Efficient KDM-CCA Secure Public-Key Encryption for Polynomial Functions

KDM $$[\mathcal {F}]$$ -CCA secure public-key encryption PKE protects the security of message fsk, with $$f\in \mathcal {F}$$ , that is computed directly from the secret key, even if the adversary has access to a decryption oracle. An efficient KDM $$[\mathcal {F}_{\text {aff}}]$$ -CCA secure PKE scheme for affine functions was proposed by Lu, Li and Jia LLJ, EuroCrypt2015. We point out that their security proof cannot go through based on the DDH assumption. In this paper, we introduce a new concept Authenticated Encryption with Auxiliary-Input $$\mathsf {AIAE}$$ and define for it new security notions dealing with related-key attacks, namely IND-RKA security and weak INT-RKA security. We also construct such an $$\mathsf {AIAE}$$ w.r.t. a set of restricted affine functions from the DDH assumption. With our $$\mathsf {AIAE}$$ ,we construct the first efficient KDM $$[\mathcal {F}_{\text {aff}}]$$ -CCA secure PKE w.r.t. affine functions with compact ciphertexts, which consist only of a constant number of group elements;we construct the first efficient KDM $$[\mathcal {F}_{\text {poly}}^d]$$ -CCA secure PKE w.r.t. polynomial functions of bounded degree d with almost compact ciphertexts, and the number of group elements in a ciphertext is polynomial in d, independent of the security parameter. Our PKEs are both based on the DDH & DCR assumptions, free of NIZK and free of pairing.