At 11:34 AM 8/2/2001 -0700, Eric Cordian wrote:
Interesting article recently posted on the Nature Web site about the normality of Pi.
http://www.nature.com/nsu/010802/010802-9.html
"David Bailey of Lawrence Berkeley National Laboratory in California and Richard Crandall of Reed College in Portland, Oregon, present evidence that pi's decimal expansion contains every string of whole numbers. They also suggest that all strings of the same length appear in pi with the same frequency: 87,435 appears as often as 30,752, and 451 as often as 862, a property known as normality."
Of cryptographic interest.
"While there may be no cosmic message lurking in pi's digits, if they are random they could be used to encrypt other messages as follows:
"Convert a message into zeros and ones, choose a string of digits somewhere in the decimal expansion of pi, and encode the message by adding the digits of pi to the digits of the message string, one after another. Only a person who knows the chosen starting point in pi's expansion will be able to decode the message."
While there's presently no known formula which generates decimal digits of Pi starting from a particular point, there's a clever formula which can be used to generate HEX digits of Pi starting from anywhere, which Bailey et al discovered in 1996, using the PSLQ linear relation algorithm.
I tried to something like this in the late '80s to allow efficient loss-less compression using "conditioned" PRNs which could generate suitable auto correlated streams. Unfortunately, I did not discover a similar method of locating the desired sequences. The search during the compression phase was to computationally difficult and I abandoned the effort. steve