Abstract
In this paper, we prove the security against related key attacks of two public key encryption schemes in the standard model. The first scheme is a variation of the scheme (KYPS09) presented by Kiltz, Pietrzak et al. in Eurocrypt 2009. While KYPS09 has been proved CCA secure under the DDH assumption, we show that it is not secure against related key attacks when the class of related key functions includes affine functions. We make a modification on KYPS09 and prove that the resulted scheme is secure against related key attacks in which the related key functions could be affine functions. We also prove the security against related key attacks of the scheme presented by Hofheinz and Kiltz in Crypto 2009 based on the HR assumption. The security proofs rely heavily on a randomness extractor called 4-wise independent hash functions.
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