Dr. Rainer Steinwandt: Quantum circuits for point addition on binary elliptic curves

Abstract:

Subgroups of elliptic curves over binary fields are a popular mathematical platform to implement cryptographic primitives and protocols. Using Shor's algorithm to tackle the discrete logarithm problem in such groups leads to the question of efficiently implementing the underlying group arithmetic on a quantum computer.

Starting out with a discussion of quantum circuits for GF(2^n)-arithmetic, the talk takes a look at different curve representations to implement point addition on binary elliptic curves, as occurring in Shor's algorithm. In addition to the 'generic' addition of a fixed point, the use of complete binary Edwards curves to add arbitrary curve points is explored.

The talk is based on joint work with Brittanney Amento and Martin Roetteler.