Perfect Hash Function for Human Readable Order Codes

Question

I am trying to generate non-sequential human readable order codes derived from (lets say) a unsigned 32bit internal id that starts at 1 and is auto incremented for each new

Answer 1

In my example code below, will every $hash be unique?

Almost. (Which, I guess, means "no, but in a way that's easily fixed".) Your function consists of a sequence of independent steps; the overall function is bijective (reversible) if and only if every single one of those steps is. (Do you see why?)

Now, each step has one of the following forms:

  $key = ($key ^ CONSTANT) ^ ($key >> NUM_BITS);
  $key = ($key ^ CONSTANT) ^ ($key << NUM_BITS);

with NUM_BITS != 0.

We can actually treat these as variants of a single form, by viewing the former as almost equivalent to this:

  $key = invert_order_of_bits($key); # clearly bijective
  $constant = invert_order_of_bits(CONSTANT);
  $key = ($key ^ $constant) ^ ($key << NUM_BITS);
  $key = invert_order_of_bits($key); # clearly bijective

So all we need is to show that this:

  $key = ($key ^ CONSTANT) ^ ($key << NUM_BITS);

is bijective. Now, XOR is commutative and associative, so the above is equivalent to this:

  $key = $key ^ ($key << NUM_BITS);
  $key = $key ^ CONSTANT;

and (x ^ y) ^ y == x ^ (y ^ y) == x ^ 0 == x, so clearly XOR-ing with a constant value is reversible (by re-XOR-ing with the same value); so all we have to show is that this is bijective:

  $key = $key ^ ($key << NUM_BITS);

whenever NUM_BITS != 0.

Now, I'm not writing a rigorous proof, so I'll just give a single reasoned-out example of how to reverse this. Suppose that $key ^ ($key << 9) is

0010 1010 1101 1110 0010 0101 0000 1100

How do we obtain $key? Well, we know that the last nine bits of $key << 9 are all zeroes, so we know that the last nine bits of $key ^ ($key << 9) are the same as the last nine bits of $key. So $key looks like

bbbb bbbb bbbb bbbb bbbb bbb1 0000 1100

so $key << 9 looks like

bbbb bbbb bbbb bb10 0001 1000 0000 0000

so $key looks like

bbbb bbbb bbbb bb00 0011 1101 0000 1100

(by XOR-ing $key ^ ($key << 9) with $key << 9), so $key << 9 looks like

bbbb b000 0111 1010 0001 1000 0000 0000

so $key looks like

bbbb b010 1010 0100 0011 1101 0000 1100

so $key << 9 looks like

0101 1000 0111 1010 0001 1000 0000 0000

so $key looks like

0111 0010 1010 0100 0011 1101 0000 1100

So . . . why do I say "almost" rather than "yes"? Why is your hash-function not perfectly bijective? It's because in PHP, the bitwise shift operators >> and << are not quite symmetric, and while $key = $key ^ ($key << NUM_BITS) is completely reversible, $key = $key ^ ($key >> NUM_BITS) is not. (Above, when I wrote that the two types of steps were "almost equivalent", I really meant that "almost". It makes a difference!) You see, whereas << treats the sign bit just like any other bit, and shifts it out of existence (bringing in a zero-bit on the right), >> treats the sign bit specially, and "extends" it: the bit that it brings in on the left is equal to the sign bit. (N.B. Your question mentions "unsigned 32bit" values, but PHP doesn't actually support that; its bitwise operations are always on signed integers.)

Due to this sign extension, if $key starts with a 0, then $key >> NUM_BITS starts with a 0, and if $key starts with a 1, then $key >> NUM_BITS also starts with a 1. In either case, $key ^ ($key >> NUM_BITS) will start with a 0. You've lost exactly one bit of entropy. If you give me $key ^ ($key >> 9), and don't tell me whether $key is negative, then the best I can do is compute two possible values for $key: one negative, one positive-or-zero.

You perform two steps that use right-shift instead of left-shift, so you lose two bits of entropy. (I'm hand-waving slightly — all I've actually demonstrated is that you lose at least one bit and at most two bits — but I'm confident that, due to the nature of the steps between those right-shift steps, you do actually lose two full bits.) For any given output value, there are exactly four distinct input-values that could yield it. So it's not unique, but it's almost unique; and it's easily fixed, by either:

• changing the two right-shift steps to use left-shifts instead; or
• moving both of the right-shift steps to the start of the function, before any left-shift steps, and saying that outputs are unique for inputs between 0 and 2³¹−1 rather than inputs between 0 and 2³²−1.