update typo and structures

pages/CSE442T/CSE442T_L10.md

@@ -1,20 +1,23 @@
 # Lecture 10
 
-## Continue
+## Chapter 2: Computational Hardness
 
-### Discrete Log Assumption
+### Discrete Log Assumption (Assumption 52.2)
 
 This is collection of one-way functions
 
 $$
 p\gets \tilde\Pi_n(\textup{ safe primes }), p=2q+1
 $$
+
 $$
 a\gets \mathbb{Z}*_{p};g=a^2(\textup{ make sure }g\neq 1)
 $$
+
 $$
 f_{g,p}(x)=g^x\mod p
 $$
+
 $$
 f:\mathbb{Z}_q\to \mathbb{Z}^*_p
 $$
@@ -35,7 +38,7 @@ $$
 P[p,q\gets \Pi_n;N\gets p\cdot q;e\gets \mathbb{Z}_{\phi(N)}^*;y\gets \mathbb{N}_n;x\gets \mathcal{A}(N,e,y);x^e=y\mod N]<\epsilon(n)
 $$
 
-#### Theorem RSA Algorithm
+#### Theorem 53.2 (RSA Algorithm)
 
 This is a collection of one-way functions
 
@@ -101,7 +104,7 @@ Let $y\in \mathbb{Z}_N^*$, letting $x=y^d\mod N$, where $d\equiv e^{-1}\mod \phi
 
 $x^e\equiv (y^d)^e \equiv y\mod n$
 
-Proof: 
+Proof:
 
 It's easy to sample from $I$:
 
@@ -175,6 +178,15 @@ So the probability of B succeeds is equal to A succeeds, which $>\frac{1}{p(n)}$
 
 Remaining question: Can $x$ be found without factoring $N$? $y=x^e\mod N$
 
+### One-way permutation (Definition 55.1)
+
+A collection function $\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$ is a one-way permutation if
+
+1. $\forall i,f_i$ is a permutation
+2. $\mathcal{F}$ is a collection of one-way functions
+
+_basically, a one-way permutation is a collection of one-way functions that maps $\{0,1\}^n$ to $\{0,1\}^n$ in a bijection way._
+
 ### Trapdoor permutations
 
 Idea: $f:D\to R$ is a one-way permutation.
@@ -196,4 +208,3 @@ $\mathcal{F}=\{f_i:D_i\to R_i\}_{i\in I}$
 #### Theorem RSA is a trapdoor
 
 RSA collection of trapdoor permutation with factorization $(p,q)$ of $N$, or $\phi(N)$, as trapdoor info $f$.
-