UNDERSTANDING HASH FUNCTIONS
by Geoff Pike

Version 0.2 --- early draft --- comments and questions welcome!
References appear in square brackets.

1 INTRODUCTION

Hashing has proven tremendously useful in constructing various fast
data structures and algorithms.  It is typically possible to simplify
the analysis of hash-based algorithms if one assumes that the relevant
hash functions are high quality.  At the other extreme, if the
relevant hash functions were always to return the same value, many
hash-based algorithms become algorithms that are slower, simpler, but still well-known.
For example, a chaining hash table devolves into a linked list.

There are many possible definitions of hash function quality.  For
example, one might want a list of keys and their hashes to provide no
pattern that would allow an opponent to predict anything about the
hashes of other keys.  Although I cannot prove it, I think I can meet
this and many other definitions of quality with

  f(s) = SHA-3(concatenation of z and s),

where z is some secret string known only to me.  This well-known trick
provides, I think, more high-quality hash functions than anyone will
need, though greater computational power in the future may push us to
replace SHA-3 from time to time.

In short, discussions about choosing a hash function are almost always
discussions about speed, energy consumption, or similar.  Concerns
about hash quality are easy to fix, for a price.

2 ANATOMY OF A HASH FUNCTION

Hash functions that input strings of arbitrary length are written in
terms of an internal state, S.  In many cases the internal state is a
fixed number of bits and will fit in machine registers.  One generic
sketch of a string hash is:

  let S = some initial value
  let c = the length of S in bits
  while (input is not exhausted) {
    let t = the next c bits of input (padded with zeroes if less than c remain)
    S = M(S xor t)
  }
  let n = the number of bytes hashed
  return F(S, n)

where M is a hash function that inputs and outputs c bits, and F is a
hash function that inputs c bits (plus, say, 64 for its second argument)
and outputs however many bits one needs to return.  In some sense we have
reduced the string-hashing problem to two integer hashing problems.

2.1 INTEGER HASHING TECHNIQUES

A hash function that inputs and outputs the same number of bits, say,
32, can use reversible bit-twiddling operations, each of which is
"onto" in the mathematical sense.  For example, multiplication by an
odd constant is reversible, as all odd numbers are relatively prime to
2^32.  Other commonly used reversible operations include:
  o  Adding or xoring a constant
  o  Bitwise rotation or other bitwise permutations
  o  bit j = (bit j) xor (bit k) for unequal constants j and k
  o  "Shift mix": S = S xor (S >> k), where k is, say, 17
  o  Replacing a fixed-length bit string with its cyclic redundancy
     checksum, perhaps via _mm_crc32_u32(f, <some constant>) [Pike]

Each of the above is a "bad" hash function that inputs and outputs
the same number of bits.  One can simply compose two or more of those
bad hash functions to construct a higher-quality hash function.

One common quality goal for integer hashing (and string hashing) is
that flipping the 19th bit, or any other small change, applied to
multiple input keys, causes a seemingly unpredictable difference each
time.  Similarly, any change to an input should lead to a seemingly
unpredictable selection of the output bits to flip.

Therefore, if we want a high-quality hash function that inputs c bits
and outputs fewer than c bits, we can simply truncate the output of a
high-quality hash function that inputs and outputs c bits.

To give a concrete example, here is Bob Jenkins' mix(), published in
1996 [Jenkins].  Its input is 96 bits in three 32-bit variables, and its output
is 96 bits.  However, one may use a subset of the output bits, as every
output bit is affected by every non-empty subset of the input bits.

  Input: a, b, and c
  Algorithm:
    a -= b; a -= c; a ^= (c>>13);
    b -= c; b -= a; b ^= (a<<8);
    c -= a; c -= b; c ^= (b>>13);
    a -= b; a -= c; a ^= (c>>12);
    b -= c; b -= a; b ^= (a<<16);
    c -= a; c -= b; c ^= (b>>5);
    a -= b; a -= c; a ^= (c>>3);
    b -= c; b -= a; b ^= (a<<10);
    c -= a; c -= b; c ^= (b>>15);
  Output: a, b, and c

2.2 VARIATIONS ON STRING HASHING

There are three variations on our initial sketch worth noting.

First, for speed, one can special-case short inputs, as the CityHash
and FarmHash algorithms do.  The number of special cases can be
reduced by using loads that may overlap: for example, a hash of a 9-
to 16-byte string can be implemented by a hash that inputs two 8-byte
values (the first 8 and last 8 bytes of the input string) and the string
length [CityHash, FarmHash].

Second, one may choose different means of incorporating input bits
into the internal state.  One example: the mixing of S and input bits
may be interleaved with the mixing of parts of S and other parts of S.
Another example: the input bits processed in a loop iteration might be
xor'ed into multiple places in S, rather than just one, or might be
hashed with each other before touching S [Murmur].  The advantages and
disadvantages of these are unclear.

Third, one may repeatedly "squeeze information" from S, by remixing it with
itself and then revealing a subset of S.  This is convenient when one would
like a family of hash functions with different output lengths.  A special
case of the idea, called the "sponge construction," has been well studied and
adopted by the authors of Keccak and others [SHA-3].

3 HASH FUNCTIONS FOR HASH TABLES

It isn't hard to find real-life examples where hash tables or the hash
functions for them take more than 5% of a program's CPU time.
Improvements to hash tables and their hash functions are therefore a
classic example of software performance tuning.  Unfortunately, the
best choice may be platform-dependent, so to avoid writing your own
collection of #ifdefs, please consider selecting something like the
FarmHash family of hash functions, that supply decent
platform-dependent logic for you.

To tune a program, often one will replace an existing hash function with a
faster, lower-quality hash function, despite the increased chance of unlucky
or pathological performance problems.  Clever algorithms can mitigate this
risk.  For example, hash tables can start with one hash function and then
switch to another if things seem to be going poorly.  Therefore, one should
rarely plan to spend much CPU time on a secure hash function (such as SHA-3)
or a near-universal hash function (such as VHASH) when seeking the best
possible performance from a hash table.  Against that, those types of hash
functions can limit the risk of pathological performance problems when one is
designing around typical hash-based algorithms that stick with a single hash
function no matter how it behaves on the data at hand.

4 

REFERENCES

[Murmur] Appleby, Austin. https://code.google.com/p/smhasher,
            https://sites.google.com/site/murmurhash/
[SMHasher] Appleby, Austin. https://code.google.com/p/smhasher
[SHA-3] Bertoni, Guido, et al. http://keccak.noekeon.org/
[Jenkins] Jenkins, Bob. http://burtleburtle.net/bob/hash/doobs.html
[VHASH] Krovetz, Ted. Message authentication on 64-bit architectures. In
            Selected Areas of Cryptography – SAC 2006.  Springer-Verlag, 2006.
[CityHash] Pike, Geoff and Alakuijala, Jyrki. https://code.google.com/p/cityhash
[FarmHash] Pike, Geoff. https://code.google.com/p/farmhash
[Pike] Pike, Geoff. http://www.stanford.edu/class/ee380/Abstracts/121017-slides.pdf