3,263 research outputs found

    Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

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    The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes O~(n3/2logq+nlog2q)\widetilde{O}(n^{3/2}\log q + n \log^2 q) time to factor polynomials of degree nn over the finite field Fq\mathbb{F}_q with qq elements. A significant open problem is if the 3/23/2 exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than 3/23/2 would yield an algorithm for polynomial factorization with exponent better than 3/23/2

    On the complexity of computing with zero-dimensional triangular sets

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    We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results

    Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic

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    The problem of computing an explicit isogeny between two given elliptic curves over F_q, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovstev & Stolbunov. While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.Comment: 21 pages, 6 figures, 1 table. Submitted to J. Number Theor

    Fast algorithms for computing isogenies between elliptic curves

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    We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree \ell (\ell different from the characteristic) in time quasi-linear with respect to \ell. This is based in particular on fast algorithms for power series expansion of the Weierstrass \wp-function and related functions
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