In this paper, we prove that the supersingular isogeny problem (Isogeny), endomorphism ring problem (EndRing) and maximal order problem (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 (HomModule). For cryptographic applications, one requires computational problems to be hard on average for random instances. It is well-known that if 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-mentionned classical problems is hard in the worst case, then all of them are hard on average. In particular, if there exist hard instances of Isogeny, then all of Isogeny, EndRing, MaxOrder and HomModule are hard on average.

翻译：本文中，我们证明在概率多项式时间归约下，超奇异同源问题（Isogeny）、自同态环问题（EndRing）和极大序问题（MaxOrder）是无条件等价的。同源密码学建立在这些问题假定困难性的基础上，而它们之间的相互关联正是SQIsign数字签名方案等密码系统设计与分析的核心。先前已知的归约依赖于广义黎曼猜想等未经证明的假设。本工作提出了无条件归约，并将这一等价关系网络扩展到计算两条超奇异椭圆曲线间所有同源构成的格问题（HomModule）。在密码学应用中，要求计算问题对随机实例具有平均情况下的困难性。众所周知，若Isogeny问题在最坏情况下困难，则其对随机实例也困难。我们通过证明若上述任一经典问题在最坏情况下困难，则所有问题在平均情况下均困难，从而扩展了这一结果。特别地，若存在Isogeny的困难实例，则Isogeny、EndRing、MaxOrder和HomModule在平均情况下都是困难的。