I am looking for the function $y=f(x)$ that would map the integer interval $[0,n)$ into itself $[0,n)$. The function must be bijective, so it is a permutation of n elements. It should "randomize" the input variable, ie.
- for every $x_1$ and $x_2$ that are close $|x_1 - x_2| < 100$ the outputs of the function should differ very much $|f(x_1) - f(x_2)| > n/{100}$
- changing any digit in the input should change most of digits in the output
- Shannon's Confusion and diffusion: http://en.wikipedia.org/wiki/Confusion_and_diffusion
- There should be very little or no fixed points (for every $x$: $f(x) \neq x$).
or any combination of the above conditions.
I don't need the inverse function. It should also be easy to compute from the closed form expression (a maximum of a few thousands simple computer operations like addition, multiplication, modulo etc.) for $n=10^{100}$.
I found some examples in Google:
- Linear congruential generator. Equation:
$X_{n+1} = (aX_n + c) \mod m$
Wikipedia states that: "Provided that the offset $c$ is nonzero, the LCG will have a full period for all seed values if and only if:
- $c$ and $m$ are relatively prime,
- $a - 1$ is divisible by all prime factors of $m$,
- $a - 1$ is a multiple of 4 if $m$ is a multiple of 4.
I've tried using another function $f(x) = (ax + c) \mod m$, because I wanted to get rid of the recursion, and as long as $m$ is prime and $a \ge 1$ I get the permutation. However the result is not that "wild" or "chaotic" as I expect it to be.
- Minimal perfect hash function
The concept is similar to what I'm looking for, but if I understand correctly the computation of this kind of function is very complicated and time-consuming and it cannot be expressed in a closed form as a simple expression for $n=10^{100}$.
- Substitution-permutation network
I am not sure how to apply S-boxes and P-boxes to the interval [0,n), but maybe it could be used.
The question is: Could you give me some examples of functions that satisfy the above conditions (or some pointers like the mathematical term for the object I am looking for)?
