Co- $$Z$$ ECC scalar multiplications for hardware, software and hardware–software co-design on embedded systems

Abstract

Recent elliptic curve scalar multiplication algorithms are based on efficient co- $$Z$$ arithmetics. These arithmetics were initially introduced by Meloni in 2007 where addition of projective points share the same $$Z$$ -coordinate. The co- $$Z$$ version algorithms are sufficiently fast and secure against a large variety of implementation attacks. This paper analyses the performance of these algorithms in hardware and then compares them against software and hardware–software co-design environments on FPGA, in terms of speed, memory, power and energy consumption. Specifically, this paper presents a survey and performance comparison of implementations of co- $$Z$$ versions of the Montgomery ladder and the Joye’s double-add algorithm in an embedded system environment.