Question

I have a use case where I need to scramble an input in such a way that:

1. Each specific input always maps to a specific pseudo-random output.
2. The output must shuffle the input sufficiently so that an incrementing input maps to a pseudo-random output.

For example, if the input is 64 bits, there must be exactly 2^64 unique outputs, and these must break incrementing inputs as much as possible (arbitrary requirement).

I will code this in C#, but can translate from Java or C, so long as there are not SIMD intrinsics. What I am looking for is some already existing code, rather than reinventing the wheel.

I have looked on Google, but haven’t found anything that does a 1:1 mapping.

Answer 1

This seems to work fairly well:

const long multiplier = 6364136223846793005;
const long mulinv_multiplier = -4568919932995229531;
const long offset = 1442695040888963407;

static long Forward(long x)
{
    return x * multiplier + offset;
}

static long Reverse(long x)
{
    return (x - offset) * mulinv_multiplier;
}

You can change the constants to whatever as long as multiplier is odd and mulinv_multiplier is the modular multiplicative inverse (see wiki:modular multiplicative inverse or Hackers Delight 10-15 Exact Division by Constants) of multiplier (modulo 2^64, obviously – and that’s why multiplier has to be odd, otherwise it has no inverse).

The offset can be anything, but make it relatively prime with 2^64 just to be on the safe side.

These specific constants come from Knuths linear congruential generator.

There’s one small thing: it puts the complement of the LSB of the input in the LSB of the result. If that’s a problem, you could just rotate it by any nonzero amount.

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For 32 bits, the constants can be multiplier = 0x4c957f2d, offset = 0xf767814f, mulinv_multiplier = 0x329e28a5.

For 64 bits, multiplier = 12790229573962758597, mulinv_multiplier = 16500474117902441741 may work better.

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Or, you could use a CRC, which is reversible for this use (ie the input is the same size as the CRC) for CRC64 it requires some modifications of course.