The final hash functions have been tested with dieharder, for each axis, in both positive and negative directions. You can see the results in the reports/ directory.
Note that some individual tests are reported as weak. This is okay! If you look at SHA-256, it even has a failed test!
It's important to understand that weak simply means the probability of the result (P-value) is less than 1%. Since hundreds of tests are performed, it is expected to receive a few weak results.
In the case of the failed SHA-256 test, I would trust SHA-256 over dieharder :-P.
The dieharder test only accepts a stream of numbers. How can we possibly test that a multi-dimensional hash is high quality when there are billions of ways of navigating the space to produce a 1D stream of numbers?
My answer is to test the functions under the worst possible scenarios, and if they pass, then surely they'll be fine under normal use.
I start by only generating hash functions that use multiply, XOR, and bit shift. Importantly, I don't use addition, bit NOT, or XOR with a constant.
This means that setting an input to 0 is the most pathological scenario. Multiplying by 0 is always 0, XORing with 0 is a no-op, and bit shifting 0 is a no-op. As a result, every hash function will return 0 if all inputs are 0!
This sounds bad!
But now we can easily test if a hash function is useful across a dimension. Just set all the other inputs to 0, and use an increasing number along the axis of interest, and see if dieharder likes the output.
I also test in the negative direction to ensure that inputs saturated with bits (i.e.,
0xFFFFFFFF) don't cause a failure.
When you read the reports/, you'll notice the functions pass along each axis, in both positive and negative directions. This ensures that each axis is contributing meaningfully to the output.