Abstract: A cryptographic method including selecting secret keys p and q, being prime numbers greater than 3, selecting public parameters for a series of data values which belong to one of a plurality of pairs of groups whereby any one of the data values in one of the pairs of groups is recovered by performing an operation kN.sub.i +1 times modulo n beginning with the any one of the data values, where k is an integer, N.sub.i is the order of the ith pair of groups and n=p.q, selecting a public encryption key e which is a factor of kN.sub.i +1 for all i, and processing communications data as a member of one of the pairs of groups by performing the operation on the communications data, whereby the order N.sub.i of the pair of groups i that the communications data belongs to can be determined on the basis of p and q, and a secret decryption key d.sub.i can be determined using e.d.sub.i =kN.sub.i +1.