Arab J Sci Eng DOI 10.1007/s13369-017-2797-3 RESEARCH ARTICLE - COMPUTER ENGINEERING AND COMPUTER SCIENCE A Fast Parallel Modular Exponentiation Algorithm Khaled A. Fathy 1 · Hazem M. Bahig 2,3 · A. A. Ragab 1 Received: 20 December 2016 / Accepted: 6 August 2017 © King Fahd University of Petroleum & Minerals 2017 Abstract Modular exponentiation is a fundamental and most time-consuming operation in several public-key cryp- tosystems such as the RSA cryptosystem. In this paper, we propose two new parallel algorithms. The first one is a fast parallel algorithm to multiply n numbers of a large number of bits. Then we use it to design a fast parallel algorithm for the modular exponentiation. We implement the parallel modular exponentiation algorithm on Google cloud system using a machine with 32 processors. We measured the performance of the proposed algorithm on data size from 2 12 to 2 20 bits. The results show that our work has a fast running time and more scalable than previous works. Keywords Binary method · Modular exponentiation · Multiplication algorithm · Parallel algorithm · Scalability 1 Introduction Modular exponentiation is one of the key computations per- formed in many public-key cryptosystems such as RSA [1]. In this type of cryptosystem, the encryption and decryption processes depend on the modular exponentiation operation. The description of the encryption and decryption processes is as follows. Given M as a plain text, E as a public key, D as a private key and N as a modulus, the cipher text is B Hazem M. Bahig hazem.m.bahig@gmail.com; hbahig@sci.asu.edu.eg 1 Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt 2 Computer Science Division, Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt 3 Computer Science and Engineering College, Hail University, Hail, Kingdom of Saudi Arabia computed as C = M E mod N in the encryption phase. The plain text is computed as M = C D mod N in the decryption phase. In cryptography, the value of E is large and therefore, the modular exponentiation is composed of a large number of modular multiplications. So, the, modular exponentiation is a time-consuming operation. There are two types of the modular exponentiation, single modular exponentiation [2–10] and multi-modular exponen- tiations [11, 12]. In this paper, we will be interested in the single modular exponentiation. Many methods were presented to compute the modular exponentiation such as the binary method, m-ary method [4, 7], residue number conversion method [9], signed digits recoding method [4–6], the sliding windows method [4, 8] and hybrid method (combination of two or more meth- ods such as the Montgomery modular reduction method, common-multiplicand-multiplication method and minimal- signed-digit exponent recoding method) [2, 3, 10]. Parallel computations play an important strategy to inc- rease the efficiency of the modular exponentiation operation. Several parallel solutions for the computation of the modular exponentiation were introducing using hardware and soft- ware. Examples of recent parallel architectures for parallel modular exponentiation are in [13–15]. In this paper, we will focus only on the parallel modular exponentiation algorithms that are designed to run on a real parallel machine. Nedjah and Mourelle [16, 17] introduced parallel modu- lar exponentiation algorithms based on the sliding window method. Wu et al. [10] presented a technique for the paral- lel computation of the modular exponentiation depending on the Montgomery multiplication algorithm. Chia-Long et al. [18] presented a parallel exponentiation algorithm based on dividing the binary representation of E into many parts. Chia- Long [19] proposed a new hybrid parallel algorithm for the modular exponentiation. The algorithm is based on different 123