Using number fields to compute logarithms in finite fields

Abstract

We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is L q [ 1 / 3 ; ( 64 / 9 ) 1 / 3 + o ( 1 ) ] , L_{q}[1/3; (64/9)^{1/3}+o(1)], where q q is the cardinality of the field, L q [ s ; c ] = exp ( c ( log ⁡ q ) s ( log ⁡ log ⁡ q ) 1 − s ) , L_{q}[s;c]={\exp }(c(\log q)^{s}(\log \log q)^{1-s}), and the o ( 1 ) o(1) is for q → ∞ q\to \infty . The number field sieve factoring algorithm is conjectured to factor a number the size of q q in the same amount of time.