A system effective to communicate a message between two devices. A first device may include a plaintext to monoid element module effective to receive a plaintext message and apply a first function to the plaintext message to produce a first monoid element. A monoid element evaluator module may be effective to receive and insert submonoid generators into a monoid expression to produce a second monoid element in response. An encryption device module may be effective to apply a second function to the first monoid element, the second monoid element, the monoid expression, and a third monoid element to produce an encrypted plaintext message. Decryption may be performed on the encrypted plaintext message knowing the private key which includes the first function, the second function, the third monoid element and the submonoid generators list.