Elliptic Curves Discrete Logarithm Problem over a Finite Prime Field Fp and p-adic Approximations

These notes summarize some computations conducted around the Elliptic Curves Discrete Logarithm Problem (ECDLP) over a finite prime field Fp. Instead of directly considering the problem of finding an integer m such that Q = [m]P, where Ē is an elliptic curve defined over Fp of order a prime l, and P, Q ∈ Ē (Fp), we rather consider two such problems. Using the p-adic elliptic logarithm, we mainly show that finding m amounts to finding good approximations, at orders for which we provide an explicit upper bound, of the liftings of the different points considered in some groups of the filtrations of neighborhoods of the point at infinity of the lifting of the elliptic curve over Qp. We do not claim to solve the ECDLP, but rather to provide an interpretation of the problem in a context that may turn out to be useful, if one can find an efficient method to concretely capture the approximations at the appropriate order.