Efficient computations in finite fields with cryptographic significance

Abstract

The increasing use of cryptographic techniques in various communication and computer systems has inspired many researchers to find ways to perform fast computations over finite fields, especially over large finite fields of characteristic two. The central theme of the thesis is an investigation of finite field computations and their architectures, such as multiplication and exponentiation. The computation of point multiples on elliptic curves is also discussed.

Three new types of finite field multipliers are given in the thesis. New bit-serial and bit-parallel multipliers using redundant bases, which is a modification of certain normal bases, are proposed. Parallel weakly dual basis multipliers are presented in F qm over Fq for any prime power q. For the polynomial basis, bit-parallel multiplication and squaring are discussed and their low-complexity constructions are investigated.

Exponentiation of a primitive element in finite fields is also considered. Structures for exponentiations using different representations of the exponent are given for both the polynomial basis and its weekly dual basis. A new signed-digit representation is proposed and used for the computation of m1P1 + m2P2 + ... + mkPk for elliptic curve cryptosystems. The performance analysis for such computations on elliptic curves using the sliding window method is also given. Other related results include closed form expressions for the average Hamming weight and length of signed-digit representations, which correspond to the numbers of multiplications and squarings in an exponentiation operation.