Subexponential algorithms for DLP in $\mathbb{Z}_s \times \mathbb{Z}_t$ Consider the accepted answer to the question: Why are elliptic curves better than cyclic groups?

It seems to suggest there are subexponential algorithms (i.e., algorithms with running time
$$
L_n[\alpha,c]=e^{(c+o(1))(\ln n)^\alpha(\ln\ln n)^{1-\alpha}},
$$
where $\ln n$ is the input size, $c &gt; 0$, and $0 &lt; \alpha &lt; 1$) for solving DLP in $\mathbb{Z}_s \times \mathbb{Z}_t$ in general.

Are there such algorithms?

If not, perhaps someone could explain the following sentence that appears in the answer:

"And working only with the representation $\mathbb{Z}_s \times \mathbb{Z}_t$ will weaken the discrete logarithm so useful for cryptographic purposes."
First of all, DLP is only defined for cyclic groups. The $\mathbb{Z}_s^+ \times \mathbb{Z}_t^+$ you are talking about is not cyclic, since (say) $s$ divides $t$. But you can define a "generalized" DLP by considering the cyclic subgroup generated by an element $g$.

Next, DLP in any subgroup of $\mathbb{Z}_t^+$ is trivial. Figuring out why is a nice exercise. It follows that any "generalized" DLP in $\mathbb{Z}_s^+ \times \mathbb{Z}_t^+$ will be trivial. So there's no need to consider subexponential algorithms.

(If we were talking about $\mathbb{Z}_t^*$, not $\mathbb{Z}_t^+$, subexponential algorithms would be interesting and important.)

An elliptic curve has a group structure that is isomorphic to $\mathbb{Z}_s^+ \times \mathbb{Z}_t^+$ (for some $s$ and $t$), which means that abstractly, the two mathematical objects have the same structure. But they have different representations.

Representation is extremely important for DLP. As far as anyone knows, "generalized" DLP is hard when the representation is a suitably chosen elliptic curve, but it is extremely easy when the representation is $\mathbb{Z}_s^+ \times \mathbb{Z}_t^+$. Abstractly, they're the same, but in practice, they are very different with respect to what we can compute.  For instance, recovering the isomorphism that connects the elliptic curve to $\mathbb{Z}_s^+ \times \mathbb{Z}_t^+$ (that maps from one representation to the other) is computationally hard: it is as hard as the discrete log problem on that elliptic curve.

PS. The above contains many mathematical subtleties, so please consider this as an invitation to study mathematics, and not as an answer to your questions.