update typo and structures

pages/CSE442T/CSE442T_L13.md

@@ -1,6 +1,10 @@
 # Lecture 13
 
-## Pseudorandom Generator (PRG)
+## Chapter 3: Indistinguishability and Pseudorandomness
+
+### Pseudorandom Generator (PRG)
+
+#### Definition 77.1 (Pseudorandom Generator)
 
 $G:\{0,1\}^n\to\{0,1\}^{l(n)}$ is a pseudorandom generator if the following is true:
 
@@ -8,7 +12,7 @@ $G:\{0,1\}^n\to\{0,1\}^{l(n)}$ is a pseudorandom generator if the following is t
 2. $l(n)> n$ (expansion)
 3. $\{x\gets \{0,1\}^n:G(x)\}_n\approx \{u\gets \{0,1\}^{l(n)}\}$
 
-### Hard-core bit (predicate) (HCB)
+#### Definition 78.3 (Hard-core bit (predicate) (HCB))
 
 Hard-core bit (predicate) (HCB): $h:\{0,1\}^n\to \{0,1\}$ is a hard-core bit of $f:\{0,1\}^n\to \{0,1\}^*$ if for every adversary $A$,
 
@@ -131,7 +135,7 @@ $G'$ is a PRG:
 
 1. Efficiently computable: since we are computing $G'$ by applying $G$ multiple times (polynomial of $l(n)$ times).
 2. Expansion: $n<l(n)$.
-3. Pseudorandomness: We proceed by contradiction. Suppose the output is not pseudorandom. Then there exists a distinguisher $D$ that can distinguish $G'$ from $U_{l(n)}$ with advantage $\frac{1}{2}+\epsilon(n)$.
+3. Pseudorandomness: We proceed by contradiction. Suppose the output is not pseudorandom. Then there exists a distinguisher $\mathcal{D}$ that can distinguish $G'$ from $U_{l(n)}$ with advantage $\frac{1}{2}+\epsilon(n)$.
 
 Strategy: use hybrid argument to construct distributions.
 
@@ -145,9 +149,9 @@ H^{l(n)}&=b_1b_2\cdots b_{l(n)}
 \end{aligned}
 $$
 
-By the hybrid argument, there exists an $i$ such that $D$ can distinguish $H^i$ and $H^{i+1}$ $0\leq i\leq l(n)-1$ by $\frac{1}{p(n)l(n)}$
+By the hybrid argument, there exists an $i$ such that $\mathcal{D}$ can distinguish $H^i$ and $H^{i+1}$ $0\leq i\leq l(n)-1$ by $\frac{1}{p(n)l(n)}$
 
-Show that there exists $D$ for 
+Show that there exists $\mathcal{D}$ for 
 
 $$
 \{u\gets U_{n+1}\}\text{ vs. }\{x\gets U_n;G(x)=u\}