Bilinear-pairings based designated-verifier multi-proxy multi-signature scheme

In a multi-proxy multi-signature scheme, one original group delegates a group of proxy signers to sign messages on behalf of the original signers. When the proxy signature is created, all of proxy signers cooperate to generate valid proxy signatures. Based on Cha and Cheon's ID-based signature scheme, a multi-proxy multi-signature scheme based on bilinear-pairs is proposed. The proposed scheme can provide the security properties of proxy protection, verifiability, strong identifiability, strong unforgeability, strong repudiability, distinguishability and prevention of misuse of proxy signing power. That is, internal attacks, external attacks, collusion attacks, equation attacks and public key substitution attacks can be resisted.