The n-Diffie-Hellman problem and multiple-key encryption

The main contributions of this paper are twofold. On the one hand, the twin Diffie-Hellman (twin DH) problem proposed by Cash et al. is extended to the n-Diffie-Hellman (n-DH) problem for an arbitrary integer n, and this new problem is shown to be at least as hard as the ordinary DH problem. Like the twin DH problem, the n-DH problem remains hard even in the presence of a decision oracle that recognizes solution to the problem. On the other hand, observe that the double-size key in the Cash et al. twin DH-based encryption scheme can be replaced by two separated keys each for one entity that results in a 2-party encryption scheme which holds the same security feature as the original scheme but removes the key redundancy. This idea is further extended to an n-party case, which is also known as n-out-of-n encryption. As examples, a variant of ElGamal encryption and a variant of Boneh-Franklin IBE have been presented; both of them have proved to be chosen ciphertext attack secure under the computational DH assumption and the computational bilinear Diffie-Hellman assumption, respectively, in the random oracle model. The two schemes are efficient, due partially to the size of their ciphertext, which is independent to the value n.