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46
Subquadratictime factoring of polynomials over finite fields
 Math. Comp
, 1998
"... Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast ..."
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Cited by 68 (11 self)
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Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field Fq with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in Fq. The new “baby step/giant step ” techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratictime methods for manipulating normal bases of finite fields. 1.
Set Reconciliation with Nearly Optimal Communication Complexity
 in International Symposium on Information Theory
, 2000
"... We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the ..."
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Cited by 59 (15 self)
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We consider the problem of efficiently reconciling two similar sets held by different hosts while minimizing the communication complexity. This type of problem arises naturally from gossip protocols used for the distribution of information. We describe an approach to set reconciliation based on the encoding of sets as polynomials. The resulting protocols exhibit tractable computational complexity and nearly optimal communication complexity. Also, these protocols can be adapted to work over a broadcast channel, allowing many clients to reconcile with one host based on a single broadcast, even if each client is missing a different subset.
Construction of secure random curves of genus 2 over prime fields
 Advances in Cryptology – EUROCRYPT 2004, volume 3027 of Lecture Notes in Comput. Sci
, 2004
"... Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken ..."
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Cited by 37 (12 self)
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Abstract. For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoof’s algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantor’s division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC. 1
Efficient Computation of Minimal Polynomials in Algebraic Extensions of Finite Fields
 In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC
, 1999
"... New algorithms are presented for computing the minimal polynomial over a finite field K of a given element in an algebraic extension of K of the form K[ff] or K[ff][fi]. The new algorithms are explicit and can be implemented rather easily in terms of polynomial multiplication, and are much more effi ..."
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Cited by 31 (0 self)
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New algorithms are presented for computing the minimal polynomial over a finite field K of a given element in an algebraic extension of K of the form K[ff] or K[ff][fi]. The new algorithms are explicit and can be implemented rather easily in terms of polynomial multiplication, and are much more efficient than other algorithms in the literature. 1 Introduction In this paper, we consider the problem of computing the minimal polynomial over a finite field K of a given element oe in an algebraic extension of K of the form K[ff] or K[ff][fi]. The minimal polynomial of oe is defined to be the unique monic polynomial OE oe=K 2 K[x] of least degree such that OE oe=K (oe) = 0. In the first case, we assume that the ring K[ff] is given as K[x]=(f) where f 2 K[x] is a monic polynomial of degree n, and that elements in K[ff] are represented in the natural way as elements of K[x] !n (the set of polynomials of degree less than n). Similarly, in the second case, we assume that K[ff] is given as a...
Fast arithmetic for triangular sets: from theory to practice
 ISSAC'07
, 2007
"... We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, ..."
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Cited by 30 (25 self)
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We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasilinear complexity. The main outcome we have in mind is the acceleration of higherlevel algorithms, by interfacing our lowlevel implementation with languages such as AXIOM or Maple. We show the potential for huge speedups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.
Fast Computation of Special Resultants
, 2006
"... We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series. ..."
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Cited by 20 (8 self)
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We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series.
ARCHITECTUREAWARE CLASSICAL TAYLOR SHIFT BY 1
, 2005
"... We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input polynomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster th ..."
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Cited by 17 (2 self)
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We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input polynomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7. Our Taylor shift algorithm requires more word additions than straightforward implementations but it reduces the number of cycles per word addition by reducing memory tra c and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers. While our algorithm is written in a highlevel language, it depends on several parameters that can be tuned to the underlying architecture.