Advanced Hash Functions

The Cyclic Redundancy Check (CRC) Algorithm

Let k[n] denote the n^th bit of the key.
Divide the corresponding polynomial k[n]*xⁿ + k[n-1]*x^n-1 + ... + k[0] by a ``magic'' polynomial.

Note: the coefficients of both polynomials are either 0 or 1.
Uses ``polynomial arithmetic mod 2'': all coefficients are calculated mod 2.
The ``magic'' (divisor) polynomial is commonly called the ``generator'' polynomial.

Use the remainder as the hash code.

The CRC algorithm is widely used for detection of errors in data storage and transmission by unreliable media.

With a well-chosen generator polynomial, it's possible to detect ALL:

single-bit errors
two-bit errors
errors where an odd number of bits are affected
``burst'' errors (up to the degree of the generator polynomial)

The CRC algorithm makes a good hash function.

A single-bit difference between two keys yields large differences between the hash codes.
Easy to apply to any size key.

The implementation can be reduced to repeated table lookup. (The table is dependent on the generator polynomial.)
Efficient implementations are widely available as part of standard libraries.

Evaluating A Hash Function: A Case Study

No hash function is perfect! For any reasonable hash table size, collisions will occur. We can only hope to minimize them.
Often, the best strategy is to test your hash function on representative input: here