Abstract
Finite fields play an important role in cryptography and the theory of error correcting codes. Three fundamental operations: additions, multiplications, and inversions must be provided. Since additions are just binary additions without carry, multiplications and inversions are the operations of most concern. In this thesis, we first review traditional approaches for multiplications and inversions. We also give an algorithm developed by Fan and Paar, which use a recursive method to implement the inversion in GF(2^{2^k}) with a very low complexity. We then use the similar recursive method to develop multiplications and inversions in GF(2^{(p-1)p^k}), where p satisfies certain conditions. Most multipliers and inverters discussed in this thesis use standard basis, but a serial approach to implement serial multipliers using dual basis is also described. We will analyze each constructed architecture and compare them based on space complexity and time complexity such as latency. The results show that our recursive approach has a lower space complexity and a better latency than conventional approaches in some conditions. It also has the advantage that it is scalable in hardware implementations. the same approach for operations can be used in more composite fields, such as GF(2^{p_1^{m_1}p_2^{m_2}\dots p_k^{m_k}}). We will take some examples for explanations.