Elliptic Curve Cryptography on Smart Cards without Coprocessors

This contribution describes how an elliptic curve cryptosystem can be implemented on very low cost microprocessors with reasonable performance. We focus in this paper on the Intel 8051 family of microcontrollers popular in smart cards and other cost-sensitive devices. The implementation is based on the use of the finite fieldGF ((2− 17)) which is particularly suited for low end 8-bit processors. Two advantages of our method are that subfield modular reduction can be performed infrequently, and that an adaption of Itoh and Tsujii’s inversion algorithm is used for the group operation. We show that an elliptic curve scalar multiplication with a fixed point, which is the core operation for a signature generation, can be performed in a group of order approximately 2 in less than 2 seconds. Unlike other implementations, we do not make use of curves defined over a subfield such as Koblitz curves.