a ring of polynomials with integer coefficients modulo the ideal generated by the polynomial x n u2212 1 an exemplary ideal of r is the ideal generated by an integer q and an exemplary quotient of r is the ring r qr z qz x x n u2212 1 of polynomials with coefficients in the finite ring z qz modulo the ideal generated by the polynomial x n u2212 1 the identification and digital signature techniques make use of the multiplication rule in the ring r given a polynomial a x a 0 a 1 x a n u2212 1 x n u2212 1 in r and a polynomial b x b 0 b 1 x b n u2212 1 x n u2212 1 in r an exemplary product may be given by c x a x b x c 0 c 1 x c n u2212 1 x n u2212 1 c i a 0 b i a 1 b i u2212 1 a i b 0 a i 1 b n u2212 1 a i 2 b n u2212 2 a n u2212 1 b i 1 this exemplary product is known as a convolution product all reference to multiplication of polynomials in the remaining description should be understood to refer to the above described exemplary multiplication in r it should also be noted that the above described multiplication rule is not a requirement of the invention and alternative embodiments may use other types of multiplication rules in other types of rings an exemplary product in the ring r is h x f 1 x g x where f x is selected from a set of constrained polynomials r f and g x is selected from a set.