Pell hyperbolas in DLP–based cryptosystems

We present a study on the use of Pell hyperbolas in cryptosystems with security based discrete logarithm problem. Specifically, after introducing group structure over generalized (and also giving explicit isomorphisms classical hyperbolas), we provide parameterization both an algebraic and geometrical approach. The particular that propose appears to be useful from cryptographic point view because product arises set parameters is connected Rédei rational functions, which can evaluated fast way. Thus, exploit these constructions for defining three different public key ElGamal scheme. show our allows obtain schemes more efficient than ones finite fields.