# Supersingular isogeny Diffie-Hellman

Supersingular isogeny graphs have been used in the Charles–Goren–Lauter cryptographic hash function and the supersingular isogeny Diffie–Hellman (SIDH) protocole of De\,Feo and Jao. A recently proposed alternative to SIDH is the commutative supersingular isogeny Diffie–Hellman (CSIDH) protocole, in which the isogeny graph is first restricted to $\FF_p$-rational curves $E$ and $\FF_p$-rational isogenies then oriented by the quadratic subring $\ZZ[\pi] \subset \End(E)$ generated by the Frobenius endomorphism $\pi: E \rightarrow E$.

We introduce a general notion of orienting supersingular elliptic curves and their isogenies, and use this as the basis to construct a general oriented supersingular isogeny Diffie-Hellman (OSIDH) protocole.

By imposing the data of an orientation by an imaginary quadratic ring $\OO$, we obtain an augmented category of supersingular curves on which the class group $\Cl(\OO)$ acts faithfully and transitively. This idea is already implicit in the CSIDH protocol, in which supersingular curves over $\FF_p$ are oriented by the Frobenius subring $\ZZ[\pi] \simeq \ZZ[\sqrt{-p}]$. In contrast we consider an elliptic curve $E_0$ oriented by a CM order $\OO_K$ of class number one. To obtain a nontrivial group action, we consider $\ell$-isogeny chains, on which the class group of an order $\OO$ of large index $\ell^n$ in $\OO_K$ acts, a structure we call a whirlpool. The map from $\ell$-isogeny chains to its terminus forgets the structure of the orientation, and the original base curve $E_0$, giving rise to a generic supersingular elliptic curve. Within this general framework we define a new oriented supersingular isogeny Diffie-Hellman (OSIDH) protocol, which has fewer restrictions on the proportion of supersingular curves covered and on the torsion group structure of the underlying curves. Moreover, the group action can be carried out effectively solely on the sequences of moduli points (such as $j$-invariants) on a modular curve, thereby avoiding expensive isogeny computations, and is further amenable to speedup by precomputations of endomorphisms on the base curve $E_0$.