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Since the Diffie-Hellman paper, asymmetric encryption has been a very important topic, and furthermore ever well studied. However, between the efficiency of RSA and the security of some less efficient schemes, no trade-off has ever been provided.  In this paper, we propose better than a trade-off: indeed, we first present a new problem, derived from the RSA assumption, the "Dependent-RSA Problem". A careful study of its difficulty is performed and some variants are proposed, namely the "Decisional Dependent-RSA Problem".  They are next used to provide new encryption schemes which are both secure and efficient. More precisely, the main scheme is proven semantically secure in the standard model. Then, two variants are derived with improved security properties, namely against adaptive chosen-ciphertext attacks, in the random oracle model. Furthermore, all those schemes are more or less as efficient as the original RSA encryption scheme and reach semantic security.

New Public Key Cryptosystems Based on the Dependent-RSA Problems