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

The problem of computing an explicit isogeny between two given elliptic curves over Fq, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovtsev & 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.