A New Isogeny Representation and Applications to Cryptography

This paper focuses on isogeny representations, defined as ways to evaluate isogenies and verify membership the language of isogenous supersingular curves (the set triples $$D,E_1,E_2$$ with a cyclic degree D between $$E_1$$ $$E_2$$ ). The tasks evaluating verifying are fundamental for isogeny-based cryptography. Our main contribution is design suborder representation, new representation targetted at case (big) prime degree. core our method revelation endomorphisms smooth norm inside well-chosen codomain’s endomorphism ring. appears be opening interesting prospects cryptography under hardness computational problem: SubOrder Ideal Problem (SOIP). As an application, we introduce pSIDH, NIKE based representation. Studying assumption particularly crucial in light recent attacks against In order manipulate efficiently develop several heuristic algorithmic tools solve equations family quaternion orders. These algorithms may independent interest.