Efficient arithmetic on elliptic curves in characteristic 2

TL;DR

This paper introduces new normal forms for elliptic curves in characteristic 2, leading to efficient algorithms for point addition, doubling, and scalar multiplication, especially useful for cryptographic applications.

Contribution

It develops elliptic curve normal forms in characteristic 2 and derives simplified, efficient algorithms for fundamental group operations, improving computational performance.

Findings

Algorithms for point doubling and addition with explicit cost analysis

Scalar multiplication algorithm with per-bit complexity

Normal forms analogous to Edwards form for characteristic 2

Abstract

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic $2$ with a $4$-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time $2M + 5S + 2m_c$ and for addition of points in time $7M + 2S$, where $M$ is the cost of multiplication, $S$ the cost of squaring, and $m_c$ the cost of multiplication by a constant. By a study of the Kummer curves $\mathcal{K} = E/\{[\pm1]\}$, we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point $P$ with $4M + 4S + 2m_c + m_t$ per bit where $m_t$ is multiplication by a constant that depends on $P$.