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Computing discrete logarithms in highgenus hyperelliptic Jacobians in provably subexponential time
 Mathematics of Computation
, 1999
"... Abstract. We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of highgenus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given ..."
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Cited by 37 (7 self)
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Abstract. We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of highgenus hyperelliptic curves over finite fields. Its expected running time for instances with genus g and underlying finite field Fq satisfying g ≥ ϑ log q for a positive constant ϑ is given by
Fast Jacobian Group Arithmetic on C_ab Curves
"... The goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jacobian group arithmetic in O(g 2 ) operations in the b ..."
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Cited by 9 (0 self)
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The goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jacobian group arithmetic in O(g 2 ) operations in the base field, where g is the genus of a curve. The main problem in this paper is whether there exists a method to perform the arithmetic in more general curves. Galbraith, Paulus, and Smart proposed an algorithm to complete the arithmetic in O(g 2 ) operations in the base field for the socalled superelliptic curves. We generalize the algorithm to the class of C ab curves, which includes superelliptic curves as a special case. Furthermore, in the case of C ab curves, we show that the proposed algorithm is not just general but more efficient than the previous algorithm as a parameter a in C ab curves grows large. Keywords: discrete logarithm problem, algebraic curve cryptography, Jacobian gr...