FPGA implementation of energy efficient multiplication over GF(2m) for ECC

Public key cryptography (PKC) is highly secure against threats compared to symmetric key cryptography (SKC). One of the PKC techniques, Elliptic curve cryptography has been gaining wider attention as compared to the popular RSA due to its lesser key size requirements in order to provide a similar security level. This paper details the hardware implementation modular multiplicative over binary field GF(2^m). Efficient scalar point multiplication is a crucial part in elliptic curve cryptography. A scalar point multiplication consists of point doubling and point addition operations. Both of these operations inherently depend on addition, multiplication, squaring and inversion. Among these, the inversion operation is the most time consuming one. The computation of multiplicative inverse primarily consists of modular multiplication and modular squaring operations. This paper proposes an efficient scalar multiplication using iterative Karatsuba-Offman multiplication algorithm (KMA) over GF(2^m). The performance comparison is based on the Xilinx Virtex-6 FPGA implementation for the NIST recommended binary field.