typographical and fixed refs

src/lec21 - Identity-Based Encryption.tex

@@ -197,15 +197,12 @@ \section{Number Theory Background}
 \begin{proof}
   The following \emph{reciprocity laws} can be proved with some
   effort.  We omit their proofs.
-  \begin{eqnarray}
-    \label{rec_1} \text{for odd $a$ and $N$}, \quad \left(\frac{a}{N}
-    \right) &=& \left(\frac{N}{a} \right) \cdot \left( -1
-    \right)^{\frac{a-1}{2}\cdot\frac{N-1}{2}} \\
-    \label{rec_2} \text{for any } N, \quad \left(\frac{-1}{N} \right) &=&
-    \left( -1 \right)^{\frac{N-1}{2}} \\
-    \label{rec_3}\left(\frac{2}{N} \right) &=& \left( -1
-    \right)^{\frac{N^2-1}{8}}
-  \end{eqnarray}
+  \begin{align}
+    \label{rec_1} \text{for odd $a$ and $N$}, \quad \parens*{\frac{a}{N}} &= \parens*{\frac{N}{a}} \cdot \parens{-1}^{\frac{a-1}{2}\cdot\frac{N-1}{2}} \; \text. \\
+    \label{rec_2} \text{for any } N, \quad
+    \parens*{\frac{-1}{N}}  &= \parens{-1}^{\frac{N-1}{2}} \\
+    \label{rec_3} \parens*{\frac{2}{N}}  &= \parens{-1}^{\frac{N^2-1}{8}} \; \text.
+  \end{align}
 
   For any even number $a$, we can reduce the problem to the case when
   $a$ is odd by using \cref{rec_3}.  Using \cref{rec_1,rec_2}, we can