Cryptography from the tropical Hessian pencil

Abstract

Abstract Recent work by Grigoriev and Shpilrain [8] suggests looking at the tropical semiring for cryptographic schemes. In this contribution we explore the tropical analogue of the Hessian pencil of plane cubic curves as a source of group-based cryptography. Using elementary tropical geometry on the tropical Hessian curves, we derive the addition and doubling formulas induced from their Jacobian and investigate the discrete logarithm problem in this group. We show that the DLP is solvable when restricted to integral points on the tropical Hesse curve, and hence inadequate for cryptographic applications. Consideration of point duplication, however, provides instances of solvable chaotic maps producing random sequences and thus a source of fast keyed hash functions.