Collusion attack and counterattack on the quantum key agreement via non-maximally entangled cluster states

Recently, Li et al. (Int J Theor Phys: DOI: 10.1007/s10773-020-04588-w, 2020) proposed a multiparty quantum key agreement protocol via non-maximally entangled cluster states. They claimed that the proposed protocol can help all the involved participants have equal influence on the final shared key. However, this study points out a loophole that makes Li et al.'s protocol suffer from a collusion attack, i.e. several dishonest participants can conspire to manipulate the final shared key without being detected by others. To avoid this loophole, an improvement is proposed here.