.1007/bfb0087999

Cited by 56 publications

(113 citation statements)

References 2 publications

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“…Note that this result is valid for all n, not merely asymptotically as the result of Shparlinski, Tsfasman, and Vladut [10] recalled above and also that the curves used are constructible in contrast to curves used in [10]. However for the asymptotic bound they obtain a better result because they use a tower which is more dense than the tower of Garcia-Stichtenoth [5] completed by intermediate steps [3].…”

Section: Known Resultsmentioning

confidence: 53%

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Multiplication algorithm in a finite field and tensor rank of the multiplication

Ballet

Rolland

2004

Journal of Algebra

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Section: Known Resultsmentioning

confidence: 53%

“…Theorem 4.2 results also when q is not a square in a better asymptotic bound than the one obtained by Shparlinski, Tsfasman, and Vladut in [10]. More precisely, …”

Section: Lemma 42 Let Us Setmentioning

confidence: 53%

Multiplication algorithm in a finite field and tensor rank of the multiplication

Ballet

Rolland

2004

Journal of Algebra

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“…Note that this notion has been studied in the context of asymptotic arithmetic complexity (see [4] and [28]). We can now state and prove our field-descent theorem.…”

Section: Theorem 6 (Chen and Cramermentioning

confidence: 99%

Asymptotically Good Ideal Linear Secret Sharing with Strong Multiplication over Any Fixed Finite Field

Cascudo

Chen

Cramer

et al. 2009

Lecture Notes in Computer Science

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Abstract. This work deals with "MPC-friendly" linear secret sharing schemes (LSSS), a mathematical primitive upon which secure multi-party computation (MPC) can be based and which was introduced by Cramer, Damgaard and Maurer (EUROCRYPT 2000). Chen and Cramer proposed a special class of such schemes that is constructed from algebraic geometry and that enables efficient secure multi-party computation over fixed finite fields (CRYPTO 2006). We extend this in four ways. First, we propose an abstract coding-theoretic framework in which this class of schemes and its (asymptotic) properties can be cast and analyzed. Second, we show that for every finite field Fq, there exists an infinite family of LSSS over Fq that is asymptotically good in the following sense: the schemes are "ideal," i.e., each share consists of a single Fq-element, and the schemes have t-strong multiplication on n players, where the corruption tolerance 3t n−1 tends to a constant ν(q) with 0 < ν(q) < 1 when n tends to infinity. Moreover, when |Fq| tends to infinity, ν(q) tends to 1, which is optimal. This leads to explicit lower bounds on τ (q), our measure of asymptotic optimal corruption tolerance. We achieve this by combining the results of Chen and Cramer with a dedicated field-descent method. In particular, in the F2-case there exists a family of binary t-strongly multiplicative ideal LSSS with 3t n−1 ≈ 2.86% when n tends to infinity, a one-bit secret and just a one-bit share for every player. Previously, such results were shown for Fq with q ≥ 49 a square. Third, we present an infinite family of ideal schemes with t-strong multiplication that does not rely on algebraic geometry and that works over every finite field Fq. Its corruption tolerance vanishes, yet still 3t n−1 = Ω(1/(log log n) log n). Fourth and finally, we give an improved non-asymptotic upper bound on corruption tolerance.

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“…Soient F q un corps fini à q éléments où q = p r est une puissance d'un nombre premier p et F q n une extension de F q de degré n. La complexité bilinéaire de la multiplication m dans F q n sur F q , notée μ q (n), est le rang du tenseur t m ∈ F * q n ⊗ F * q n ⊗ F q n associé à m, où F * q n désigne le dual de F q n sur F q (voir [7,1]). Winograd [9] et De Groote [4] ont montré que μ q (n) 2n − 1, avec égalité si et seulement si n 1 2 q + 1.…”

Section: Complexité Bilinéaire De La Multiplicationunclassified

Complexité bilinéaire de la multiplication dans des petits corps finis

Chaumine

2006

Comptes Rendus. Mathématique

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Curves with many points and multiplication in finite fileds

Cited by 56 publications

References 2 publications

Multiplication algorithm in a finite field and tensor rank of the multiplication

Multiplication algorithm in a finite field and tensor rank of the multiplication

Asymptotically Good Ideal Linear Secret Sharing with Strong Multiplication over Any Fixed Finite Field

Complexité bilinéaire de la multiplication dans des petits corps finis

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