## Abstract

Given a computational model with registers of *unlimited size* that is equipped with the set $\{+,-,\times ,\xf7,\&\}=:\mathrm{\U0001d5ae\U0001d5af}$
of unit cost operations, and given a safe prime number *q*, we present the first explicit algorithm that computes discrete
logarithms in ${\mathbb{Z}}_{q}^{*}$ to a base *g* using only $\mathcal{\mathcal{O}}({(\mathrm{log}q)}^{2})$ operations from $\mathrm{\U0001d5ae\U0001d5af}$. For a random *n*-bit prime number *q*, the algorithm is successful as long as the subgroup of ${\mathbb{Z}}_{q}^{*}$
generated by *g* and the subgroup generated by the element $p={2}^{\lfloor {\mathrm{log}}_{2}(q)\rfloor}$ share a subgroup of size at least
${2}^{(1-\mathcal{\mathcal{O}}(\mathrm{log}n/n))n}$.

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