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Computing zeta functions over finite fields
 Contemporary Math
, 1999
"... Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subj ..."
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Cited by 14 (3 self)
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Abstract. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1.
Finding Points on Curves over Finite Fields
 Comp. Compl
, 1996
"... We solve two computational problems concerning plane algebraic curves over finite fields: generating an (approximately) uniform random point, and finding all points deterministically in amortized polynomial time (over a prime field, for nonexceptional curves). ..."
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Cited by 5 (3 self)
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We solve two computational problems concerning plane algebraic curves over finite fields: generating an (approximately) uniform random point, and finding all points deterministically in amortized polynomial time (over a prime field, for nonexceptional curves).
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