[Kranar]

The decision problem for ECDLP is not known to be NP-complete, in fact it's likely not NP-complete but rather NP-hard and therefore proving that P = NP would not provide any insight into finding a polynomial time solution for it.

[tromp]

Having an efficient algorithm for SAT (or any NP complete problem) immediately gives you an efficient algorithm for ALL problems in NP, including ECDLP.

[Kranar]

It is not known if the decision problem for ECDLP is in NP, in fact it's likely not in NP.

NP-hard does not mean NP, it means a problem that is ATLEAST as hard as NP. NP-hard problems that are in NP are known as NP-Complete.

Finally ECDLP itself is not a decision problem, so even if the decision problem for ECDLP is in NP and P = NP, that does not mean that ECDLP itself has a polynomial time solution.

NP, NP-Complete, NP-Hard etc... only refer to decision problems, not to general purpose computations.

[tromp]

The decision problem for ECDLP

  { (Curve_spec, P, x) : x is a prefix of the discrete log of point P on the specified elliptic curve }

is trivially in NP.

[Kranar]

I think you're mixing up quite a lot of things here. For one, apart from the impreciseness of your statement of the decision problem, it wouldn't be a decision problem for ECDLP but rather for the more general discrete logarithm problem (DLP). ECDLP is assumed to be in a higher complexity class than DLP.

With that said, even if the decision problem for ECDLP is in NP and P = NP, that does not mean that a solution to ECDLP is in P.

[tromp]

> ECDLP is assumed to be in a higher complexity class than DLP.

No, it's not. All instances of DLP, including ECDLP, are just asking for an x such that b^x = a, as Wikpedia will tell you [1]. They only differ in the choice of group, but exponentiation within the group is assumed to be an efficient operation.

> With that said, even if the decision problem for ECDLP is in NP, that does not mean that a solution to ECDLP is in P.

It trivially does. You simply start with an empty prefix x and extend it 1 bit at a time by repeatedly asking if (Curve_spec, P, x0) is in the language. If so you continue with x0, if not you continue with x1.

[1] https://en.wikipedia.org/wiki/Discrete_logarithm

[tgflynn]

A decision problem as used to define the classes NP and P is of the form: does there exist an x such that F(x) = 1, where F is a P-time computable function.

If you can solve that problem in P-time then you can also solve the functional version (find an x for which F(x) = 1) in P-time simply by setting each bit to a value for which the decision problem is satisfiable, one by one.