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11
Computing tropical varieties
 Journal of Symbolic Computation
, 2007
"... Abstract. The tropical variety of a ddimensional prime ideal in a polynomial ring with complex coefficients is a pure ddimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementatio ..."
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Cited by 38 (11 self)
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Abstract. The tropical variety of a ddimensional prime ideal in a polynomial ring with complex coefficients is a pure ddimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms. 1.
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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Cited by 18 (1 self)
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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.
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
 Journal of Number Theory
"... Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the se ..."
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Cited by 15 (7 self)
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Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log DE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed. 1.
MODULAR POLYNOMIALS VIA ISOGENY VOLCANOES
, 2010
"... We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese R ..."
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Cited by 9 (2 self)
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We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of lisogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(l3 (log l) 3 log log l), and compute Φl mod m using O(l2 (log l) 2 + l2 log m) space. We have used the new algorithm to compute Φl with l over 5000, and Φl mod m with l over 20000. We also consider several modular functions g for which Φ g l is smaller than Φl, allowing us to handle l over 60000.
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
Computing Gröbner fans
, 2007
"... This paper presents algorithms for computing the Gröbner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are based on a uniform definition of the Gröbner fan that applies to both homogeneous and nonhomogeneous idea ..."
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Cited by 2 (1 self)
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This paper presents algorithms for computing the Gröbner fan of an arbitrary polynomial ideal. The computation involves enumeration of all reduced Gröbner bases of the ideal. Our algorithms are based on a uniform definition of the Gröbner fan that applies to both homogeneous and nonhomogeneous ideals and a proof that this object is a polyhedral complex. We show that the cells of a Gröbner fan can easily be oriented acyclically and with a unique sink, allowing their enumeration by the memoryless reverse search procedure. The significance of this follows from the fact that Gröbner fans are not always normal fans of polyhedra, in which case reverse search applies automatically. Computational results using our implementation of these algorithms in the software package Gfan are included.
Number Theory Meets Cache Locality – Efficient Implementation of a Small Prime FFT for the GNU Multiple Precision Arithmetic Library
"... When multiplying really large integer operands, the GNU Multiple Precision Arithmetic Library uses a method based on the Fast Fourier Transform. To make an algorithm execute quickly on a modern computer, data has to be available in the cache memory. If that is not the case, a large portion of the ex ..."
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Cited by 1 (0 self)
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When multiplying really large integer operands, the GNU Multiple Precision Arithmetic Library uses a method based on the Fast Fourier Transform. To make an algorithm execute quickly on a modern computer, data has to be available in the cache memory. If that is not the case, a large portion of the execution time will be spent accessing the main memory. It might pay off to perform much extra work to achieve good cache locality. In extreme cases, 500 primitive operations may be performed in the time of a single memory access. This report describes the implementation of a cache friendly variant of the Fast Fourier Transform and its application to integer multiplication. The variant uses arithmetic modulo primes near machine wordsize. The multiplication method is shown to be competitive with its counterpart in version 4.1.4 of the GNU Multiple Precision Arithmetic Library for interesting platforms. Talteori möter cachelokalitet Effektiv implementation av småprimtalsFFT
A GENERIC APPROACH TO SEARCHING FOR JACOBIANS
 MATHEMATICS OF COMPUTATION
, 2009
"... We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution ..."
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We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N 1/12) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over lowdegree extension fields, where in genus 2 we find Jacobians over F p 2 and trace zero varieties over F p 3 with nearprime orders up to 372 bits in size. For p =2 61 − 1, the average time to find a group with 244bit nearprime order is under an hour on a PC.
A Note on the Space Complexity of Fast DFinite Function Evaluation
"... Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear di ..."
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Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2 −p in timeO(p(lgp) 3+o(1) ) and spaceO(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p lgp) bits of memory. 1.