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**2019/1404**

**Title:**

CSIDH on the surface

**Authors:**
Wouter Castryck, 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.

**ePrint:**
https://eprint.iacr.org/2019/1404

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