A method, system and apparatus for performing user identification, digital signatures and other secure communication functions in which keys are chosen essentially at random from a large set of vectors and key lengths are comparable to the key lengths in other common identification and digital signature schemes at comparable security levels. The signing technique of an embodiment of the identification/digital signature scheme hereof uses a mixing system based on multiplication in a ring and reduction modulo an ideal q in that ring; while the verification technique uses special properties of products of elements whose validity depends on elementary probability theory. The security of the identification/digital signature scheme comes from the interaction of reduction modulo q and the difficulty of forming products with special properties.