FPGA Implementation of Elliptic Curve Point Multiplication over GF(2191)

Abstract

Hardware acceleration of cryptographic algorithms is beneficial because considerable performance improvements can be attained compared to software implementations. Thus, hardware implementations can be used in critical applications requiring high encryption or decryption speeds. Parallel architecture with efficient hardware implementation of Galois field arithmetic operations is used to produce high speed computation time for the scalar multiplication operation which is the main operation in Elliptic Curve Cryptography (ECC) system. This work proposed a modification in karatsuba-ofman algorithm which is one of the best algorithms used to perform multiplication operation over Galois field. The modification contrasted on truncating karatsuba-ofman algorithm in a low level and using the classic polynomial multiplication algorithm. In addition, this work proposed architecture for implementing ECC on hardware using Montgomery algorithm in projective coordinates. The results show that the proposed architecture is able to compute GF(2^191) elliptic curve scalar multiplication operations in 72.939 μs on Xilinx Virtex-II XC2V6000 FPGA device and 100.68 μs on Xilinx VirtexE 2600. Also, the proposed architecture can be changed to be suitable for any arbitrary Galois field size with little modifications.KeywordsGalois fieldElliptic curve cryptographyKaratsuba-Ofman multiplierfield programmable gate arrayspolynomial multiplicationpolynomial inversion