Algorithm for generating a three dimensional random number space

Question

I am looking for an algorithm to generate pseudo random numbers in a three (or better n) dimensional space of large extents. When initialized with a seed the generator should be able to repeatedly produce the same numbers for the same seed.

But unlike most generators available in programming languages, it should not just return the next random number in a sequence, but instead generate numbers for specific coordinates, no matter in what sequence the values are requested.

The size of the space should be regarded as too big to generate all the numbers at initialization time. In Java it could look something like this:

Random3D gen = new Random3D(seed);
int n1 = gen.getInt(3,0,6);
int n2 = gen.getInt(2,-3,1);
...

How would I do something like this?

I tried it in Java by writing some code using java.util.Random, but the quality of the results was not very good.

Answer 1

If you want to receive always the same result for the same coordenate, when you specify the seed, them you're not looking for a true random generator.

You want a fast algorithm, a reliable one? For a fast one, that a look at Mersenne twister . For a stronger one, you could look Blum Blum Shub .

You could use your n-dimensional coordinates, plus your seed, to generate the pseudo-random-number-generator. You could, for example, calculate the sha1 or md5 or any other hash of the coordinates + seed and use it in the PRNG.

Edit: for a simple solution, the math.random can receive a seed of 48 bits (smaller than the md5 output), which can be a bit small for your question (you mentionated to have high dimensionality, right? with larges coordinates?)

Answer 2

I believe you can solve your problem by using your input numbers as values used to calculate the seed which is passed into a standard RNG; if I read your question right, you want the SAME "random" number resultant from the same input, which this solution would provide.

Answer 3

I implemented the idea given by the answer from woliveirajr : using the Blum Blum Shub pseudo random number generator in its explicit (non-iterative) form, together with a message digest to produce the right index from the arguments.

(You also can take this source from my github repository .)

package de.fencing_game.paul.examples;

import java.math.BigInteger;
import java.security.MessageDigest;
import java.security.NoSuchAlgorithmException;
import java.util.Random;


/**
 * A pseudo random number generator, which does not
 * produce a series of numbers, but each number determined by
 * some input (and independent of earlier numbers).
 *<p>
 * This is based on the
 * <a href="http://en.wikipedia.org/wiki/Blum_Blum_Shub">Blum Blum Shub
 *  algorithm</a>, combined with the SHA-1 message digest to get the
 *  right index.
 *</p>
 *<p>
 * Inspired by the question
 *  <a href="https://stackoverflow.com/q/6586042/600500">Algorithm
 *   for generating a three dimensional random number space</a> on
 * Stack Overflow, and the answer from woliveirajr.
 */
public class PseudoRandom {

    /**
     * An instance of this class represents a range of
     * integer numbers, both endpoints inclusive.
     */
    public static final class Range {

        public int min;
        public int max;

        public Range(int min, int max) {
            this.min = min;
            this.max = max;
        }

        /**
         * clips a (positive) BigInteger to the range represented
         * by this object.
         * @returns an integer between min and max, inclusive.
         */
        final int clip(BigInteger bigVal) {
            BigInteger modulus =
                BigInteger.valueOf(max + 1L - min);
            return (int)(min + bigVal.mod(modulus).longValue());
        }
    }


    /* M = p * q =
       510458987753305598818664158496165644577818051165198667838943583049282929852810917684801057127 *
       1776854827630587786961501611493551956300146782768206322414884019587349631246969724030273647
     */
    /**
     * A big number, composed of two large primes.
     */
    private static final BigInteger M =
        new BigInteger("90701151669688414188903413878244126959941449657"+
                       "82009133495922185615411523457607691918744187485"+
                       "10492533485214517262505932675573506751182663319"+
                       "285975046876611245165890299147416689632169");

    /* λ(M) = lcm(p-1, q-1) */
    /**
     * The value of λ(M), where λ is the Carmichael function.
     * This is the lowest common multiple of the predecessors of
     * the two factors of M.
     */
    private static final BigInteger lambdaM =
        new BigInteger("53505758348442070944517069391220634799707248289"+
                       "10045667479610928077057617288038459593720911813"+
                       "73249762745139558184229125081884863164923576762"+
                       "05906844204771187443203120630003929150698");

    /**
     * The number 2 as a BigInteger, for use in the calculations.
     */
    private static final BigInteger TWO = BigInteger.valueOf(2);



    /**
     * the modular square of the seed value.
     */
    private BigInteger s_0;

    /**
     * The MessageDigest used to convert input data
     * to an index for our PRNG.
     */
    private MessageDigest md;



    /**
     * Creates a new PseudoRandom instance, using the given seed.
     */
    public PseudoRandom(BigInteger seed) {
        try {
            this.md = MessageDigest.getInstance("SHA-1");
        }
        catch(NoSuchAlgorithmException ex) {
            throw new RuntimeException(ex);
        }
        initializeSeed(seed);
    }

    /**
     * Creates a new PseudoRandom instance, seeded by the given seed.
     */
    public PseudoRandom(byte[] seed) {
        this(new BigInteger(1, seed));
    }

    /**
     * Creates a new PseudoRandom instance,
     * seeded by the current system time.
     */
    public PseudoRandom() {
        this(BigInteger.valueOf(System.currentTimeMillis()));
    }

    /**
     * Transforms the initial seed into some value that is
     * usable by the generator. (This is completely deterministic.)
     */
    private void initializeSeed(BigInteger proposal) {

        // we want our seed be big enough so s^2 > M.
        BigInteger s = proposal;
        while(s.bitLength() <= M.bitLength()/2) {
            s = s.shiftLeft(10);
        }
        // we want gcd(s, M) = 1
        while(!M.gcd(s).equals(BigInteger.ONE)) {
            s = s.add(BigInteger.ONE);
        }
        // we save s_0 = s^2 mod M
        this.s_0 = s.multiply(s).mod(M);
    }

    /**
     * calculates {@code x_k = r.clip( s_k )}.
     */
    private int calculate(Range r, BigInteger k) {
        BigInteger exp = TWO.modPow(k, lambdaM);
        BigInteger s_k = s_0.modPow(exp, M);
        return r.clip(s_k);
    }


    /**
     * returns a number given by a range, determined by the given input.
     */
    public int getNumber(Range r, byte[] input) {
        byte[] dig;
        synchronized(md) {
            md.reset();
            md.update(input);
            dig =  md.digest();
        }
        return calculate(r, new BigInteger(1, dig));
    }


    /**
     * returns a number given by a range, determined by the given input.
     */
    public int getNumber(Range r, int... input) {
        byte[] dig;
        synchronized(md) {
            md.reset();
            for(int i : input) {
                md.update(new byte[]{ (byte)(i >> 24), (byte)(i >> 16),
                                      (byte)(i >> 8), (byte)(i >> 0)} );
            }
            dig = md.digest();
        }
        return calculate(r, new BigInteger(1, dig));
    }

    /**
     * Test method.
     */
    public static void main(String[] test) {
        PseudoRandom pr = new PseudoRandom("Hallo Welt".getBytes());

        Range r = new Range(10, 30);
        for(int i = 0; i < 10; i++) {
            System.out.println("x("+i+") = " + pr.getNumber(r, i));
        }
        for(int i = 0; i < 5; i++) {
            for(int j = 0; j < 5; j++) {
                System.out.println("x("+i+", "+j+") = " +
                                   pr.getNumber(r, i, j));
            }
        }
        // to show that it really is deterministic:
        for(int i = 0; i < 10; i++) {
            System.out.println("x("+i+") = " + pr.getNumber(r, i));
        }
    }
}

I arbitrarily selected these big prime numbers - I don't know if they are really cryptographically secure (eg whether p-1 and q-1 have the necessary factorization properties). If you really need security, you should keep these numbers secret (eg generate them yourself).

Also, I use the input seed to generate s (and s_0 ) - instead one could have used a fixed s (with known good properties, like a large period), and use the seed as input to the message digest (together with the input I'm using here).

Of course, one also could have directly used the message digest's output, instead of using it only as an index to BBS.