We propose two types of key management systems that use complementary exponential calculation, in which users in the system divide into groups, and the different modulus numbers are assigned to each group and edges between groups. Key generation information over the modulus numbers is issued to a user by a trusted center. The user who receives the information can generate shared encryption keys between users in the system without using key exchange protocol. In our proposed system, the number of primes is one of the parameters for generating key generation information. The number decreases in-inverse proportion to the square of the number of groups compared to the original method. Our proposed technique enabled us to extend the number of users in the system to more than one million, which is not possible with the original method.