In this paper, we prove that the supersingular isogeny problem (\({{ \textsc {Isogeny}}}\)), endomorphism ring problem (\({{ \textsc {EndRing}}}\)) and maximal order problem (\({{ \textsc {MaxOrder}}}\)) are equivalent under probabilistic polynomial time reductions, unconditionally.
Isogeny-based cryptography is founded on the presumed hardness of these problems, and their interconnection is at the heart of the design and analysis of cryptosystems like the SQIsign digital signature scheme. Previously known reductions relied on unproven assumptions such as the generalized Riemann hypothesis. In this work, we present unconditional reductions, and extend this network of equivalences to the problem of computing the lattice of all isogenies between two supersingular elliptic curves (\({{ \textsc {HomModule}}}\)).
For cryptographic applications, one requires computational problems to be hard on average for random instances. It is well-known that if \({{ \textsc {Isogeny}}}\) is hard (in the worst case), then it is hard for random instances. We extend this result by proving that if any of the above-mentioned classical problems is hard in the worst case, then all of them are hard on average. In particular, if there exist hard instances of \({{ \textsc {Isogeny}}}\), then all of \({{ \textsc {Isogeny}}}\), \({{ \textsc {EndRing}}}\), \({{ \textsc {MaxOrder}}}\) and \({{ \textsc {HomModule}}}\) are hard on average.
