One of the fundamental hardness assumptions underlying isogeny-based cryptography is the problem of finding a non-trivial endomorphism of a given supersingular elliptic curve. We show that this problem is related to the problem of finding a good splitting of a principally polarized superspecial abelian surface. We provide formal security reductions, as well as a proof-of-concept implementation of an algorithm to compute endomorphisms of elliptic curves by solving the splitting problem.