Paper 2019/1404

CSIDH on the surface

Wouter Castryck and Thomas Decru

Abstract

For primes \(p \equiv 3 \bmod 4\), we show that setting up CSIDH on the surface, i.e., using supersingular elliptic curves with endomorphism ring \(Z[(1 + \sqrt{-p})/2]\), amounts to just a few sign switches in the underlying arithmetic. If \(p \equiv 7 \bmod 8\) then the availability of very efficient horizontal 2-isogenies allows for a noticeable speed-up, e.g., our resulting CSURF-512 protocol runs about 5.68% faster than CSIDH-512. This improvement is completely orthogonal to all previous speed-ups, constant-time measures and construction of cryptographic primitives that have appeared in the literature so far. At the same time, moving to the surface gets rid of the redundant factor \(Z_3\) of the acting ideal-class group, which is present in the case of CSIDH and offers no extra security.