In this paper, we provide a practical review with numerical example and complexity analysis for greatest common divisor (GCD) and Least Common Multiple (LCM) algorithms that are commonly used in the computing coprocessors design such as Cryptoprocessor design. The paper discusses four common GCD algorithms: Dijkstra's algorithm, Euclidian algorithm, Binary GCD algorithm, Lehmer's algorithm, and two LCM algorithms: LCM based Prime Factorizations algorithm and LCM based GCD reduction. It was found that Lehmer's algorithm can be used efficiently to compute GCD and LCM with time complexity of ( ( ) ) which enhances the linear time ( ( )) complexity of well-known Euclidian algorithm.

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Public key cryptographic schemes are vastly used to ensure confidentiality, integrity, authentication and nonrepudiation. Schmidt-Samoa cryptosystem (SSC) is a public key cryptosystem, which depends on the difficulty of large integer factorization problem. The implementation of SSC to secure different recent communication technologies such as cloud and fog computing is on demand due to the assorted security services offered by SSC such as data encryption/decryption, digital signature and data integrity. In this paper, we provide a systematic review of SSC public key cryptosystem to help crypto-designers to implement SSC efficiently and adopt it in hardware or software-based applications. According to the literature, the effective utilization and design SSC can place it as a viable alternative of RSA cryptosystems and many others.

In this paper, we proposed different comparable reconfigurable hardware implementations for the radix-8 fast two operands multiplier coprocessor using Karatsuba method and Booth recording method by employing carry save (CSA) and kogge stone adders (KSA) on Wallace tree organization. The proposed designs utilized family with target chip device along with simulation package. Also, the proposed designs were synthesized and benchmarked in terms of the maximum operational frequency, the total path delay, the total design area and the total thermal power dissipation. The experimental results revealed that the best multiplication architecture was belonging to Wallace Tree CSA based Radix-8 Booth multiplier ( ) which recorded: critical path delay of , maximum operational frequency of , hardware design area (number of logic elements) of , and total thermal power dissipation estimated as . Consequently, method can be efficiently employed to enhance the speed of computation for many multiplication based applications such embedded system designs for public key cryptography.