A Variant of the Cramer-Shoup Cryptosystem for Groups with Unknwon Order

The Cramer-Shoup cryptosystem for groups of prime order is a practical public-key cryptosystem, provably secure in the standard model under standard assumptions. This paper extends the cryptosystem for groups of unknown order, namely the group of quadratic residues modulo a composed N. Two security results are: In the standard model, the scheme is provably secure if both the Decisional Diffie-Hellman assumption for QR_N *and* the factorisation assumption for N hold. In the random oracle model, the security of the scheme is provable by a quite efficient reduction