FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers

We describe algorithms for point multiplication on Koblitz curves using multiple-base expansions of the form k = ∑ ±τ(τ − 1) and k = ∑ ±τ(τ −1)(τ− τ −1). We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first provably sublinear point multiplication algorithm on Koblitz curves. For the first type, we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of τ -adic non-adjacent form expansions. We present details of an innovative FPGA implementation of our algorithm and performance data demonstrating the efficiency of our method.