Computing individual discrete logarithms in non-prime finite fields

This talk is about computing discrete logarithms in non-prime finite fields. These fields arise in pairing-based cryptography. In this setting, the pairing-friendly curve is defined over GF(q) and the pairing takes its values in an extension GF(q^k), where k is the embedding degree.<br/> Fr example, GF(p^2) is the embedding field of supersingular elliptic curves in large characteristic; GF(p^3), GF(p^4), GF(p^6) are the embedding fields of MNT curves, and GF(p^12) is the embedding field of the well-known Barreto-Naehrig curves. In small characteristic, GF(2^(4*n)), GF(3^(6*m)) are considered. To compute discrete logarithms in these fields, one uses the Number Field Sieve algorithm (NFS) in large characteristic (e.g. GF(p^2)), the NFS-High-Degree variant (NFS-HD) in medium characteristic (e.g. GF(p^12)) and the Quasi Polynomial-time Algorithm (QPA) in small characteristic when applicable. These algorithms are made of four steps: polynomial selection, relation collection, linear algebra modulo the prime order of the target group and finally, individual logarithm computation.<br/> All these finite fields are extensions, hence have subfields. We use their structure to speed-up the individual discrete logarithm computation. We obtain an important speed-up in practice and the best case is when the embedding degree k is even. We will illustrate the improvements with the practical case of GF(p^4) with p^4 of 400 bits (120 decimal digits).