Research Article Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications Shahram Jahani, Azman Samsudin, and Kumbakonam Govindarajan Subramanian School of Computer Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia Correspondence should be addressed to Azman Samsudin; azman@cs.usm.my Received 15 November 2013; Revised 4 July 2014; Accepted 5 July 2014; Published 24 July 2014 Academic Editor: Jin L. Kuang Copyright © 2014 Shahram Jahani et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Public-key cryptosystems are broadly employed to provide security for digital information. Improving the efciency of public-key cryptosystem through speeding up calculation and using fewer resources are among the main goals of cryptography research. In this paper, we introduce new symbols extracted from binary representation of integers called Big-ones. We present a modifed version of the classical multiplication and squaring algorithms based on the Big-ones to improve the efciency of big integer multiplication and squaring in number theory based cryptosystems. Compared to the adopted classical and Karatsuba multiplication algorithms for squaring, the proposed squaring algorithm is 2 to 3.7 and 7.9 to 2.5 times faster for squaring 32-bit and 8-Kbit numbers, respectively. Te proposed multiplication algorithm is also 2.3 to 3.9 and 7 to 2.4 times faster for multiplying 32-bit and 8-Kbit numbers, respectively. Te number theory based cryptosystems, which are operating in the range of 1-Kbit to 4-Kbit integers, are directly benefted from the proposed method since multiplication and squaring are the main operations in most of these systems. 1. Introduction Te growth of digital technologies has an exponential trend and as a consequence the need of information security also increases even more than before [1, 2]. Cryptography is an essential tool in providing a reasonable solution for this necessity. Te modern feld of cryptography consists of two main areas, the symmetric-key cryptography and the public- key cryptography. Te same key is used in symmetric-key cryptosystems to encrypt and decrypt a message, while the public-key cryptosystems use two keys in their protocols. Most of the public-key cryptosystems [3] use modular expo- nentiation in their calculation. For example, Dife and Hell- man introduced the frst key exchange scheme in 1967 that is based on the modular exponentiation [4]. Few years later in 1978, one of the most used public-key cryptosystems, RSA [3], is also based on the modular exponentiation. ElGamal key exchange [5] is another example of public key that has been developed based on the modular exponentiation. Modular exponentiation, =  mod , is a one-way function because the inverse of a modular exponentiation (= log  ) is a known hard problem [6–8]. To achieve a comfortable level of security, the length of the key material for these cryptosystems must be larger than 1024 bits [9], and in the near future, it is predicted that 2048-bit and 4096-bit systems will become standard [10]. Calculating modular exponentiation for a large exponent and large modulo is a costly operation and therefore improv- ing its efciency has become an important research issue for researchers in cryptography and mathematics. Tere are two main approaches currently being employed in order to improve the efciency of modular exponentiation: improving the involved operations, exponentiation, and division, sepa- rately, and improving both of the operations simultaneously. Residue number system (RNS) [11] and Montgomery mod- ular multiplication [12] are examples of the frst approach, while binary and m-ary exponentiation or Barrett reduction [13] are instances from the second approach. Tis paper focuses on the second approach, by proposing a new number representation, which will improve the squaring and mul- tiplication operations, two of the three main operations in calculating modular exponentiation [14]. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 107109, 9 pages http://dx.doi.org/10.1155/2014/107109