**A NEW APPROACH TO THE DISCRETE LOGARITHM PROBLEM WITH AUXILIARY INPUTS**

*Taechan Kim and Jung Hee Cheon*

**Abstract: **The discrete logarithm problem with auxiliary inputs is to
solve~$\alpha$ for given elements $g, g^\alpha, \ldots, g^{\alpha^d}$
of a cyclic group $G=\langle g \rangle$ of prime order~$p$.
The best-known algorithm, proposed by Cheon in 2006,
solves $\alpha$ in the case of $d | (p\pm 1)$
with running time of $O\left( \sqrt{p/d} + d^i \right)$
group exponentiations~($i=1$ or $1/2$ depending on the sign).
There have been several attempts to generalize this algorithm
in the case of $\Phi_k(p)$ for $k \ge 3$,
but it has been shown, by Kim, Cheon and Lee, that
they cannot have better complexity than the usual square root algorithms.

We propose a new algorithm to solve the DLPwAI. The complexity of the algorithm is determined by a chosen polynomial $f \in \F_p[x]$ of degree $d$. We show that the proposed algorithm has a running time of $\widetilde O\left( \sqrt{p / \tau_f} +d \right)$ group exponentiations, where~$\tau_f$ is the number of absolutely irreducible factors of $f(x)-f(y)$. We note that it is always smaller than $\widetilde O(p^{1/2})$.

To obtain a better complexity of the algorithm, we investigate an upper bound of $\tau_f$ and try to find polynomials that achieve the upper bound. We can find such polynomials in the case of $d|(p\pm 1)$. In this case, the algorithm has a running time of $\widetilde O\left(\sqrt{p/d} +d \right)$ group operations which corresponds with the lower bound in the generic group model. On the contrary, we show that no polynomial exists that achieves the upper bound in the case of $d \vert\Phi_3(p)=p^2+p+1$.

As an independent interest, we present an analysis of a non-uniform birthday problem. Precisely, we show that a collision occurs with a high probability after $O\big( \frac{1}{ \sqrt{\sum_{k} {w_k}^2} } \big)$ samplings of balls, where the probability $w_k$ of assigning balls to the bin $k$ is arbitrary.

**Category / Keywords: **discrete logarithm problem, Cheon's algorithm, birthday problem

**Date: **received 29 Oct 2012, last revised 3 Jun 2014

**Contact author: **yoshiki1 at snu ac kr

**Available format(s): **PDF | BibTeX Citation

**Version: **20140604:003610 (All versions of this report)

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