Simple tabulation dates back to Zobrist in 1970. Keys are viewed as c
characters from some alphabet A. We initialize c tables h_0, ..., h_{c-1}
mapping characters to random hash values. A key x=(x_0, ..., x_{c-1}) is hashed
to h_0[x_0] xor...xor h_{c-1}[x_{c-1}]. The scheme is extremely fast when the
character hash tables h_i are in cache. Simple tabulation hashing is not
4-independent, but we show that if we apply it twice, then we get high
independence. First we hash to intermediate keys that are 6 times longer than
the original keys, and then we hash the intermediate keys to the final hash
values.
The intermediate keys have d=6c characters from A. We can view the hash
function as a degree d bipartite graph with keys on one side, each with edges
to d output characters. We show that this graph has nice expansion properties,
and from that we get that with another level of simple tabulation on the
intermediate keys, the composition is a highly independent hash function. The
independence we get is |A|^{Omega(1/c)}.
Our space is O(c|A|) and the hash function is evaluated in O(c) time. Siegel
[FOCS'89, SICOMP'04] proved that with this space, if the hash function is
evaluated in o(c) time, then the independence can only be o(c), so our
evaluation time is best possible for Omega(c) independence---our independence
is much higher if c=|A|^{o(1)}.
Siegel used O(c)^c evaluation time to get the same independence with similar
space. Siegel's main focus was c=O(1), but we are exponentially faster when
c=omega(1).
Applying our scheme recursively, we can increase our independence to
|A|^{Omega(1)} with o(c^{log c}) evaluation time. Compared with Siegel's scheme
this is both faster and higher independence.
Our scheme is easy to implement, and it does provide realistic
implementations of 100-independent hashing for, say, 32 and 64-bit keys
