The paper presents overview of the most interesting GF (2^m) algorithms and proposes efficient hardware solutions applicable to elliptic curve cryptosystems. It focuses on fields of size m = 233, one of the sizes recommended by NIST (National Institute of Standards and Technology). We perform analysis of most popular algorithms used for multiplication over finite fields; suggest efficient hardware solutions and point advantages and disadvantages of each algorithm. The article overviews and compares classic, Mastrovito and Montgomery multipliers. Hardware solutions presented here, implement their modified versions to gain on efficiency of the solutions. Moreover we try to present a fair comparison with existing solutions. It is hard to compare different hardware finite field multipliers designs due to the fact that the most important improvements are described only theoretically. The designs presented here are targeted to FPGA devices.