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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
The 2adic CM method for genus 2 curves with application to cryptography
 in ASIACRYPT ‘06, Springer LNCS 4284
, 2006
"... Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method ..."
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Cited by 18 (1 self)
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Abstract. The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field Q(i p 75 + 12 √ 17), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1, j2, j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestre’s algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography. 1
and Tsujii’s algorithm
, 2010
"... Abstract. We present an algorithm based on the birthday paradox, which is a lowmemory parallel counterpart to the algorithm of Matsuo, Chao and Tsujii. This algorithm computes the group order of the Jacobian of a genus 2 curve over a finite field for which the characteristic polynomial of the Frobe ..."
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Abstract. We present an algorithm based on the birthday paradox, which is a lowmemory parallel counterpart to the algorithm of Matsuo, Chao and Tsujii. This algorithm computes the group order of the Jacobian of a genus 2 curve over a finite field for which the characteristic polynomial of the Frobenius endomorphism is known modulo some integer. The main tool is a 2dimensional pseudorandom walk that allows to heuristically choose random elements in a 2dimensional space. We analyze the expected running time based on heuristics that we validate by computer experiments. Compared with the original algorithm by Matsuo, Chao and Tsujii, we lose a factor of about 3 in running time, but the memory requirement drops from several GB to almost nothing. Our method is general and can be applied in other contexts to transform a babystep giantstep approach into a low memory algorithm. 1