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Cryptography is one of the most prominent application areas of the finite field arithmetic. Almost all public-key cryptographic algorithms including the recent algorithms such as elliptic curve and pairing-based cryptography rely heavily on finite field arithmetic, which needs to be performed efficiently to meet the execution speed and design space constraints. These objectives constitute massive challenges that necessitate interdisciplinary research efforts that will render the best algorithms, architectures, implementations, and design practices. This paper aims to provide a concise perspective on designing architectures for efficient finite field arithmetic for usage in cryptography. We present different architectures, methods and techniques for fast execution of cryptographic operations as well as high utilization of resources in the realization of cryptographic algorithms. While it is difficult to have a complete coverage of all related work, this paper aims to reflect the current trends and important implementation issues of finite field arithmetic in the context of cryptography.