Nick Szabo's Papers and Concise Tutorials

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The ideas of "absolute compression" has on a Turing machine the same meaning as the idea of "truly random numbers", and for the same reason. The assumption of randomness used in proving that one-time pads and other protocols are "unconditionally" secure is very similar to the assumption that a string is "absolutely compressed". The problem is that determining the absolute entropy of a string, as well as t he equivalent problem of determining whether it is "real random", is both uncomputable and language-dependent.

Empirically, it seems likely that generating truly random numbers is much more practical than absolute compression. If one has access to certain well-observed physical phenomena, one can make highly confident, if still mathematically unproven, assumptions of "true randomness", but said phenomena don't help with absolute compression.

If we restrict ourselves to Turing machines, we can do something *close* to absolute compression and tests of true randomness -- but not quite. And *very* slow. From a better physical source there is still the problem that if we can't sufficiently test them, how can we be so confident they are random anyway? Such assumptions are based on the extensive and various, but imperfect, statistical tests physicists have done (has anybody tried cryptanalyzing radioactive decay? :-)

We can come close to testing for true randomness and and doing absolute compression on a Turing machine. For example, here is an algorithm that, for sufficiently long but finite number of steps t, will *probably* give you the absolute compression (I believe the probability converges on a number related to Chaitin's "Omega" halting probability as t grows, but don't quote me -- this would make an interesting research topic).

probably_absolute_compress(data,t) { for all binary programs smaller than data { run program until it halts or it has run for time t if (output of program == data AND length(program) < length(shortest_program)) { shortest_program = program } } print "the data: ", data print "the (probably) absolute compression of the data", shortest_program return shortest_program }(We have to makes some reasonable assumption about what the binary programming language is -- see below).

We can then use our probably-absolute compression algorithm as a statstical test of randomness as follows:

probably_true_random_test(data,t) { if length(probably_absolute_compress(data,t)) = length(data) then print "data is probably random" else print "pattern found, data is not random" }We can't *prove* that we've found the absolute compression. However, I bet we can get a good idea of the *probability* that we've found the absolute compression by examining this algorithm in terms of the algorithmic probability of the data and Chaitin's halting probability.

Nor is the above algorithm efficient. Similarly, you can't prove that you've found truly random numbers, nor is it efficient to generate such numbers on a Turing machine. (Pseudorandom numbers are another story, and numbers derived from non-Turing physical sources are another story).

We can distill probably-true-random numbers from data of sufficient entropy as follows:

probably_true_random_distill(data,t) { return probably_absolute_compress(seed,t) }For cryptographic applications there are two important ideas, one-wayness and expanding rather than contracting the seed, that are not captured here. Probably_true_random_distill is more like the idea of hashing an imperfect entropy source to get the "core" entropy one believes exists in it. Only probably_true_random_distill far more reliable, as one can actually formally analyze the probability of having generated truly random numbers. It is, alas, much slower than hashing. :-(

Back to the theoretical point about whether there is such a thing as "absolute" entropy or compression. The Kolmogorov complexity (the smallest program that, when run, produce the decompressed data) is clearly defined and fully general for Turing machines. If we could determine the Kolmogorov complexity we wouldn't need to invoke any probability distribution to determine the absolute minimum possible entropy of any data to be compressed on a Turning machine.

It is, alas, uncomputable. To find the Kolmogorov complexity we could simply search through the space of all programs smaller than the data. But due to the halting problem we cannot always be certain that there does not exist a smaller program that, run for a sufficiently long period of time, will produce the decompressed data. When we can't prove that there is no smaller program than the data which generates the data, we also can't prove that there is not a pattern hidden in the data which makes it less than "truly random". The finite version of this search process, in the program probably_perfect_compression, circumvents the halting problem by arbitrarily halting programs that have already run for t steps.

Also, since the length of the program depends on what language it's written in, absolute Kolmogorov complexity is good only for analyzing growth rates. The choice of language adds a constant length to the program. We'd have to look at probably_perfect_compression in this context to see if the choice of binary language is a reasonable one or if other languages would give better compressions on the data we are likely to encounter.

One consequence is that one-time pads themselves have a big problem when we assume "truly random" numbers. This assumption is, in terms of the provability of security, no weaker or stronger than than an assumption of "perfect compression". (Which assumption is more practical is a different question -- as per above, if one has access to certain well-observed physical phenomena, one can make highly confident, if still mathematically unproven, assumptions of "true randomness", but said phenomena don't help with perfect compression). Similar problems occur in other designs for cryptographic protocols in which statistical tests are abused.

Nick Szabo's Papers and Concise Tutorials