Computing supersingular isogenies on Kummer surfaces

We apply Scholten’s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over \(\mathbb {F}_{p^2}\) and corresponding abelian surfaces over \(\mathbb {F}_{p}\). Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2, 2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography.