# Exercise 11.3.2

Suppose that we hash a string of $r$ characters into $m$ slots by treating it as a radix-128 number and then using the division method. We can easily represent the number $m$ as a 32-bit computer word, but the string of $r$ characters, treated as a radix-128 number, takes many words. How can we apply the division method to compute the hash value of the string without using more than a constant number of words of storage outside the string itself?

Yes, this follows pretty easily from the laws of modulo arithmetic. We need to observe that if the string is $s = \langle a_n, \ldots, a_1, a_0 \rangle$, then its hash $h(s)$ is going to be:

$$ \begin{aligned} h(s) &= \left( \sum_{i=0}^{n}{a_i \cdot {128}^{i}} \right) \bmod m \\ &= \sum_{i=0}^{n}{ \Big( \left( a_i \cdot {128}^{i} \right) \bmod m \Big) } \\ &= \sum_{i=0}^{n}{ \Big( ( a_i \bmod m ) ( {128}^{i} \bmod m ) \Big) } \\ \end{aligned} $$

We can easily compute $a_i \bmod m$ without extra memory. To compute ${128}^{i} \bmod m$ without extra memory, we just need to observe that $k^i \bmod m = k(k^{i-1} \bmod m) \bmod m$, that is, we can compute the module for each power incrementally, without ever needing unbound memory.

### Python code

K = 128 def consthash(digits, m): result = 0 power = 1 for d in reversed(digits): result += ((d % m) * power) % m result %= m power = (power * K) % m return result