A NEW RSA-TYPE SCHEME BASED ON SINGULAR CUBIC CURVES Y(2)EQUIVALENT-TO-X(3)+BX(2) (MOD-N)

We propose an RSA-type scheme over the non-singular part of a singular cubic curve E(n)(0, b): y2 = x3 + bx2 (mod n), where n is a product of form-free primes p and q. Our new scheme encrypts/decrypts messages of 2 log n bits by operations of the x and y coordinates. The decryption is carried out over F(p) or a subgroup of a quadratic extension of F(p), depending on quadratic residuosity of message-dependent parameter b. The decryption speed in our new scheme is about 4.6 and 5.8 times faster than that in the KMOV scheme and the Demytko scheme, respectively. We prove that if b is a quadratic residue in Z(n), breaking our new scheme over E(n) (0,b) is not easier than breaking the RSA scheme.