Suppose you have an efficient algorithm for calculating discrete logs in some base "a" would it be possible to use this algorithm to find discrete logs in a different base for which we don't know an efficient algorithm ? And if yes, how ?


If $a$ generates the same sized subgroup as the different base $g$, then yes, it's easy. Note: in $Z_p^*$, there's only one subgroup of a given size, so if $a$ and $g$ generate the same sized subgoup, they must generate the same group, and hence $a^y = g$ for some $y$).

To solve the problem $g^x = h$, we first solve $a^y = g$ and $a^z = h$ (which are both easy, they're problems with our preferred base. We then know $a^{xy} = a^z$, or $xy = z \pmod{p-1}$ (or $\pmod{q}$ if you know the size of the subgroup $q$); computing $x = y^{-1}z \pmod {p-1}$ is easy.

If $g$ generates a supergroup of $a$ that's only slightly larger, then this can be solved as well; if $g$ generates a significantly larger subgroup (e.g. the subgroup that $a$ generates is tiny), then this isn't of much help.

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