Up to present, proposed are many multi-signature schemes in which signers use respective moduli in the signature generation process. The FDH-based schemes are proposed by Mitomi et al. and Lysyanskaya et al.. The PSS-based schemes are proposed by Kawauchi et al. and Komano et al.. The FDH-based schemes have the advantage that the signature size is independent of the number of the signers. However, since the signature generation algorithm is deterministic, it has a bad reduction rate as a defect. Consequently, the signers must unfortunately use the keys large enough to keep the security. On the other hand, in the PSS-based schemes, good reduction rates can be obtained since the signature generation algorithms are probabilistic. However, the size of the random component shall overflow the security parameter, and thereby the signature size shall grow by the total size of the random components used the signers. That means, if the size of the random component is smaller, the growth of the signature size can be kept smaller. In this paper, we propose new probabilistic multi-signature scheme, which can be proven secure despite that smaller random components are used. We compare the proposed scheme and two existing schemes. Finally, we conclude that the proposed scheme is so-called optimal due to.

We proposed a one-way trapdoor permutation f based multi-signature scheme which can keep tighter reduction rate. Assuming the underlying hash functions are ideal, our proposed scheme is not only provably secure, but are so in a tight. An ability to forge multi-signatures with a certain amount of computational resources implies the ability to invert a one-way trapdoor permutation f (on the same size modulus) with about the same computational effort. The proposed scheme provides the exact security against Adaptive-Chosen-Message-Attack and Adaptive-Insider-Attack by . can also attack in key generation phase, and act in collusion with corrupted signers.