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Computing Hilbert class polynomials with the Chinese Remainder Theorem
, 2010
"... We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O ..."
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We present a spaceefficient algorithm to compute the Hilbert class polynomial HD(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(D  1/2+ɛ log P) space and has an expected running time of O(D  1+ɛ). We describe practical optimizations that allow us to handle larger discriminants than other methods, with D  as large as 1013 and h(D) up to 106. We apply these results to construct pairingfriendly elliptic curves of prime order, using the CM method.
CONSTRUCTING ELLIPTIC CURVES OF PRIME ORDER
"... Abstract. We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time e O((log N) 3), and it is so fast that it may profitably be used to ta ..."
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Abstract. We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time e O((log N) 3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained. 1.
GENUS2 CURVES AND JACOBIANS WITH A GIVEN NUMBER OF POINTS
"... Abstract. We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of Frational points. In the case of the Jacobian, we show that any ‘CMconstruction ’ to produce the required genus2 c ..."
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Abstract. We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of Frational points. In the case of the Jacobian, we show that any ‘CMconstruction ’ to produce the required genus2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus2 curve having exactly 102014 + 9703 (prime) points, and two genus2 curves each having exactly 102013 points. In an appendix we provide a complete parametrization, over an arbitrary base field k of characteristic neither 2 nor 3, of the family of genus2 curves over k that have krational degree3 maps to elliptic curves, including formulas for the genus2 curves, the associated elliptic curves, and the degree3 maps. 1.