Titlebook: Algorithmic Number Theory; First International Leonard M. Adleman,Ming-Deh Huang Conference proceedings 1994 Springer-Verlag Berlin Heidel

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Lattice sieving and trial division,gorithm. Our experiments indicate that we have achieved a substantial speed-up compared to other implementations that are reported in the literature. The main improvements are a new lattice sieving technique and a trial division method that is based on lattice sieving in a hash table. This also allo

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A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of ls do not appear to be easily adaptable for finding discrete logarithms in the groups associated with elliptic curves and the Jacobians of hyperelliptic curves. This has led to the development of cryptographic systems based on the discrete logarithm problem for such groups [12, 7, 8]. In this paper a

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A new modular interpolation algorithm for factoring multivariate polynomials,ariate factorizations. Whereas other interpolation algorithms for polynomial factorization depend on various extensions of the Hilbert irreducibility theorem, our approach is the first to depend only upon the classical formulation. The key to our technique is the interpolation scheme for multivalued

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Heegner point computations, in the “congruent number” problem. We begin by briefly reviewing the cyclotomic construction of units in real quadratic number fields, which is analogous in many ways to the Heegner-point approach to the arithmetic of elliptic curves, and allows us to introduce some of the key ideas in a more famil

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An analysis of the Gaussian algorithm for lattice reduction,when applied to random inputs, the complexity is asymptotically constant, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and funct