A group law on the projective plane with applications in Public Key Cryptography

Abstract

We present a new group law defined on a subset of the projective plane $\mathbb{F}{P}^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. Finally, we present a variant of the proposed group law but over the ring $\mathbb{Z}/pq\mathbb{Z}$, and explain how the security becomes enhanced, though at the cost of a longer key length.

* Updated abstract. * Updated security considerations in section 3. * Added brand new section 4, considering an analogous cryptosystem over a ring, thus adding security. * Update conclusions, taking the new section into account. * Updated references. * Corrected typos