Given a number, produce another random number that is the same every time and distinct from all other results

Question

Basically, I would like help designing an algorithm that takes a given number, and returns a random number that is unrelated to the first number. The stipulations being that a) the given output number will always be the same for a similar input number, and b) within a certain range (ex. 1-100), all output numbers are distinct. ie., no two different input numbers under 100 will give the same output number.

I know it's easy to do by creating an ordered list of numbers, shuffling them randomly, and then returning the input's index. But I want to know if it can be done without any caching at all. Perhaps with some kind of hashing algorithm? Mostly the reason for this is that if the range of possible outputs were much larger, say 10000000000, then it would be ludicrous to generate an entire range of numbers and then shuffle them randomly, if you were only going to get a few results out of it.

Doesn't matter what language it's done in, I just want to know if it's possible. I've been thinking about this problem for a long time and I can't think of a solution besides the one I've already come up with.

Edit: I just had another idea; it would be interesting to have another algorithm that returned the reverse of the first one. Whether or not that's possible would be an interesting challenge to explore.

Answer 1

This sounds like a non-repeating random number generator. There are several possible approaches to this.

As described in this article, we can generate them by selecting a prime number p and satisfies p % 4 = 3 that is large enough (greater than the maximum value in the output range) and generate them this way:

int randomNumberUnique(int range_len , int p , int x)
    if(x * 2 < p)
       return (x * x) % p
    else
       return p - (x * x) % p

This algorithm will cover all values in [0 , p) for an input in range [0 , p).

Answer 2

Here's an example in C#:

    private void DoIt()
    {
        const long m = 101;
        const long x = 387420489; // must be coprime to m

        var multInv = MultiplicativeInverse(x, m);

        var nums = new HashSet<long>();
        for (long i = 0; i < 100; ++i)
        {
            var encoded = i*x%m;
            var decoded = encoded*multInv%m;
            Console.WriteLine("{0} => {1} => {2}", i, encoded, decoded);
            if (!nums.Add(encoded))
            {
                Console.WriteLine("Duplicate");
            }
        }
    }

    private long MultiplicativeInverse(long x, long modulus)
    {
        return ExtendedEuclideanDivision(x, modulus).Item1%modulus;
    }

    private static Tuple<long, long> ExtendedEuclideanDivision(long a, long b)
    {
        if (a < 0)
        {
            var result = ExtendedEuclideanDivision(-a, b);
            return Tuple.Create(-result.Item1, result.Item2);
        }
        if (b < 0)
        {
            var result = ExtendedEuclideanDivision(a, -b);
            return Tuple.Create(result.Item1, -result.Item2);
        }
        if (b == 0)
        {
            return Tuple.Create(1L, 0L);
        }
        var q = a/b;
        var r = a%b;
        var rslt = ExtendedEuclideanDivision(b, r);
        var s = rslt.Item1;
        var t = rslt.Item2;
        return Tuple.Create(t, s - q*t);
    }

That generates numbers in the range 0-100, from input in the range 0-100. Each input results in a unique output.

It also shows how to reverse the process, using the multiplicative inverse.

You can extend the range by increasing the value of m. x must be coprime with m.

Code cribbed from Eric Lippert's article, A practical use of multiplicative inverses, and a few of the previous articles in that series.

Answer 3

You can not have completely unrelated (particularly if you want the reverse as well).

There is a concept of modulo inverse of a number, but this would work only if the range number is a prime, eg. 100 will not work, you would need 101 (a prime). This can provide you a pseudo random number if you want.

Here is the concept of modulo inverse:

If there are two numbers a and b, such that

(a * b) % p = 1

where p is any number, then

a and b are modular inverses of each other.

For this to be true, if we have to find the modular inverse of a wrt a number p, then a and p must be co-prime, ie. gcd(a,p) = 1

So, for all numbers in a range to have modular inverses, the range bound must be a prime number.

A few outputs for range bound 101 will be:

1 == 1
2 == 51
3 == 34
4 == 76
etc.

EDIT:

Hey...actually you know, you can use the combined approach of modulo inverse and the method as defined by @Paul. Since every pair will be unique and all numbers will be covered, your random number can be:

random(k) = randomUniqueNumber(ModuloInverse(k), p)      //this is Paul's function