We first model the formal security model of multisignature scheme following that of group signature scheme. Second, we prove that the following three probabilistic multisignature schemes based on a trapdoor permutation have tight security; PFDH (probabilistic full domain hash) based multisignature scheme (PFDH-MSS), PSS (probabilistic signature scheme) based multisignature scheme (PSS-MSS), and short signature PSS based multisignature scheme (S-PSS-MSS). Third, we give an optimal proof (general result) for multisignature schemes, which derives the lower bound for the length of random salt. We also estimate the upper bound for the length in each scheme and derive the optimal length of a random salt. Two of the schemes are promising in terms of security tightness and optimal signature length. In appendix, we describe a multisignature scheme using the claw-free permutation and discuss its security.

This paper deals with broadcast encryption schemes, in which a sender can send information securely to a group of receivers excluding some receivers over a broadcast channel. In this paper we propose modifications of the Complete Subtree (CS), the Subset Difference (SD) and the Layered Subset Difference (LSD) methods based on the Master Key Tree (MKT). Our modifications eliminate log N keys or labels from receivers' storage, in exchange for an increase in the computational overhead, where N is the total number of receivers. We also propose modifications of the SD and LSD methods by applying the Trapdoor One-way Permutation Tree (TOPT) which is originally proposed in order to modify the CS method. Our modifications based on TOPT also eliminate log N labels, and the computational cost is much smaller than MKT based methods.