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4

There are several obvious ways to optimize the search. The easiest approach would be to take advantage of the identity $g^e \cdot g = g^{e+1}$. That is, if we have already computed $g^e$, and verified that it is not the value we're looking for, then to step to $g^{e+1}$, we don't compute $g^{e+1}$ from scratch; instead, we take the $g^e$ value we have, and ...


0

Here's a cyclic group of any order $q \ge 1$: Identity: $0$. Generator: $1$. Group operation: $a \cdot b$ is (a + b) % q.



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