The aim of the discrete logarithm problem with auxiliary inputs is to solve for
${\it\alpha}$
, given the elements
$g,g^{{\it\alpha}},\ldots ,g^{{\it\alpha}^{d}}$
of a cyclic group
$G=\langle g\rangle$
, of prime order
$p$
. The best-known algorithm, proposed by Cheon in 2006, solves for
${\it\alpha}$
in the case where
$d\mid (p\pm 1)$
, with a running time of
$O(\sqrt{p/d}+d^{i})$
group exponentiations (
$i=1$
or
$1/2$
depending on the sign). There have been several attempts to generalize this algorithm to the case of
${\rm\Phi}_{k}(p)$
where
$k\geqslant 3$
. 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
$\widetilde{O}(\sqrt{p/{\it\tau}_{f}}+d)$
group exponentiations, where
${\it\tau}_{f}$
is the number of absolutely irreducible factors of
$f(x)-f(y)$
. We note that this number is always smaller than
$\widetilde{O}(p^{1/2})$
.
In addition, we present an analysis of a non-uniform birthday problem.