A Cramer-Shoup Variant Related to the Quadratic Residuosity Problem

At Eurocrypt ’02 Cramer and Shoup [7] proposed a general paradigm to construct practical public-key cryptosystems secure against the adaptive chosen ciphertext attack as well as several concrete examples. Using the construction, we present a new variant of the Cramer-Shoup encryption scheme, which is secure against the adaptive chosen ciphertext attack. Our variant is based on the problem related to the quadratic residuosity (QR). In [7], they also proposed the encryption scheme based on the QR, but this scheme is less efficient than those based on the decision Diffie-Hellman problem or the decision composite residuosity. In this paper, we define the new assumptions related to the QR assumption and propose a new public-key encryption scheme whose security is based on the new assumption. Our scheme is more efficient than the QR-based scheme proposed by Cramer and Shoup.