International Association
for Cryptologic Research

<p>The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields (except for the weak small characteristic case). The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. A precomputation, solely dependent on an approximate finite field size and an extension degree, allows to efficiently compute discrete logarithms in a constant proportion of the finite fields of the given approximate size and extension degree. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination improves the asymptotic complexity. We also notice that combining our approach with the MNFS variant would be an unnecessary complication as all the potential gain of MNFS is subsumed by our Factory variant anyway. Furthermore, we demonstrate how Chebotarev's density theorem allows to compute the density of finite fields that can be solved with a given precomputation. Finally, we provide experimental data in order to assess the practical reach of our approach. </p>

We report on two new records: the factorization of RSA-240, a 795-bit number, and a discrete logarithm computation over a 795-bit prime field. Previous records were the factorization of RSA-768 in 2009 and a 768-bit discrete logarithm computation in 2016. Our two computations at the 795-bit level were done using the same hardware and software, and show that computing a discrete logarithm is not much harder than a factorization of the same size. Moreover, thanks to algorithmic variants and well-chosen parameters, our computations were significantly less expensive than anticipated based on previous records. The last page of this paper also reports on the factorization of RSA-250.