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At TCC 2005, Groth underlined the usefulness of working in small RSA subgroups of hidden order. In assessing the security of the relevant hard problems, however, the best attack considered for a subgroup of size 22?? had a complexity of O (2 ??). Accordingly, ?? = 100 bits was suggested as a concrete parameter. This paper exhibits an attack with a complexity of roughly 2??/2 operations, suggesting that Groth's original choice of parameters was overly aggressive. It also discusses the practicality of this new attack and various implementation issues.
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