# Why are computations (isogeny) in SIDH done in an extended prime field?

While reading the SIDH key exchange protocol, I noticed that all the isogeny computations and curves are defined over the extended prime field $$\mathbb{F}_{p^2}$$. Does it make the problem computationally hard for the attacker or what is the reasoning?

I am an inventor of SIDH. The computations take place in $$\operatorname{GF}(p^2)$$ just because all supersingular elliptic curves are defined over $$\operatorname{GF}(p^2)$$, up to isomorphism. It's just the way the mathematics works out. For general mathematical background, see any standard reference, such as Silverman "The Arithmetic of Elliptic Curves."