Dave Korn wrote:
Of course, I could point out that there is precisely *1* bit of information in that huge GIF, so even compressing it to 35 bytes isn't a great achievement... it's one of the set of less-common inputs that grow bigger as a compromise so that real pictures, which tend to have at least *some* variation, grow smaller.
OK, according to Shannon's definition of entropy, how much information is there in \pi or e or other transcendent number? AFAIK all finite strings are in fractional expansion of \pi (take positive integer base other than 10 if you want). Sure, the numbers are correlated, because we can express \pi or e as infinite series, though only couple of bytes is necessary.
But: if you take any finite string, there exists a finite natural number as index where the string can be found in \pi. So are there any "random strings" at all? What if I can express the digits starting at 2^(2^k)-th index analytically in a short, fast algorithm? In case no other can, then I have perfect one-time pad, at least for some time, because conventional computers will not reach the place in some reasonable time (yes, I know, there may be other correlations). The point I am trying to make: if I take e.g. a transcendent number and can analytically express a part of its fractional expansion, I *might* have a strong key.
The point for this: I am researching an authenticating scheme where keys are *programs*, not data. It means that key can change itself over time, for example. The strongest keys would therefore be somewhat recursive polymorphic code, that enumerates all possible permutations on some finite set in some deterministic, though seemingly random order.
I know that there are "short" descriptions for lots of infinite structures, the mentioned transcendent numbers, then take Mandelbrot set, various IFSs (iterated function systems), quaternions, unmeasurable sets, there are lots of possibilities. What I know for sure is that for many mathematical structures short descriptions (programs or (partially) recursive functions) exist. What I conjecture is that for each finite ordered set a short "compression function" exists. The whole trick is that it is almost impossible (meaning computationally infeasible) to look for the compression functions using a finite deterministic algorithm. Though a human can do it.
It will yet take some time until I have some more results about the categories of key programs.
Cheers, OM --------------------------------------------------------------------- The Cryptography Mailing List Unsubscribe by sending "unsubscribe cryptography" to [EMAIL PROTECTED]
