MurmurHash2Flaw

Let's look at what happens in Murmur2's inner loop when processing 'bad' keys of the form "uint32_t key[2] = {x,x};" -

Here's the loop as written

    while(len >= 4)
    {
        uint32_t k = *(uint32_t*)data;

        // we'll call this the 'pre-mix' step

        k *= m;
        k ^= k >> r;
        k *= m;

        // and this the 'merge' step

        h *= m;
        h ^= k;

        data += 4;
        len -= 4;
    }

Since the two uint32_t's from the key are the same value 'x' the results of the pre-mix for both passes through the loop will be the same, so we can unroll this as

x *= m;
x ^= x >> r;
x *= m;

h *= m;
h ^= x;
h *= m;
h ^= x;

Now, what happens if m == 1?

x ^= x >> r;

h ^= x;
h ^= x;

X completely cancels out and does not affect the hash. All keys hash to the same value.

Luckily we're not using m == 1, right? Well, unfortunately we're not that lucky - there's still a lot of cancellation. If I evaluate

uint32_t test ( uint32_t x )
{
    const uint32_t m = 0x5bd1e995;
    const int r = 24;

   
uint32_t h = 0;

    x *= m;
    x ^= x >> r;
    x *= m;

    h *= m
    h ^= x;
    h *= m
    h ^= x;


    return h;
}

for all possible values of x, I only get 172,013,942 unique results instead of the 2^32 expected. That means we're getting ~4.6 bits cancelled out on average. Not good.

Will this flaw cause your program to fail? Probably not - what this means in real-world terms is that if your keys contain repeated 4-byte values AND they differ only in those repeated values AND the repetitions fall on a 4-byte boundary, then your keys will collide with a probability of about 1 in 2^27.4 instead of 2^32. Due to the birthday paradox, you should have a better than 50% chance of finding a collision in a group of 13115 bad keys instead of 65536.

Can this be patched up by choosing a different value of 'm'? Unfortunately not. Different values produce different amounts of cancellation, but there is always cancellation - the low bit of h will always end up 0 no matter which multiplier you use.

MurmurHash3 (not yet published) uses a much different mix setup that eliminates this problem and runs considerably faster than MurmurHash2, so if this flaw does prove to be a problem for your application you should be able to switch to MurmurHash3 without losing performance.
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