Computer and network security systems rely on the privacy and authenticity of information, which requires implementation of cryptographic functions. Software implementations of these functions are often desired because of their exibility and cost eff ectiveness. In this study, we concentrate on developing high-speed and area-efficient modular multiplication and exponentiation algorithms for number-theoretic cryptosystems. The RSA algorithm, the Die-Hellman key exchange scheme and Digital Signature Standard require the computation of modular exponentiation, which is broken into a series of modular multiplications. One of the most interesting advances in modular exponentiation has been the introduction of Montgomery multiplication. We are interested in two aspects of modular multiplication algorithms: development of fast and convenient methods on a given hardware platform, and hardware requirements to achieve high-performance algorithms. Arithmetic operations in the Galois eld GF(2k) have several applications in coding theory, computer algebra, and cryptography. We are especially interested in cryptographic applications where k is large, such as elliptic curve cryptosystems.