Many fast variants of RSA are designed to speed up encryption and decryption. Prime power RSA (PP-RSA) is one of the most important variants. It is widely used in electronic money trading systems and high-speed programs. At present, the security of PP-RSA with moduli N = prqshas not been fully studied. In this paper, we give three powerful attacks based on Coppersmith’s method, applying to the cases when the most significant bits or the least significant bits of the private key are known. Our attacks work in polynomial time. This is the first work on partial key exposure attacks of PP-RSA with moduli N = prqs.

Accelerating scalar multiplication has always been a significant topic when people talk about the elliptic curve cryptosystem. Many approaches have been come up with to achieve this aim. An interesting perspective is that computers nowadays usually have multicore processors which could be used to do cryptographic computations in parallel style. Inspired by this idea, we present a new parallel and efficient algorithm to speed up scalar multiplication. First, we introduce a new regular halve-and-add method which is very efficient by utilizing λ projective coordinate. Then, we compare many different algorithms calculating double-and-add and halve-and-add. Finally, we combine the best double-and-add and halve-and-add methods to get a new faster parallel algorithm which costs around 12.0% less than the previous best. Furthermore, our algorithm is regular without any dummy operations, so it naturally provides protection against simple side-channel attacks.