A babystep-giantstep method for faster deterministic integer   factorization
  In 1977, Strassen presented a deterministic and rigorous algorithm for solving the problem of computing the prime factorization of natural numbers $N$. His method is based on fast polynomial arithmetic techniques and runs in time $\widetilde{O}(N^{1/4})$, which has been state of the art for the last forty years. In this paper, we will combine Strassen's approach with a babystep-giantstep method to improve the currently best known bound by a superpolynomial factor. The runtime complexity of our algorithm is of the form \[ \widetilde{O}\left(N^{1/4}\exp(-C\log N/\log\log N)\right). \] 
Markus Hittmeir
math.NT