Utilizing two subfields to accelerate individual logarithm computation in extended tower number field sieve

The hardness of discrete logarithm problem (DLP) over finite fields forms the security foundation of many cryptographic schemes. When the characteristic is not small, the state-of-the-art algorithms for solving the DLP are the number field sieve (NFS) and its variants. NFS first computes the logarithms of the factor base, which consists of elements of small norms. Then, for a target element, its logarithm is calculated by establishing a relation with the factor base. Although computing the factor-base elements is the most time-consuming part of NFS, it can be performed only once and treated as pre-computation for a fixed finite field when multiple logarithms need to be computed. In this paper, we present a method for accelerating individual logarithm computation by utilizing two subfields. We focus on the case where the extension degree of the finite field is a multiple of 6 within the extended tower number field sieve framework. Our method allows for the construction of an element with a lower degree, while maintaining the same coefficient bound compared to Guillevic’s method, which uses only one subfield. Consequently, the element derived from our approach enjoys a smaller norm, which will improve the efficiency in individual logarithm computation.