Applications of elliptic curves in cryptography and computational number theory

Then two people, traditionally called Alice and Bob, can communicate in privacy even if their messages are being intercepted (e.g., over the internet!). To send a message m ∈ MP , Alice sends m′ = E(m) ∈ MC, and Bob decodes m′ by applying D to it. This assumes that Alice and Bob have decided on D and E, which they usually keep private between themselves. The cryptographer (Cathy) only sees the message m′, but in principle does not know how to apply D. A reasonable model is to assume that Alice and Bob have sent many messages m, encoded and decoded using E, D, and that Cathy has seen all the corresponding coded forms m′. In addition, Cathy may have found out by other