Complete Example 45

chapters/algebra-moonmath.tex

@@ -457,9 +457,9 @@ \subsubsection{Hash functions}\label{sec:hash-functions} Generally speaking, a h
 If $H$ is a hash function that maps binary strings of arbitrary length onto binary strings of length $k$, and $s\in\{0,1\}^*$ is a binary string, we write $H(s)_j$ for the bit at position $j$ in the image $H(s)$.
 \end{notation}
 
-\begin{example}[$k$-truncation hash]\label{ex:k-truncation-hash} One of the most basic hash functions $H_k:\{0,1\}^*\to \{0,1\}^k$ is given by simply truncating every binary string $s$ of size $|s|> k$ to a string of size $k$ and by filling any string $s'$ of size $|s'|<k$ with zeros. To make this hash function deterministic, we define that both truncation and filling should happen ``on the left''.
+\begin{example}[$k$-truncation hash]\label{ex:k-truncation-hash} One of the most basic hash functions $H_k:\{0,1\}^*\to \{0,1\}^k$ is given by simply truncating every binary string $s$ of size $|s|> k$ to a string of size $k$ and by filling any string $s'$ of size $|s'|<k$ with zeros. To make this hash function deterministic, we define that both truncation and filling should happen on the highest bits, or ``on the left''.
 
-For example, if the parameter $k$ is given by $k=3$, $s_1=<0,0,0,0,1,0,1,0,1,1,1,0>$ and $s_2=1$, then \smecomb{$H_3(s_1)=<1,1,0>$ and $H_3(s_2)=<0,0,1>$}.
+For example, if the parameter $k$ is given by $k=3$, $s_1=<0,0,0,0,1,0,1,0,1,1,1,0>$ and $s_2=1$, then $H_3(s_1)=<1,1,0>$ and $H_3(s_2)=<0,0,1>$.
 \end{example}
 
 A desirable property of a hash function is \term{uniformity}, which means that it should map input values as evenly as possible over its output range. In mathematical terms, every string of length $k$  from $\{0,1\}^k$ should be generated with roughly the same probability.