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This paper proposes fast RSA-type public-key schemes based on singular cubic curves y2 + axy = x3 over the ring Zn. The x and y coordinates of a 2 log n-bit long plaintext/ciphertext are transformed to a log n-bit long shadow plaintext/ciphertext by isomorphic mapping. Decryption is carried out by exponentiating this shorter shadow ciphertext over Zn. The decryption speed of the proposed schemes is about 2.0 times faster than that of the RSA scheme for a K-bit long message if [K/log n] is even. We prove that breaking each of the proposed schemes is computationally equivalent to breaking the RSA scheme in one-to-one communication circumstances. We also prove that the proposed schemes have the same security as the RSA scheme against the Hastad attack when linearly related plaintexts are encrypted i n broadcast applications.

Fast RSA-type Schemes Based on Singular Cubic Curves y 2 + axy ≡ x 3 (mod n)