Public key cryptography with matrices

We discuss how public key cryptography can be achieved using simple multiplication of matrices over a given commutative ring. We propose a trapdoor function, which is one of the fastest one in the cryptographic literature known to the author. Using this trapdoor function we describe algorithms for key agreement and public key encryption whose security is based on solving a system of multivariate quadratic equations over the given commutative ring. This is the first public key cryptosystem with constant complexity (fixed number of multiplications) irrespective of the key size taken for the case of commutative ring of integers modulo a composite.