Optimal Dispersal of Certificate Chains

We consider a network where users can issue certificates that identify the public keys of other users in the network. The issued certificates in a network constitute a set of certificate chains between users. A user u can obtain the public key of another user v from a certificate chain from u to v in the network. For the certificate chain from u to v, u is called the source of the chain and v is called the destination of the chain. Certificates in each chain are dispersed between the source and destination of the chain such that the following condition holds. If any user u needs to securely send messages to any other user v in the network, then u can use the certificates stored in u and v to obtain the public key of v (then u can use the public key of v to set up a shared key with v to securely send messages to v). The cost of dispersing certificates in a set of chains among the source and destination users in a network is measured by the total number of certificates that need to be stored in all users. A dispersal of a set of certificate chains in a network is optimal if no other dispersal of the same chain set has a strictly lower cost. In this paper, we show that the problem of computing optimal dispersal of a given chain set is NP-complete. Thus, minimizing the total number of certificates stored in all users is NP-complete. We identify three special classes of chain sets that are of practical interests and devise three polynomial-time algorithms that compute optimal dispersals for each class. We also present two polynomial-time extensions of these algorithms for more general classes of chain sets