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11
Counting points on varieties over finite fields of small characteristic
 ALGORITHMIC NUMBER THEORY
, 2008
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Computing Zeta Functions of Hyperelliptic Curves over Finite Fields of Characteristic 2
 Advances in CryptologyCRYPTO 2002, LNCS 2442
, 2002
"... We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm ..."
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Cited by 10 (1 self)
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We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field Fq of characteristic 2, thereby extending the algorithm of Kedlaya for small odd characteristic. For a genus g hyperelliptic curve over n , the asymptotic running time of the algorithm is O(g ) and the space complexity is O(g ).
Irreducible Polynomials of Given Forms
, 1999
"... We survey under a unified approach on the number of irreducible polynomials of given forms: x + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little ..."
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Cited by 7 (4 self)
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We survey under a unified approach on the number of irreducible polynomials of given forms: x + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little is know when q is small or fixed. We present computer experiments on several classes of polynomials over F 2 and compare our data with the results that hold for large q. We also mention some related applications and problems of (irreducible) polynomials with special forms.
Algorithmic theory of zeta functions over finite fields
 ALGORITHMIC NUMBER THEORY
, 2008
"... We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field. ..."
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Cited by 7 (3 self)
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We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field.
Modular Counting of Rational Points over Finite Fields
"... Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p ..."
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Cited by 3 (0 self)
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Let Fq be the finite field of q elements, where q = p h. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper, we give a deterministic algorithm which computes the reduction of N(f) modulo p b in O(n(8m) (h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed in [5]. 1
The BlackBox Niederreiter Algorithm and its Implementation over
 the Binary Field”, Math. Comp
"... Abstract. The most timeconsuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the socalled “blackbox ” Niederreiter algorithm, in which these elements are found ..."
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Abstract. The most timeconsuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the socalled “blackbox ” Niederreiter algorithm, in which these elements are found by using a method developed by Wiedemann. The main advantages over an approach based on Gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower. The blackbox Niederreiter algorithm for factoring polynomials over the binary field was implemented in the C programming language, and benchmarks for factoring highdegree polynomials over this field are presented. These benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations. In addition, the Wan algorithm, which was recently introduced, is described, and connections between (implementation aspects of) Wan’s and Niederreiter’s algorithm are given. 1.
Computing zeta functions of nondegenerate hypersurfaces with few monomials
 SUBMITTED EXCLUSIVELY TO THE LONDON MATHEMATICAL SOCIETY
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PRIMARY DECOMPOSITION OF ZERODIMENSIONAL IDEALS OVER FINITE FIELDS
, 2006
"... Abstract. A new algorithm is presented for computing primary decomposition of zerodimensional ideals over finite fields. Like Berlekamp’s algorithm for univariate polynomials, the new method is based on the invariant subspace of the Frobenius map acting on the quotient algebra. The dimension of the ..."
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Abstract. A new algorithm is presented for computing primary decomposition of zerodimensional ideals over finite fields. Like Berlekamp’s algorithm for univariate polynomials, the new method is based on the invariant subspace of the Frobenius map acting on the quotient algebra. The dimension of the invariant subspace equals the number of primary components, and a basis of the invariant subspace yields a complete decomposition. Unlike previous approaches for decomposing multivariate polynomial systems, the new method does not need primality testing nor any generic projection, for it reduces the general decomposition problem directly to root finding of univariate polynomials over the ground field. Also, it is shown how Gröbner basis structure can be used to get partial primary decomposition without any root finding. 1.
COMPUTING ZETA FUNCTIONS OF SPARSE NONDEGENERATE HYPERSURFACES
"... Abstract. Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly wellsuited to work with polynomials in small character ..."
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Abstract. Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly wellsuited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the Lfunction of an exponential sum. Let p be prime and let Fq be a finite field with q = p a elements. Let V be a variety defined over Fq, described by the vanishing of a finite set of polynomial equations with coefficients in Fq. We encode the number of points #V (Fqr) on V over the extensions Fqr of Fq in an exponential generating series, called the zeta function of V:
on Factoring Polynomials Over Finite Fields: A Survey
"... This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. c ○ 2001 Academic Press 1. ..."
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This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. c ○ 2001 Academic Press 1.