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Date: Thu, 2 Aug 2001 11:34:57 -0700 (PDT)

From: Eric Cordian <emc@artifact.psychedelic.net>

To: cypherpunks@einstein.ssz.com

Subject: Pi

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.

If you sum the following series for k=0 to k=infinity, its limit is Pi.

1/16^k[4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6)]

(Exercise: Prove this series sums to Pi)

Since this is an expression for Pi in inverse powers of 16, it is easy to

multiply this series by 16^d and take the fractional part, evaluating

terms where d>k by modular exponentiation, and evaluating a couple of

terms where d<k to get a digit's worth of precision, yielding the (d+1)th

hexadecimal digit of Pi.

Presumedly, if one could express PI as a series in inverse powers of 10,

one could do the same trick to get decimal digits. Such a series has so

far eluded researchers.

-- Eric Michael Cordian 0+ O:.T:.O:. Mathematical Munitions Division "Do What Thou Wilt Shall Be The Whole Of The Law"

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