I'm currently going through the book "Guide to Elliptic Curve Cryptography" by Darrel Hankerson, Scott Vanstone, and Alfred Menezes. In the book, the authors state that
[…] there is no mathematical proof that the ECDLP is intractable. That is, no one has proven that there does not exist an efficient algorithm for solving the ECDLP. Indeed, such a proof would be extremely surprising. For example, the non-existence of a polynomial-time algorithm for the ECDLP would imply that P ≠ NP thus settling one of the fundamental outstanding open questions in computer science.1
1 P is the complexity class of decision (YES/NO) problems with polynomial-time algorithms. NP is the complexity class of decision problems whose YES answers can be verified in polynomial-time if one is presented with an appropriate proof. While it can readily be seen that P ⊆ NP, it is not known whether P = NP.
However, I can't understand why the non-existence of a polynomial-time algorithm for the ECDLP implies that P ≠ NP.
I would like to ask for clarification on this topic and if anyone knows of any resources that could help me understand this better. I have searched online, but couldn't find much information. Any insights or resources would be greatly appreciated.