Simple extractors via constructions of cryptographic pseudo-random generators

Trevisan has shown that constructions of pseudo-random generators from hard functions (the Nisan-Wigderson approach) also produce extractors. We show that constructions of pseudo-random generators from one-way permutations (the Blum-Micali-Yao approach) can be used for building extractors as well. Using this new technique we build extractors that do not use designs and polynomial-based error-correcting codes and that are very simple and efficient. For example, one extractor produces each output bit separately in $O(\log^2 n)$ time. These extractors work for weak sources with min entropy $\lambda n$, for arbitrary constant $\lambda > 0$, have seed length $O(\log^2 n)$, and their output length is $\approx n^{\lambda/3}$.Comment: 21 pages, an extended abstract will appear in Proc. ICALP 2005; small corrections, some comments and references adde

Some comments on C. S. Wallace's random number generators

We outline some of Chris Wallace's contributions to pseudo-random number generation. In particular, we consider his idea for generating normally distributed variates without relying on a source of uniform random numbers, and compare it with more conventional methods for generating normal random numbers. Implementations of Wallace's idea can be very fast (approximately as fast as good uniform generators). We discuss the statistical quality of the output, and mention how certain pitfalls can be avoided.Comment: 13 pages. For further information, see http://wwwmaths.anu.edu.au/~brent/pub/pub213.htm

Measuring sets in infinite groups

We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a property P with probability p". The validity of a statement like that does, of course, heavily depend on how one defines probability on groups, or, equivalently, how one measures sets in a group (in particular, in a free group). We hope that new approaches to defining probabilities on groups outlined in this paper create, among other things, an appropriate framework for the study of the "average case" complexity of algorithms on groups.Comment: 22 page

Pseudo noise code and data transmission method and apparatus

Pseudo noise ranging codes, having a predetermined chipping rate, and a pair of binary data sources, each having a bit rate no greater than one tenth the chipping rate, quadriphase, digitally modulate a suppressed carrier wave having a first frequency are examined. Two additional binary data sources, each having a bit rate that is not restricted by the chipping rate of the first carrier, quadriphase, digitally modulate a suppressed carrier wave having a second frequency. The first and second frequencies are only slightly displaced so that there is overlap in the frequency bands which modulate the two carriers. The two suppressed carrier waves are linearly combined and transmitted from a first station to a second station so that the amplitude of the transmitted first wave is controlled so as not to degrade the detectability of the second wave at the second station