Friday, 5 April 2013

..the finite field notion, the "wraparound" feature .. finite fields the systems existing

Finite fields the role of parameters, of constants, prediction.

In Ben Goertzel's 'Chaotic Logic', page 35, Chapter 3.4.1 Discrete Logarithms, mentions about finite fields. Its intuitive content generates ideas in my mind.

"3.4.1. Discrete Logarithms (*)
For a simple mathematical example, let us look to the theory of finite fields. A finite field is a way of doing arithmetic on a bounded set of integers. For instance, suppose one takes the field of size 13 (the size must be a prime or a prime raised to some power). Then, in this field the largest number is 12. One has, for example, 12 + 1 = 0, 10 + 5 = 2, 3 x 5 = 2, and 8 x 3 = 12. One can do
division in a finite field as well, although the results are often counterintuitive -- for instance, 12/8 = 3, and 2/3 = 5 (to see why, just multiply both sides by the denominator).
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..the finite field notion, the "wraparound" feature .. finite fields the systems existing .. for all systems possible and plausible .. physical but not only .. mental, social, psychological, emotional .. all to be seen as finite fields .. scale being not the limit .. and prediction worked out in the same way it is described in this chapter of Ben Goerzel's, chaotic logic.

The size of the fields given

"In finite field theory there is something called the "discrete logarithm" of a number, written dlogb(n). The discrete logarithm is defined just like the ordinary logarithm, as the inverse of exponentiation. But in a finite field, exponentiation must be defined in terms of the "wraparound" arithmetic illustrated in the previous paragraph. For instance, in the field of size 7, 34 = 4. Thus one has dlog3(4) = 4. But how could one compute the log base 3 of 4, without knowing what it was? The powers of 3 can wrap around the value 7 again and again -- they could wrap around many times before hitting on the correct value, 4.
The problem of finding the discrete logarithm of a number is theoretically easy, in the sense that there are only finitely many possibilities. In our simple example, all one has to do is take 3 to higher and higher powers, until all possibilities are covered. But in practice, if the size of the field is not 7 but some much larger number, this finite number of possibilities can become prohibitively large.
So, what if one defines the dynamical system nk = dlogb(nk-1)? Suppose one is given n1, then how can one predict n1000? So far as we know today, there is better way than to proceed in order: first get n2, then n3, then n4, and so on up to n999 and n1000. Working on n3 before one knows n2 is essentially useless, because a slight change in the answer for n2 can totally change the answer for n3. The only way to do all 1000 steps in parallel, it seems, would be to first compute a table of all
possible powers that one might possibly need to know in the course of calculation. But this would require an immense number of processors; at least the square of the size of the field.
This example is, incidentally, of more than academic interest. Many cryptosystems in current use are reliant on discrete logarithms. If one could devise a quick method for computing them, one could crack all manner of codes; and the coding theorists would have to come up with something better.
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