Multiplication in GF(2m): area and time dependency/efficiency/complexity analysis.

Efficient arithmetic units are crucial for cryptographic hardware design. The cryptographic systems are based on mathematical theories thus they strongly depend on the performance of the arithmetic units comprising them. If the arithmetic operator does not take a considerable amount of resources or is time non efficient it negatively impacts the performance of the whole cryptosystem. This work is intended to analyse the hardware possibilities of the algorithms performing multiplication in finite field extensions GF(2m). Such multipliers are used in Elliptic Curve Cryptography (ECC) applications. There are only two operations defined in the field: addition and multiplication. Addition is considered as a trivial operation - it is a simple bitwise XOR. On the other hand multiplication in the field is a very complex operation. To conform to the requirements of ECC systems it should be fast, area efficient and what is the most important perform multiplication of large numbers (100 - 600 bits). The paper presents analysis of GF(2m) two-step modular multiplication algorithms. It considers classical (standard, school-book) multiplication, matrix-vector approach algorithm and Karatsuba-Ofman algorithm, exploring thoroughly their advantages and disadvantages.