On Rotation Group and Encryption of Analog Signals

The problem of generating randon elements in groups has direct application to cryptography. For instance, we like to know whether the DES permutations are randon permutations of the 264 possible 64-bit words. The whole symmetric group is known to be generated by certain k-functions (7). Another example is the Wyner voice encryption scheme (12) which requires the production of large numbers of random real orthogonal matrices. N. J. A. Sloane has given a survey on this problem (11) which has led to the following questions for a given group G: 1.How does one generate elements of G at random?2.How can one test if certain given elements of G really are random?3.Does a given subset H generate the whole group G?4.If so, how long does it take? In this paper, we consider these questions for the orthogonal group O(d) for any positive integer d. By looking at the Lie algebra o(d) of O(d) and one-parameter subgroups of O(d), we can find the generation of an arbitrary element in terms of one-parameter rotation groups in uniform fashion. The length of generation can be determined. Random elements are generated using random number generator on the real parameter space of each one-parameter subgroup.