Pell hyperbolas in DLP–based cryptosystems

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.     

An algorithm for DLP on anomalous elliptic curves over Fp

This paper improves the method of discrete logarithm on anomalous elliptic curves, and establishes an isomorphism from E(Fp) to Fp which can be more easily implemented. Fruthermore, we give an optimized algorithm for discrete logarithm on anomalous elliptic curves E(Fp).     

An algorithm for DLP on anomalous elliptic curves over Fp

This paper improves the method of discrete logarithm on anomalous elliptic curves, and establishes an isomorphism from E(F_p) to F_p which can be more easily implemented. Fruthermore, we give an optimized algorithm for discrete logarithm on anomalous elliptic curves E(F_p).