# Approaching Encryption through Complex Number Logarithms

### George Stergiopoulos, Miltiadis Kandias, Dimitris Gritzalis

#### Abstract

In this paper, we approach encryption through the properties of complex logarithm and the complex plane. We introduce a mathematical concept to be used in cryptography. As an example, we propose a new cryptosystem, by mixing known robust techniques such as chain-block encryption and AES-like structures together with complex exponentiation to provide robust encryption of plaintext messages. The proposed method implements encryption by transforming complex numbers into position vectors in a two-dimensional Cartesian coordinate system called the complex plane and utilizes the properties of the complex logarithm together with well-defined techniques from global standards (such as AES), in order to ensure robustness against cryptanalysis. This is made possible without implementing any computational costly algorithm. This has two important consequences: First, it may open up viable solutions to known limitations in cryptography such as relatively complex key schedules (i.e. in Feistel ciphers) and the need for relatively large keys used in encryption methods (bit-wise). Second, it proposes a new mathematical concept that can be used in future cryptosystems. An example of this is the preliminary cryptosystem found in this paper. We present its algorithm and show that it can be implemented using fast mechanisms for encryption and decryption.

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#### Paper Citation

#### in Harvard Style

Stergiopoulos G., Kandias M. and Gritzalis D. (2013). **Approaching Encryption through Complex Number Logarithms** . In *Proceedings of the 10th International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2013)* ISBN 978-989-8565-73-0, pages 574-579. DOI: 10.5220/0004604005740579

#### in EndNote Style

TY - CONF

JO - Proceedings of the 10th International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2013)

TI - Approaching Encryption through Complex Number Logarithms

SN - 978-989-8565-73-0

AU - Stergiopoulos G.

AU - Kandias M.

AU - Gritzalis D.

PY - 2013

SP - 574

EP - 579

DO - 10.5220/0004604005740579

#### in Bibtex Style

@conference{secrypt13,

author={George Stergiopoulos and Miltiadis Kandias and Dimitris Gritzalis},

title={Approaching Encryption through Complex Number Logarithms},

booktitle={Proceedings of the 10th International Conference on Security and Cryptography - Volume 1: SECRYPT, (ICETE 2013)},

year={2013},

pages={574-579},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004604005740579},

isbn={978-989-8565-73-0},

}