The digital world is kept secure by cryptography. The idea behind most modern cryptographic systems is that if I have a public number (some sort of seed) and a secret number (usually called a private key) I can perform some operation with them to generate a third number (which usually called a public key). The public key is easy to generate by combining a seed and a private key, but hard to generate any other way; this allows us to verify the authenticity of a private key very easily. One such operation involves finding collinear points on an elliptic curve, giving rise to elliptic curve cryptography. This talk introduced elliptic curves over finite fields, explained how we can use their geometry to define an appropriate operation, and touched on why this operation is appropriate for cryptography. Slides for the talk can be found here.
(Skyler Marks, Notes; written April 19, 2025)
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