Efficient Algorithms for Elliptic Curve Cryptosystems on Embedded Systems

This thesis describes how an elliptic curve cryptosystem can be implemented on low cost microprocessors without coprocessors with reasonable performance. We focus in this paper on the Intel 8051 family of microcontrollers popular in smart cards and other cost-sensitive devices, and on the Motorola Dragonball, found in the Palm Computing Platform. The implementation is based on the use of the Optimal Extension Fields GF ((2 − 17)) for low end 8-bit processors, and GF ((2 − 1)) for 16-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 may be 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 on an 8-bit smart card. On a 16-bit microcontroller, signature generation in a group of order approximately 2 can be performed in under 700 milliseconds. Unlike other implementations, we do not make use of curve parameters defined over a subfield such as Koblitz curves.