Im following these slides from "An Introduction to the Theory of Elliptic Curves" http://www.math.brown.edu/johsilve/Presentations/WyomingEllipticCurve.pdf, but I'm having some difficulty understanding how the ECDLP can be solved in anomalous curves.
On the slides it says: "If #E(Fp) = p, then there is a “p-adic logarithm map” that gives an easily computed homomorphism logp-adic : E(Fp) -> Z/pZ. It is easy to solve the discrete logarithm problem in Z/pZ, so if #E(Fp) = p, then we can solve ECDLP in time O(log p)."
But I'm having trouble understanding some concepts. I understand that there exists an homomorphism between the elliptic curve E(Fp) and the ring of integers Z/pZ. I might be wrong but from what I understand this homomorphism is map phi that satisfies this properties
- phi(O) = 0
- phi(P + Q) = phi(P) + phi(Q)
- phi(kP) = k.phi(P)
What I don't understand is why is it easy to solve the discrete logarithm problem in Z/pZ. Isn't the Diffie Hellman key exchange based on the difficulty of computing discrete logarithms?
But even assuming that it is easy do solve the DLP in Z/pZ, how could I get to the solution to the ECDLP assuming I have the solution to the DLP?
Finally, does anyone know any books or papers where I can read more about this? I tried looking but didnt find anything