fix errors and update news

content/CSE4303/CSE4303_L6.md

@@ -0,0 +1,5 @@
+# CSE4303 Introduction to Computer Security (Lecture 6)
+
+Refer to this lecture notes
+
+[CSE442T Lecture 3](https://notenextra.trance-0.com/CSE442T/CSE442T_L3/)

content/CSE4303/CSE4303_L7.md

@@ -0,0 +1,65 @@
+# CSE4303 Introduction to Computer Security (Lecture 7)
+
+## Cyptography in Symmetric Systems
+
+### Symmetric systems
+
+Symmetric (shared-key) encryption
+
+- Classical techniques
+- Computer-aided techniques
+- Formal reasoning
+- Realizations:
+  - Stream ciphers
+  - Block ciphers
+
+#### Stream ciphers
+
+1. Operate on PT one bit at a time (usually), as a bit "stream"
+2. Generate arbitrarily long keystream on demand
+
+Security abstraction:
+
+1. XOR transfers randomness of keystream to randomness of CT regardless of PT’s content
+2. Security depends on G being “practically” indistinguishable from random string and “practically” unpredictable
+3. Idea: shouldn’t be able to predict next bit of generator given all bits seen so far
+
+Keystream $G(k)$
+
+- Idea: shouldn’t be able to predict next bit of generator given all bits seen so far
+- Strategies and challenges: many!
+- Idea that doesn’t quite work: Linear Feedback Shift Register (LFSR)
+  - Choice of feedback: by algebra
+  - Pro: fast, statistically close to random
+  - Problem: susceptible to cryptanalysis (b/c linear)
+  - LFSR-based
+- Modifications to basic LFSR:
+  - Use non-linear combo of multiple LFSRs
+  - Use controlled clocking (e.g. only cycle the LFSR when another LFSR outputs a 1)
+  - Etc.
+- Others: mod arithmetic-based, other algebraic constructions
+
+#### Block ciphers
+
+1. Operate on PT one block at a time
+2. Use same key for multiple blocks (with caveats)
+3. Chaining modes intertwine successive blocks of CT (or not)
+
+View cipher as a Pseudo-Random Permutation (PRP)
+
+- PRP defined over $(K, X)$:
+
+$$
+E: K \times X \to X
+$$
+
+such that:
+
+1. There exists an “efficient” deterministic algorithm to evaluate $E(k,x)$.
+2. The function $E( k, \cdot )$ is one-to-one.
+3. There exists an “efficient” inversion algorithm $D(k,y)$.
+
+- i.e. a PRF that is an invertible 1-to-1 mapping from message space to
+message space
+
+

content/CSE4303/_meta.js

@@ -8,4 +8,6 @@ export default {
     CSE4303_L3: "Introduction to Computer Security (Lecture 3)",
     CSE4303_L4: "Introduction to Computer Security (Lecture 4)",
     CSE4303_L5: "Introduction to Computer Security (Lecture 5)",
+    CSE4303_L6: "Introduction to Computer Security (Lecture 6)",
+    CSE4303_L7: "Introduction to Computer Security (Lecture 7)",
 }

content/Math4201/Math4201_L34.md

@@ -21,7 +21,7 @@ If $\mathbb{R}_l$ is second countable, then for any real number $x$, there is an
 
 Any such open sets is of the form $[x,x+\epsilon)\cap A$ with $\epsilon>0$ and any element of $A$ being larger than $\min(U_x)=x$.
 
-In summary, for any $x\in \mathbb{R}$, there is an element $U_x\in \mathcal{B}$ with $(U_x)=x$. In particular, if $x\neq y$, then $U_x\neq U_y$. SO there is an injective map $f:\mathbb{R}\rightarrow \mathcal{B}$ sending $x$ to $U_x$. This implies that $\mathbb{B}$ is uncountable.
+In summary, for any $x\in \mathbb{R}$, there is an element $U_x\in \mathcal{B}$ with $(U_x)=x$. In particular, if $x\neq y$, then $U_x\neq U_y$. So there is an injective map $f:\mathbb{R}\rightarrow \mathcal{B}$ sending $x$ to $U_x$. This implies that $\mathcal{B}$ is uncountable.
 
 </details>
 

content/Math4201/Math4201_L4.md

@@ -27,7 +27,7 @@ $$
 Let $(X,\mathcal{T})$ be a topological space. Let $\mathcal{C}\subseteq \mathcal{T}$ be a collection of subsets of $X$ satisfying the following property:
 
 $$
-\forall U\in \mathcal{T}, \exists C\in \mathcal{C} \text{ such that } U\subseteq C
+\forall U\in \mathcal{T}, \exists C\in \mathcal{C} \text{ such that } C\subseteq U
 $$
 
 Then $\mathcal{C}$ is a basis and the topology generated by $\mathcal{C}$ is $\mathcal{T}$.