Pell hyperbolas in DLP-based cryptosystems
This work addresses efficiency improvements in cryptography for secure communication systems, though it appears incremental as it builds on existing ElGamal schemes with a new parameterization.
The authors tackled the problem of improving efficiency in discrete logarithm problem (DLP)-based cryptosystems by introducing a parameterization using Pell hyperbolas, which leverages Rédei rational functions for fast evaluation. They developed three ElGamal-based cryptosystems that are shown to be more efficient than classical finite field schemes.
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's structure over generalized Pell conics (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.