Lattice Enumeration and Automorphisms for Tower NFS: A 521-Bit Discrete Logarithm Computation

Abstract

The tower variant of the number field sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimension greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows. In this article, we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field \({{{\mathbb {F}}}}_{p^6}\). The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS.