The Generic Hardness of Subset Membership Problems under the Factoring Assumption

We analyze a large class of subset membership problems related to integer factorization. We show that there is no algorithm solving these problems efficiently without exploiting properties of the given representation of ring elements, unless factoring integers is easy. Our results imply that problems with high relevance for a large number of cryptographic applications, such as the quadratic residuosity and the subgroup decision problems, are generically equivalent to factoring.