| Finite field multiplication and inversion are two basic operations involved in Elliptic Cure Cryptosystem (ECC), high performance of field operations can be applied to provide efficient computation of ECC. In this thesis, two classes of fields are proposed for multipliers with much reduced time delay. A most-significant-digit first and a least-significant-digit first digit-serial Montgomery multiplications are also proposed, using novel fixed elements R(x) which are different from xm and xm-1. Architectures of the proposed Montgomery multipliers are studied and obtained for the fields generated by the irreducible pentanomials, which are selected based on the proposed special finite fields. Complexities of the Montgomery multipliers in term of critical path delay and gate count of the architectures are investigated; the critical path delay of the proposed multipliers are found to be as good as or better than the existing works for the same class of fields. Then, implementation of the proposed multipliers (m=233) using Field Programmable Gate Array (FPGA) is provided. In addition, an FPGA implementation of an efficient normal basis inversion algorithm is also presented (m=163). The normal basis multiplication unit is implemented using a digit-level structure, and a C-code is written to generate the first coordinate of the product of two normal basis elements for all field size m. |
