Design of low complexity GF(2m) multiplier based on Karatsuba algorithm

The multiplication is the basic arithmetic operation of others over the finite field GF(2^m) which have widespread applications. In this paper, we analyze the complexity of the classic multiplier mixed Karatsuba multiplier for 2-term polynomials in the finite field GF(2^m). By using a method to transform the multiplication operands, we can obtain the complexity of Karatsuba multiplier with any iteration step for any given length m and the constraint when the iteration should be stopped. The results show that this method has a low space complexity and is especially suitable for elliptic curve cryptography (ECC) in which m is suggested to be prime.