TopIntroduction
The Internet, a channel of communication for exchanging data between sender and receiver, is a public place which needs to be protected against intruders, spammers, and malicious attacks. This virtual place is a basis for many applications such as e-/m- commerce (Tan, 2014), e-voting (Chamberlain, 2011), e-banking (Pravettoni, Leotta, Lucchiari, & Misuraca, 2007; Rezai-Rad, Vaezi, & Nattagh, 2012), tele-communications such as tele-EEG (Coates, Clarke, Davison, & Patterson, 2012), wireless networking (Johnson, Green, & Leeson, 2013; Tan, Lee, Lam, & Yoo, 2013; Zakerolhosseini, Sokouti, & Pezeshkian, 2013), and security for smart phones/tablets messaging (Aktas et al., 2013; Black, 2006; Curioso et al., 2005; McCreadie & McGregory, 2005). By growth of networking technology and the number of spies and hackers on the Internet, the insecure channel becomes unsafe for transmitting private data. The encryption, the best way for exchanging data safely and securely, is a security service for providing confidentiality. However, there are two general methods for encryption and decryption of transmitting data which are symmetric and asymmetric cipher algorithms. In the symmetric cipher algorithm, the encryption and decryption processes are done by only one shared key between sender and receiver while in the asymmetric cipher algorithm, the encryption is performed by a public key and the decryption is done by its corresponding private key. Lattice based cryptographies are in the group of public key ciphers which are faster than other versions of public key ciphers. The first lattice based cryptography was invented by Ajtai (Ajtai, 1996). The other two known lattice based ciphers are GGH (Goldreich, Goldwasser, & Halevi, 1997) and NTRU (Hoffstein, Pipher, & Silverman, 1998). This article focuses on major attacks and issues of GGH based on arithmetic matrices and proposes two methods (i.e., C-GGH based on complex number algebra and Q-GGH based on quaternion algebra) for improving the original GGH to encounter the existing attacks and increases the security of this cipher against lattice attacks in low dimensions. GGH is known to consume more memory and performs equally as fast as NTRU while it is implemented, and the interesting topic of this article is focused on improving its memory usage, its speed and strength of the security.