An identity-based non-interactive public key distribution system is presented that is based on a novel trapdoor one-way function allowing a trusted authority to compute the discrete logarithms modulo a publicly known composite number m while this is infeasible for an adversary not knowing the factorization of m. Without interaction with a key distribution center or with the recipient of a given message, a user can generate a mutual secure cipher key based solely on the recipient's identity and his own secret key, and subsequently send the message, encrypted with the generated cipher used in a conventional cipher, over an insecure channel to the recipient. In contrast to previously proposed identitybased systems, no public keys, certificates for public keys or other information need to be exchanged and thus the system is suitable for certain applications that do not allow for interaction. The paper solves an open problem proposed by Shamir in 1984.