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Evgueni Doukhnitch (Istanbul Aydin University, Turkey), Alexander G. Chefranov (Eastern Mediterranean University, North Cyprus) and Ahmed Mahmoud (Al-Azhar University-Gaza, Palestine)

Copyright: © 2013
|Pages: 23

DOI: 10.4018/978-1-4666-4030-6.ch005

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TopThere are two hyper-complex number systems which are used in modern encryption systems: quaternions and octonions (Ward, 1997). The quaternion number system was discovered by a physicist Hamilton (1843) (Hamilton, 1847); it is an extension of the complex number system (so called hyper complex number). It has two parts, a scalar part and a vector part which is a vector in three-dimensional space Since the introduction of quaternion, it has been applied in several areas in computer science and engineering problems such as graphics and robotics (Kuipers 1999; Marins et al., 2001). It can be used to control rotations in three-dimensional space. The application of the quaternion number system is attractive in computation models due to its matrix representation. It has been applied as a mathematical model in encryption by several researchers. In (Nagase et al., 2004, 2005), a new Quaternion Encryption Scheme (QES) is proposed for signal encryption providing good hiding properties.

The octonion (Cayley numbers or octaves) number system was suggested by John T. Graves (1843) and discovered by Arthur Cayley (1845) (Ward, 1997). An octonion has two parts, a scalar part and a vector part which is a vector in seven-dimensional space It is used in physics for 8-D rotation description and for quaternion valued matrix decomposition (Doukhnitch & Ozen, 2011). Recently in (Malekian & Zakerolhosseini, 2010), new encryption schemes based on non-associative octonion algebra were proposed for signal encryption with better security against lattice attack and/or more capability for protocol design.

Hyper-complex number based ciphers are attractive not only because they may be represented using matrix-vector multiplication but also that the inverse matrix for such transformation is a transpose of the original matrix. Matrix operations are rather simple and can be efficiently implemented that is especially important for multimedia data transmission.

The QES works as follows. A sequence of signal samples is arranged as a sequence of frames containing three three-component vectors, represented as a 3x3 matrix -th column of which is the -th mentioned above sample-vector (). Each vector in a frame is encrypted by applying to it one and the same transformation represented by its multiplication from one side by some quaternion and from the other side by its inverse producing the ciphertext vector

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