In the last 20-30 years, the world of modern cryptography has been largely dominated by traditional systems such as the Data Encryption Standard and the RSA algorithm. Such systems have provided a secure way for storing and transmitting information and they are nowadays incorporated in many network protocols and secure storage media. More recently, the increasing advance of crypto-analytical techniques and tools and the emergence of new applications, for example wireless communications and mobile computing, have stimulated the research and development of innovative cryptographic algorithms. These newer systems require a more detailed and sophisticated mathematical formalization and operations, which are not normally supported by general-purpose processors. For example, many basic operations required to implement recently proposed cryptographic algorithms, such as the Advanced Encryption Standard or Elliptic Curve Cryptosystems, are based on arithmetic in finite fields (or Galois fields). This chapter is, thus, intended to give an overview of such developments in modern cryptography. In particular, it aims at giving the reader a comprehensive understanding of innovative cryptosystems, their basic structure, alternative existing hardware architectures to implement them, and their performance requirements and characterizations. Emphasis will be made throughout on two important cases: the Advanced Encryption Standard and Elliptic Curve Cryptosystems.