On Friday I gave a geometry club seminar, speaking about some of the computational aspects of discrete-logarithm cryptography in general and as implemented for elliptic curves. My notes supplement rather than completely describe the talk, being heavier on the formalities and lighter on the narrative.

The topics covered are: Diffie-Hellman and one-way functions for key exchange; the generic Discrete Logarithm Problem and BSGS algorithm; scalar multiplication- addition chains, fast exponentiation, m-ary methods and windowing; group law implementations, Side-channel attacks and the Edwards form.

I also refered to a recent real-world example of a side-channel attack; see this story from The Register for details.