Efficient algorithms for speeding up the computations of elliptic curve cryptosystems

Abstract

As we know, the performance of the elliptic curve cryptosystem (ECC) deeply depends on the computation of scalar multiplication. Thus, how to speed up the computation of the elliptic curve scalar multiplication is a significant issue. In 1994, Lim and Lee proposed a more flexible precomputation method used in wireless networks environments for speeding up the computation of exponentiation. This method can be also used for speeding up the scalar multiplication of elliptic curves. We call it LLECC method. However, the less storage is equipped with the computing devices, the less efficient it is. For this reason, we propose a more efficient algorithm than LLECC’s in this paper. First, we modify LLECC method to reduce the storage of precomputed values, and then propose an efficient algorithm based on the nonadjacent form (NAF) representation and multidoubling method. Furthermore, the proposed algorithm can be also used for speeding up the multi-point multiplication of elliptic curves. According to the simulation results, the proposed algorithm can reduce 11% and 21% in the aspect of the computational complexity and storage cost, respectively, in an elliptic curve of size 160-bit over finite fields with characteristic greater than 3, as compared with LLECC’s. Finally, we implement the elliptic curve digital signature algorithm-like (ECDSA-like) system in the personal digital assistant (PDA) using the proposed algorithms to improve the scalar multiplication.