The Exact Security of a Stateful IBE and New Compact Stateful PKE Schemes

Recently, Baek et al. proposed a stateful identity based encryption scheme with compact ciphertext and commented that the security of the scheme can be reduced to the Computational Bilinear Diffie Hellman (CBDH) problem. In this paper, we formally prove that the security of the stateful identity based encryption scheme by Baek et al. cannot be reduced to the CBDH problem. In fact, we show that the challenger will confront the Y-Computational problem while providing the decryption oracle access to the adversary. We provide the exact and formal security proof for the scheme, assuming the hardness of the Gap Bilinear Diffie Hellman (GBDH) problem. We also propose two new stateful public key encryption scheme with ciphertext verifiability. Our schemes offer more compact ciphertext when compared to all existing stateful public key encryption schemes with ciphertext verifiability. We have proved all the schemes in the random oracle model.