Efficient method to find the multiplicative inverse in GF (2m) using FPGA by exponentiation to (2k)

Multiplicative inverse in GF (2
^m
) is a complex step in some important application such as Elliptic Curve Cryptography (ECC) and other applications. It operates by multiplying and squaring operation depending on the number of bits (m) in the field GF (2
^m
). In this paper, a fast method is suggested to find inversion in GF (2
^m
) using FPGA by reducing the number of multiplication operations in the Fermat's Theorem and transferring the squaring into a fast method to find exponentiation to (2
^k
). In the proposed algorithm, the multiplicative inverse in GF(2
^m
) is achieved by number of multiplications depending on log
₂
(m) and each exponentiation is operates in a single clock cycle by generating a reduction matrix for high power of two exponentiation. The number of multiplications is in range between (log
₂
(m) and 2log
₂
(m)-2). If m equals 163 then the number of multiplication operations is 9 and number of exponentiation operation each one with one clock cycle equals 10.