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In the model of identity-based encryption, first proposed by Shamir, one can use arbitrary strings as public keys. The research on IBE is an active and stimulating field of cryptography, and so far IBE schemes have been constructed under several assumptions: pairing-related assumptions, quadratic residue-related assumptions and lattice-related assumptions. Akavia et al. and Alwen et al. showed that some variants of them are secure against private key leakage attacks. The security of these schemes is either analyzed in the random oracle model or is based on "Non-static" assumption in the standard model.
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