Symmetric Subgroup Membership Problems

We define and discuss symmetric subgroup membership problems and their properties, including a relation to the Decision DiffieHellman problem. We modify the Cramer-Shoup framework, so that we can derive a chosen ciphertext secure cryptosystem in the standard model from symmetric subgroup membership problems. We also discuss how chosen ciphertext secure hybrid cryptosystems based on a symmetric subgroup membership can be constructed in the standard model, giving a very efficient cryptosystem whose security relies solely on the symmetric subgroup membership problem.