Cryptanalysis of a key exchange protocol based on the endomorphisms ring End $${(\mathbb{Z}_{p} \times \mathbb{Z}_{p^2})}$$

Abstract

Climent et al. (Appl Algebra Eng Commun Comput 22:91–108, 2011) identified the elements of the endomorphisms ring End\({(\mathbb{Z}_p \times \mathbb{Z}_{p^2})}\) with elements in a set, E p , of matrices of size 2 × 2, whose elements in the first row belong to \({\mathbb{Z}_{p}}\) and the elements in the second row belong to \({\mathbb{Z}_{p^2}}\). By taking advantage of matrix arithmetic, they proposed a key exchange protocol using polynomial functions over E p defined by polynomials in \({\mathbb{Z}[X]}\). In this note, we show that this protocol is insecure; it can be broken by solving a set of 10 consistent homogeneous linear equations in 8 unknowns over \({\mathbb{Z}_{p^2}}\).