Reviewing and Analyzing Efficient GCD/LCM Algorithms for Cryptographic Design

Abstract

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.

Implementing a lightweight Schmidt-Samoa cryptosystem (SSC) for sensory communications

“…For the arithmetic multiplication module, we developed our own multiplier by using Wallace Tree CSA Based Radix-8 Booth Multiplier (Asad et al, 2019). For the least common multiple (LCM) using the GCD reduction method with pulse minus GCD (Marouf et al, 2017b) used to implement the greatest common divisor operation. For modular exponentiation, we have implemented the right-to-left modular exponentiation based on NAF representation (Marouf et al, 2017c).…”

“…To verify the proposed architecture, we have implemented the proposed crypto algorithm using VHDL (LaMeres, 2017) to describe the compressor on the Altera Cyclone IV FPGA chip family (Altera Corporation, 2012a). The completed design of SSC composes several design modules including the random number generation (Tian et al, 2009;Abu Al-Haija et al, 2018b), primality testing (Ishmukhametov and Mubarakov, 2013;Asad et al, 2017a), arithmetic addition units (Ercegovac and Lang, 2004;Marouf et al, 2017a), arithmetic multiplication unit (Karatsuba and Ofman, 1963;Asad et al, 2017b;Asad et al, 2019), greatest common divisor (GCD), and least common multiple (LCM) units (Brent and Kung, 1984;Stein, 2009;Marouf et al, 2017b), modular exponentiation unit (Walter, 2010;Marouf et al, 2017c), and modular inverse unit (Hlaváč and Lórencz, 2013;Al-Haija et al, 2018). Finally, we have synthesized the resulting hardware coding using Quartus II CAD design tool (Altera Corporation, 2012b), which confirms that SSC can be used as an efficient and comparable alternative to RSA for securing the wireless sensor networks (Abu Al-Haija et al, 2014).…”

A Systematic Expository Review of Schmidt-Samoa Cryptosystem

“… Number Theory Algorithms: Because of the modular factors (p, q) must be prime, therefore, two components are contributing here generate test a prime number with desired length: a random number generator (RNG) [2] and a prime number tester PNT) [14]. Also, to test the co-prime relativity, a greatest common devisor (GCD) unit [15] is required in Schmidt-Samoa. In addition, to generate the private key modulus, a Least common multiple (LCM) [15] unit is needed.…”

“…Also, to test the co-prime relativity, a greatest common devisor (GCD) unit [15] is required in Schmidt-Samoa. In addition, to generate the private key modulus, a Least common multiple (LCM) [15] unit is needed.…”