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

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, Ep, 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 Ep 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 $${\mathbb{Z}_{p^2}}$$.