Strategies for VLSI implementations of finite field inversion algorithms

Inversion of a finite field element is the most time consuming of all field arithmetic operations which is why it is avoided as much as possible in elliptical curve cryptosystem implementations. Unfortunately, there exists only two methods for performing inversion: the Euclidean algorithm and inversion through multiplication based on Fermat´s theorem. VLSI implementations of these methods are examined in detail using TSMC SCN6M 0.18μm technology in GF(2¹⁶³) using polynomial basis representation. Observations are made comparing the variants of each method and strategies are presented to improve VLSI implementations.