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Steven H. Low and Nicholas F. Maxemchul formulated in 1998 a collusion
problem, which is to determine whether it is possible for a group of
colluders to reveal secret data by combining their knowledge. However, their
formulation of collusion has some drawbacks that each colluder must expose
her own secret key, so that it cannot properly express the way how collusion
is formed. Fortheremore, their formulation models attakers to collude after
the communication has finished, so it cannot take into considerration more
effective attack, which is perfomed during protocol execution. Therefore we
propose two collusion models, the one is introduced that the secret keys of
colluders are not shared and the collusion condition is represented in a
directed graph, the other enables to carry out the collusion attack during
the communication protocol runs.
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