An L(1/3) algorithm for discrete logarithm computation and principality   testing in certain number fields
  We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge, Gaudry and Thom\'{e} in the context of algebraic curves over finite fields. 
Jean-Fran\c{c}ois Biasse
math.NT cs.CC