Combinatorial Arithmetic on Elliptic Curves

Abstract

We propose a scalar multiplication technique on an elliptic curve, which operates on triples of collinear points. The computation of this operation requires a new approach to operation chains, with similarities to Montgomery ladders for x-only scalar multiplication. We develop a diagrammatic calculus with a combinatorial flavor for the purpose of developing these operation chains. Some interesting algebra arises when studying this diagrammatic calculus, which leads us to improvements to our algorithms. We propose some cryptographic applications for our scalar multiplication technique.