A Group Law on the Projective Plane with Applications in Public Key Cryptography

Raúl Durán Díaz, Luis Hernández Encinas and Jaime Muñoz Masqué
Mathematics, 8(5), 734 (2020), 20pp., Special Issue "Mathematics Cryptography and Information Security (Q1, Mathematics, F.I. 1.105)

In the context of new threats to Public Key Cryptography arising from a growing computational power both in classic and in quantum worlds, we present a new group law defined on a subset of the projective plane $F$$P$2 over an arbitrary field $F$ , which lends itself to applications in Public Key Cryptography and turns out to be more efficient in terms of computational resources. In particular, we give explicitly the number of base field operations needed to perform the mentioned group law. Based on it, we present 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. We present an experimental setup in order to show real computation times along a comparison with the group operation in the group of points of an elliptic curve. Based on current state-of-the-art algorithms, we provide parameter ranges suitable for real world applications. Finally, we present a promising variant of the proposed group law, by moving from the base field $F$ to the ring $Z/pqZ$ , and we explain how the security becomes enhanced, though at the cost of a longer key length.