A mathematical explanation of the DES encryption system I need a mathematical explanation of what does the DES encryption system really do.

This means I need more explanation than the one that offers FIPS, which is more an explanation for computer specialists.

Among other things, I want to know where do these permutation tables come from:
$$IP\\
\newcommand\T{\Rule{0pt}{1em}{.3em}}
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
  58 \T &amp; 50 \T &amp; 42 \T &amp; 34 \T &amp; 26 \T &amp; 18 \T &amp; 10 \T &amp;  2 \\\hline
  60 \T &amp; 52 \T &amp; 44 \T &amp; 36 \T &amp; 28 \T &amp; 20 \T &amp; 12 \T &amp;  4 \\\hline
  62 \T &amp; 54 \T &amp; 46 \T &amp; 38 \T &amp; 30 \T &amp; 22 \T &amp; 14 \T &amp;  6 \\\hline
  64 \T &amp; 56 \T &amp; 48 \T &amp; 40 \T &amp; 32 \T &amp; 24 \T &amp; 16 \T &amp;  8 \\\hline
  57 \T &amp; 49 \T &amp; 41 \T &amp; 33 \T &amp; 25 \T &amp; 17 \T &amp;  9 \T &amp;  1 \\\hline
  59 \T &amp; 51 \T &amp; 43 \T &amp; 35 \T &amp; 27 \T &amp; 19 \T &amp; 11 \T &amp;  3 \\\hline
  61 \T &amp; 53 \T &amp; 45 \T &amp; 37 \T &amp; 29 \T &amp; 21 \T &amp; 13 \T &amp;  5 \\\hline
  63 \T &amp; 55 \T &amp; 47 \T &amp; 39 \T &amp; 31 \T &amp; 23 \T &amp; 15 \T &amp;  7 \\\hline
\end{array}
$$

I know this kind of encryption systems have a lot to do with the mathematical concept of a field.

So I would like to know if someones has a PDF or any other real mathematical explanation of the DES encryption system.
The DES standard (FIPS 46-3) is actually a rather straightforward description of DES. It tells with precision and detail where each bit goes. It is a specification for implementers (who can be thought as "computer specialists" but anybody who wants to learn about DES should be able to understand that specification). What FIPS 46-3 does not tell is why DES was designed that way. If you want more mathematics, you can have a look at the Handbook of Applied Cryptography (free download !), in particular chapter 7, which goes over the case of DES.

For the initial and final permutations (called "IP" in FIPS 46-3), they were defined not for security (they are fixed, key-less permutations which anyone can readily invert), but to ease implementations in hardware contexts of the 1970s' era: they make it easier to make an hardware implementation which plugs on an 8-bit bus. See this question for some details.

There is nothing about fields in DES. It is all bit-by-bit manipulations; to some extent you could say that a bit is a value in GF(2), the field with two elements (0 and 1), but that's quite stretching it.