On 06/30/2014 06:37 PM, Herman Rubin wrote: > On 2014-06-29, Vincent Granville <firstname.lastname@example.org> wrote: > >> Most random number generators use an algorithm a(k+1) = f(a(k)) > to produce a sequence of integers a(1), a(2), etc. that behaves like > random numbers. The function f is integer-valued and bounded; because > of these two conditions, the sequence a(k) eventually becomes periodic > for k large enough. This is an undesirable property, and many public > random number generators (those built in Excel, Python, and other > languages) are poor and not suitable for cryptographic applications, > Markov Chains Monte-Carlo associated with hierarchical Bayesian models, > or large-scale Monte-Carlo simulations to detect extreme events (example: > fraud detection, big data context). > >> This challenge involves identifying natural transcendental numbers with > expansions (series, continued fractions) converging very fast to easily > produce the decimals of the numbers in question. It starts with Log 10 - > 2 Log 3 an a series that produces one new digit for each new term. > >> In this challenge, you will learn how to build non-periodic, > high-quality random number generators, test the strength of your random > number generator, and design generators that are easy to implement, fit > for big data and extreme events simulation, and run fast. Dr. Vincent > Granville has promised to participate, so this should add more fun to > this challenge. > > The above condition, combined with the standard mathematical definition > of a random number, is impossible. If one can compute the number ti a > specified accuracy in a finite number of steps, one can easily give > a stopping rule so that the digits at the stopped position are not > asymptotically uniform. But what does asymptotic tell one about the > first 10^10 anyhow? > > I mistrust all computer genrated sequences as random; von Neumann > stated that to even think of using systematic numbers as random > puts one in a state of sin. For SOME purposes of computing integrals > over the unit k-cell, there are what are called quasi-random numbers, > which will, at least asymptotically, do better than random numbers > for sufficiently smooth functions. > > What I have suggested is that one take a "good" pseudo-random sequence, > possibly using some of Mr. Granville's suggestions, as a binary sequence > and exclusive OR it with a reasonable random sequence. The pseudo-random > part will guarantee the uniformity and more, and the physical random numbers > will break up any periodicities of the pseudo-random ones. > > >> Here's the URL: > http://www.analyticbridge.com/forum/topics/challenge-of-the-week-random-numbers > >
I take your idea seriously:
mix: (a) "strong" pseudo-random bit generator (deterministic) with: (b) a "high-entropy" source of (what else?) physically generated "random bits" . (non-deterministic).