Pell hyperbolas in DLP–based cryptosystems |
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| Institution: | 1. Politecnico di Torino, Italy;2. Università di Trento, Italy;1. TÜB?TAK B?LGEM UEKAE, National Research Institute of Electronics and Cryptology, Gebze, Kocaeli, Turkey;2. Department of Mathematics, Hacettepe University, Beytepe, Ankara, Turkey;1. University of Bristol, England, UK;2. University of Glasgow, Scotland, UK;3. The University of New South Wales Canberra, Australia;1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, China;2. Guangdong Province Key Laboratory of Information Security Technology, Guangzhou 510006, China;3. School of Computer Science and Engineering, Hunan University of Science and Technology, Xiangtan 411201, China;4. School of Mathematics and Statistics, Central South University, Changsha 410083, China;5. School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China;1. Department of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China;2. Hubei Key Laboratory of Intelligent Wireless Communications, South-Central University for Nationalities, Wuhan 430074, China;3. Department of Informatics, University of Bergen, N-5020 Bergen, Norway |
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| Abstract: | We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group structure over generalized Pell hyperbolas (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields. |
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| Keywords: | ElGamal scheme Pell hyperbola Public cryptography Rédei function |
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