Short-Memory Scalar Multiplication for Koblitz Curves

Abstract

This paper presents a scalar multiplication method for Koblitz curves. Koblitz curves are elliptic curves where the scalar multiplication can be computed in a much faster way than with other curves, allowing designs and implementations without arithmetic coprocessor. The new method is as fast as the fastest known techniques on Koblitz curves but requires much less memory; therefore, it is of particular interest for environments with low resources. Our technique is well suited for both hardware and software implementations. In hardware, we show that a normal basis implementation reduces memory consumption by 85 percent compared to conventional methods, but this still has exactly the same computational cost. In software, thanks to a mixed normal-polynomial bases approach, our technique allows memory savings up to 70 percent and, depending on the instruction set of the CPU, can be as fast as the fastest known scalar multiplication methods or can even beat them by a large margin. Therefore, in software and in hardware, our scalar multiplication technique offers high performance without sacrifice in view of memory.