Pi

Phillip H. Zakas pzakas at toucancapital.com
Thu Aug 2 13:05:35 PDT 2001


this is truly interesting...do you have a link to the original 1996 paper?
do you know if anyone has incorporated this into a program?
phillip

> -----Original Message-----
> From: owner-cypherpunks at Algebra.COM
> [mailto:owner-cypherpunks at Algebra.COM]On Behalf Of Eric Cordian
> Sent: Thursday, August 02, 2001 2:35 PM
> To: cypherpunks at 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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