pseudo random distribution which guarantees all possible permutations of value sequence - C++

Random question.

I am attempting to create a program which would generate a pseudo-random distribution. I am trying to find the right pseudo-random algorithm for my needs. These are my concerns:

1) I need one input to generate the same output every time it is used.

2) It needs to be random enough that a person who looks at the output from input 1 sees no connection between that and the output from input 2 (etc.), but there is no need for it to be cryptographically secure or truly random.

3)Its output should be a number between 0 and (29^3200)-1, with every possible integer in that range a possible and equally (or close to it) likely output.

4) I would like to be able to guarantee that every possible permutation of sequences of 410 outputs is also a potential output of consecutive inputs. In other words, all the possible groupings of 410 integers between 0 and (29^3200)-1 should be potential outputs of sequential inputs.

5) I would like the function to be invertible, so that I could take an integer, or a series of integers, and say which input or series of inputs would produce that result.

The method I have developed so far is to run the input through a simple halson sequence:

while (input>0) {
    denominator *=3;
    numerator = numerator * 3 + (input%3);
    input = input/3;
}

and multiply the result by 29^3200. It meets requirements 1-3, but not 4. And it is invertible only for single integers, not for series (since not all sequences can be produced by it). I am working in C++, using boost multiprecision.

Any advice someone can give me concerning a way to generate a random distribution meeting these requirements, or just a class of algorithms worth researching towards this end, would be greatly appreciated. Thank you in advance for considering my question.