A new RSA-type scheme based on singular cubic curves (y-alpha x)(y-beta x)x(3) (mod n)

This paper proposes a new RSA-type scheme over non-singular parts of singular cubic curves E(n)(alpha,beta): (y-alpha x)(y-beta x) = x(3) (mod n). In usual one-to-one communication, we prove that breaking the proposed scheme is not easier than breaking the RSA scheme for the whole ciphertexts. If encryption key e is larger than 19 for 512 bits modulus n, then the proposed scheme is secure against the Hastad attack in broadcast applications. A plaintext of two blocks, i.e., x and y coordinates in E(n)(alpha,beta), is encrypted to a ciphertext of three blocks, where the size of one block is log(2) n bits. The decryption speed of the proposed scheme is as fast as that of the RSA scheme for the even block plaintext.