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Mar 5, 2021 at 23:10 comment added J. Doe ty for answering this as well. This was the first question and the linked was a 2nd trial but it generally also answering this question. I didn't knew how to proceed with this question. I think I know the difference now (except maybe the relation for some special primes which are somewhere in between those cases but not needed now). Thank you again.
Mar 5, 2021 at 23:10 comment added J. Doe as far as I understood EC can be embedded in a $\mathbb{F}_{p^k}$ DLP. If the embedding degree $k$ is small NFS can be used at this. Depending at $k$ it can be faster. see question1 and question2
Mar 5, 2021 at 22:47 vote accept J. Doe
Mar 5, 2021 at 21:43 history edited fgrieu CC BY-SA 4.0
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Mar 5, 2021 at 21:35 history answered fgrieu CC BY-SA 4.0