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.

Pell hyperbolas in DLP-based cryptosystems / Alecci, Gessica; Dutto, Simone; Murru, Nadir. - In: FINITE FIELDS AND THEIR APPLICATIONS. - ISSN 1071-5797. - 84:(2022), pp. 10211201-10211218. [10.1016/j.ffa.2022.102112]

Pell hyperbolas in DLP-based cryptosystems

Murru, Nadir
2022-01-01

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.
2022
Alecci, Gessica; Dutto, Simone; Murru, Nadir
Pell hyperbolas in DLP-based cryptosystems / Alecci, Gessica; Dutto, Simone; Murru, Nadir. - In: FINITE FIELDS AND THEIR APPLICATIONS. - ISSN 1071-5797. - 84:(2022), pp. 10211201-10211218. [10.1016/j.ffa.2022.102112]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/364636
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