Cunningham project

Abstract: We searched the Cunningham tables for new algebraic factorizations similar to those discovered by Aurifeuille. A naive search would have been too slow. We accelerated it enough to make it feasible. Many interesting results were found. Dedicated to the memory of John Selfridge, who loved the integers.

Abstract: Since Crypto 83 we have had considerably more experience in factoring large integers. Implementation of various modifications to the quadratic sieve algorith have enabled the factorization of hard 70- digit numbers in times comparable to 50 digits one year ago. These modifications include: 1) Subsequences with large divisors (Special q's). 2) Multipliers to improve quadratic properties. 3) Increased size of prime base using segmented Gaussian Elimination. 4) Optimization of the code with respect to Cray hardware.Using this code in its various stages of development the 10 most wanted numbers from the Cunningham Project have been factored. Details will be published elsewhere.

Abstract: Constructing elliptic curves with a known number of points over a prime field by A. Agashe, K. Lauter, and R. Venkatesan On a generalization of the radix representation-A survey by S. Akiyama, T. Borbely, H. Brunotte, A. Petho, and J. M. Thuswaldner Multiplicative structure of values of the Euler function by W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski Congruences and exponential sums with the Euler function by W. D. Banks and I. E. Shparlinski On the existence of cyclic difference sets with small parameters by L. D. Baumert and D. M. Gordon Doubly focused enumeration of locally square polynomial values by D. J. Bernstein Cubic reciprocity and explicit primality tests for $h\cdot 3^k\pm1$ by W. Bosma Algorithms for finding almost irreducible and almost primitive trinomials by R. P. Brent and P. Zimmermann Commentary on Lucas' test by J. Brillhart Number field cryptography by J. Buchmann, T. Takagi, and U. Vollmer Reconfigurable computing machines and their applications in computational number theory by D. A. Buell, S. Devarkal, and H. A. Wake Heuristics for class numbers of prime-power real cyclotomic fields by J. Buhler, C. Pomerance, and L. Robertson Counting $A_4$ and $S_4$ number fields with given resolvent cubic by H. Cohen On a conjecture of Feit and Thompson by K. Dilcher and J. Knauer Amicable pairs, a survey by M. Garcia, J. M. Pedersen, and H. te Riele On the research contributions of Hugh C. Williams by A. Granville Hilbert modular fourfolds of arithmetic genus one by H. G. Grundman and L. E. Lippincott Values of ternary recurrence sequences and torsion on certain curves arising from the work of Hugh Williams by E. Herrmann and P. G. Walsh Divisibility of exponents of class groups of pure cubic number fields by S. Hernandez and F. Luca New conjectures and results for small cycles of the discrete logarithm by J. Holden and P. Moree Hyperelliptic curves and cryptography by M. Jacobson, Jr., A. Menezes, and A. Stein Remarks on S. Chowla's hypothesis implying that $L(s,\chi)>0$ for $s>0$ and for real characters $\chi$ by S. Louboutin On the computation of cube roots modulo $p$ by S. Muller Jeepers, creepers, $\ldots$ by R. D. Patterson and A. J. van der Poorten Algebraic tori in cryptography by K. Rubin and A. Silverberg An analysis of the generalized binary GCD algorithm by J. P. Sorenson An elliptic curve trapdoor system (extended abstract) by E. Teske Periodic continued fractions and elliptic curves by A. J. van der Poorten The Cunningham project by S. S. Wagstaff, Jr. Extensions and improvements for the CM method for genus two by A. Weng Advice to referees of submissions to High Primes and Misdemeanours by A. J. van der Poorten and A. Stein.