Edgar Costa

Faster deterministic integer factorization

The best known unconditional deterministic complexity bound for computing the prime factorization of an integer N is O(M_int(N^(1/4) log N)), where M_int(k) denotes the cost of multiplying k-bit integers. This result is due to Bostan--Gaudry--Schost, following the Pollard--Strassen approach. We show that this bound can be improved by a factor of (log log N)^(1/2).

A search for Wilson primes

A Wilson prime is a prime p such that (p-1)! = -1 mod p^2. We report on a search for Wilson primes up to 2 * 10^13, and describe several new algorithms that were used in the search. In particular we give the first known algorithm that computes (p-1)! mod p^2 in average polynomial time per prime.

Variation of Néron–Severi Ranks of Reductions of K3 Surfaces

We study the behavior of geometric Picard ranks of K3 surfaces over under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes p < 216 in several representative examples and investigate the resulting statistics.

Rigorous computation of the endomorphism ring of a Jacobian

We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.