PSEC { 3 : Provably Secure Elliptic Curve Encryption Scheme { V 3 (

We describe a new version of the elliptic curve encryption schemes PSEC (Provably Secure Elliptic Curve). PSEC{3 is a public-key encryption system that uses the elliptic curve El Gamal trapdoor function and two random functions (hash functions) as well as any semantically secure symmetric encryption scheme, such as the one-time pad, or any classical block-cipher. Furthermore, we de ne a new problem, the Elliptic Curve Gap Di e-Hellman problem (EC|Gap{ DH) which is likely stronger than the more classical Elliptic Curve Decision Di e-Hellman (EC{DDH) problem. Indeed, its tractability would imply the equivalence between the Computational and the Decisional versions of the Elliptic Curve Di e-Hellman problem. PSEC{3 therefore has several outstanding properties as follows: 1. with the one-time pad, PSEC{3 is semantically secure or non-malleable against chosen-ciphertext attacks (IND-CCA2 or NM-CCA2), in the random oracle model, under the Elliptic Curve Gap Di e-Hellman (EC{Gap-DH) assumption. 2. with any symmetric encryption, PSEC{3 is semantically secure or non-malleable against chosenciphertext attacks (IND-CCA2 or NM-CCA2), in the random oracle model, under the Elliptic Curve Gap Di e-Hellman (EC{Gap-DH) assumption, if the underlying symmetric encryption is simply semantically secure against passive attacks. 3. if the underlying random functions are replaced by practical random-like functions (e.g., SHA and MD5), PSEC{3 is as e cient as the basic Elliptic Curve El Gamal scheme, for the encryption process but also for the decryption process, which is the major novelty of this new proposal. The encryption scheme described in this contribution is obtained by using a new result on conversion techniques using random functions by the authors.