Permutation Domain Research Articles

Fast Monte Carlo Algorithms for Permutation Groups

We introduce new, elementary Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups (given by a list of generators). The methods are of a combinatorial character, using only elementary group theory. The key idea is that under certain conditions, "random subproducts" of the generators successfully emulate truly random elements of a group. We achieve a nearly optimal O(n3 logcn) asymptotic running time for membership testing, where n is the size of the permutation domain. This is an improvement of two orders of magnitude compared to known elementary algorithms and one order of magnitude compared to algorithms which depend on heavy use of group theory. An even greater asymptotic speedup is achieved for normal closures, a key ingredient in group-theoretic computation, now constructible in Monte Carlo time O(n2 logcn), i.e., essentially linear time (as a function of the input length). Some of the new techniques are sufficiently general to allow polynomial-time implementations in the very general model of "black box groups" (group operations are performed by an oracle). In particular, the normal closure algorithm has a number of applications to matrix-group computation. It should be stressed that our randomized algorithms are not heuristic: the probability of error is guaranteed not to exceed a bound ϵ > 0, prescribed by the user. The cost of this requirement is a factor of |log ϵ| in the running time.

A Random Base Change Algorithm for Permutation Groups

A new random base change algorithm is presented for a permutation group G acting on n points whose worst case asymptotic running time is better for groups with a small to moderate size base than any known deterministic algorithm. To achieve this time bound, the algorithm requires a random generator Rand(G) producing a random element of G with the uniform distribution and so that the time for each call to Rand (G) is bounded by some function f(n, G). The random base change algorithm has probability 1 - 1/|G| of completing in time O(f(n, G) log |G|) and outputting a data structure for representing the point stabilizer sequence relative to the new ordering. The algorithm requires O(n log |G|) space and the data structure produced can be used to test group membership in time O(n log |G|). Since the output of this algorithm is a data structure allowing generation of random group elements in time O(n log |G|), repeated application of the random base change algorithms for different orderings of the permutation domain of G will always run in time O (n log2 |G|). An earlier version of this work appeared in Cooperman and Finkelstein (1992b).

Arrovian theorems for economic domains: Assignments, matchings and pairings

In this paper, we show that Arrow's and Wilson's theorems hold for permutation, symmetric permutation and symmetric domains. Permutation domains concern assignments: n indivisible private goods are distributed betweenn selfish individuals. Symmetric permutation domains concern matchings: two sets ofn selfish individuals being given, pairs with an individual from each set are to be made. Symmetric domains concern pairings: a set of 2n selfish individuals being given,n pairs of individuals are to be made.

Speech processing techniques for HF radio security

The paper presents a review of the techniques used to date to provide secure duplex or half-duplex conversational speech communications over HF radio channels, and gives an assessment of likely future developments. The paper begins with a brief discussion of the mechanisms of speech production and hearing, with particular reference to the factors which are important for communications over the difficult HF medium. Various methods for speech scrambling based on time and frequency domain permutations are then described, and the effects of HF channel distortions on their speech quality performance is assessed. Digital vocoding techniques based on the long established channel vocoder and the more recent linear predictive coder are described, and their performance advantages over scrambling techniques are discussed. The importance of the type of modem employed in conjunction with vocoders is emphasised, particularly with respect to the techniques used to overcome the effects of fading and cochannel interference. In the future, continuing developments in semiconductor technology will have a significant impact on the amount of processing which can be performed within moderately priced equipment. The paper is concluded with some prognostications as to how the techniques used to provide narrowband secure speech will change to yield improved performance, both in terms of the speech quality and the reliability of obtaining communications over HF channels.