A low-time-complexity and secure dual-field scalar multiplication based on co-Z protected NAF

In this paper, we incorporate the co-Z arithmetic with Protected NAF to build a dual-field scalar multiplication method in ECC (elliptic curve cryptography) with lower time complexity and higher security, called co-Z Protected NAF. The Protected NAF is the alteration of the original NFA against SPA (simple power analysis) attack. But the employed dummy operations, double-and-add-always, often results in two severe problems: the high time complexity and the vulnerability of safe-error attack. So, the speed advantage of co-Z point addition is leveraged to greatly compensate the time penalty incurred by Protected NAF. Meanwhile, not only does the co-Z not change the SPA immunity existed in Protected NAF, but the property of updating point P in it improves the security to resist the safe-error attack. Experiment results show that the co-Z Protected NAF can obtain 1.36 times speedup with respect to Protected NAF over GF(p), and is even faster than original NAF. And it can also counteract 30.7% time loss over GF(2m) caused by dummy operations. Furthermore, because the co-Z Protected NAF is only the optimization on scalar multiplication, only less than 1% extra area cost is generated to achieve its improvements in time complexity and security.