On security notions for verifiably encrypted signature

First we revisit three (BGLS, MBGLS and GZZ) verifiably encrypted signature schemes [2], [6], [9]. We find that they are all not strong unforgeable. We remark that the notion of existential unforgeable is not sufficient for fair exchange protocols in most circumstances. So we propose three new (NBGLS, MBGLS and NGZZ) verifiably encrypted signature schemes which are strong unforgeable. Also we reconsider other two (ZSS and CA) verifiably encrypted signature schemes [8], [11], we find that they both cannot resist replacing public key attack. So we strongly suggest that strong unforgeable for verifiably encrypted signature be a better notion than existential unforgeable and checking adjudicator knowing its private key is a necessary step for secure verifiably encrypted signature scheme.