Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic

TL;DR

This paper introduces optimized algorithms for computing isogenies between elliptic curves over small characteristic fields, including the first complete implementation and improvements for better asymptotic complexity.

Contribution

It provides the first full implementation of Couveignes' second algorithm and enhancements that improve its asymptotic efficiency in small characteristic.

Findings

Comparative experiments show differences between existing algorithms.

The new implementation demonstrates practical feasibility.

Improved algorithm has the best asymptotic complexity in isogeny degree.

Abstract

The problem of computing an explicit isogeny between two given elliptic curves over F_q, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovstev & Stolbunov. While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.