Post-Quantunm Scheme using Montgomery+ curves

Isogenies between elliptic curves are popular objects in post-quantum cryptography. The Commutative Supersingular Isogeny Diffie-Hellman key exchange (CSIDH) used supersingular elliptic curves of Montgomery form over F
p
with p ≡ 3 (mod 8) whose endomorphism rings are $Z[\sqrt { - p} ]$, while CSURF considered those of Montgomery
^-
form with p ≡ 7 (mod 8) and endomorphism rings $\mathbb{Z}[(1 + \sqrt { - p} )/2]$. In both protocol, the coefficients can uniquely represent the ${\mathbb{F}_p}$ - isomorphism class because they have ratio (1:1) between the coefficients and the ${\mathbb{F}_p}$ -isomorphism classes. Castryck and Decru showed that the ratio is (2:1) when Montgomery supersingular curves with endomorphism ring $\mathbb{Z}[(1 + \sqrt { - p} )/2]$ and p ≡ 7 (mod 8) were considered. So the major obstacle is that the coefficients are not unique for each ${\mathbb{F}_p}$-isomorphism class. To get a new hard homogeneous space, we divide the coefficients into two orbits to deal with (2:1), which can get the unique representatives of ${\mathbb{F}_p}$-isomorphism classes in one orbit, leading to our resulting protocol. Compared with CSURF, the protocol has the same security level and the startup 3 parameter $ \pm \frac{3}{{\sqrt 2 }}$ in ${\mathbb{F}_p}$ need to be precomputed.