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Why is the discrete logarithm problem assumed to be hard?

Someone else asked the same question but the answers only explain that exponentiation is in $O(\log(n))$ while the fastest known algorithms to compute discrete logarithms is in $O(n)$. (I'm glossing over details like the runtime of index calculus here.)

Somewhere else I read: "We assume discrete logarithms to be hard because for over 40 years very smart people failed to find a fast algorithm."

Now, I wonder if there are any better arguments. Can you actually explain why discrete logarithms are hard?

LinusK
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1 Answers1

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Now, I wonder if there are any better arguments.

Ultimately, no, not really.

We don't have any proof that computing discrete logs is hard. For that matter, we don't have any proof that any problem within $NP$ (that is, any problem where, if the answer is "yes", there is a quickly checkable proof of that) is hard.

We do have some partial proofs, for example, that in the "black-box" model, a discrete log on a prime-order group is hard. On the other hand, the assumptions that makes is known to be false for finite-fields, and so that's less useful than one would hope.

poncho
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