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In this paper, the authors present a different CPA-secure counterexample (under SXDH) as well as the first CCA-secure counterexample (under SXDH and the existence of certain NIZK proof systems) for n > 1. Moreover, the 2-circular attacks recover the secret keys of both parties and thus exhibit a catastrophic failure of the system whereas the attack in Acar et al. provides a test whereby the adversary can distinguish whether it is given a 2-cycle or two random ciphertexts. These negative results are an important step in answering deep questions about which attacks are prevented by commonly-used definitions and systems of encryption.
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