Universal Hash Proofs and a Paradigm for Adaptive Chosen Ciphertext
Secure Public-Key Encryption
Ronald Cramer
October 2001 |

## Abstract:
We present several new and fairly practical public-key
encryption schemes and prove them secure against adaptive chosen ciphertext
attack. One scheme is based on Paillier's Decision Composite Residuosity
(DCR) assumption, while another is based in the classical Quadratic
Residuosity (QR) assumption. The analysis is in the standard cryptographic
model, i.e., the security of our schemes does not rely on the Random Oracle
model.
We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems. Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof system as a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory
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