On the asymptotic complexity of computing discrete logarithms in the field $\operatorname{\mathit{GF}}(p)$

We analyse the modification of an algorithm for finding discrete logarithms over the field $\mathit{GF}(p)$ ($p$ is a prime number) which has been described by the author previously. It is shown that this modification gives the best estimate at the present time of the complexity of finding discrete logarithms over finite prime fields which coincides with the best known estimate of the complexity of factoring integers obtained by D. Coppersmith.