At Eurocrypt '02 Cramer and Shoup 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, 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.