The aim of the discrete logarithm problem with auxiliary inputs is to solve for
, given the elements
of a cyclic group
, of prime order
. The best-known algorithm, proposed by Cheon in 2006, solves for
in the case where
$d\mid (p\pm 1)$
, with a running time of
group exponentiations (
depending on the sign). There have been several attempts to generalize this algorithm to the case of
. However, it has been shown by Kim, Cheon and Lee that a better complexity cannot be achieved than that of the usual square root algorithms.
We propose a new algorithm for solving the DLPwAI. We show that this algorithm has a running time of
group exponentiations, where
is the number of absolutely irreducible factors of
. We note that this number is always smaller than
In addition, we present an analysis of a non-uniform birthday problem.