upgrade structures and migrate to nextra v4

content/CSE442T/CSE442T_L1.md

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+# Lecture 1
+
+## Chapter 1: Introduction
+
+### Alice sending information to Bob
+
+Assuming _Eve_ can always listen
+
+Rule 1. Message, Encryption to Code and Decryption to original Message.
+
+### Kerckhoffs' principle
+
+It states that the security of a cryptographic system shouldn't rely on the secrecy of the algorithm (Assuming Eve knows how everything works.)
+
+**Security is due to the security of the key.**
+
+### Private key encryption scheme
+
+Let $M$ be the set of message that Alice will send to Bob. (The message space) "plaintext"
+
+Let $K$ be the set of key that will ever be used. (The key space)
+
+$Gen$ be the key generation algorithm.
+
+$k\gets Gen(K)$
+
+$c\gets Enc_k(m)$ denotes cipher encryption.
+
+$m'\gets Dec_k(c')$ $m'$ might be null for incorrect $c'$.
+
+$P[k\gets K:Dec_k(Enc_k(M))=m]=1$ The probability of decryption of encrypted message is original message is 1.
+
+*_in some cases we can allow the probability not be 1_
+
+### Some examples of crypto system
+
+Let $M=\text{all five letter strings}$.
+
+And $K=[1,10^{10}]$
+
+Example:
+
+$P[k=k']=\frac{1}{10^{10}}$
+
+$Enc_{1234567890}("brion")="brion1234567890"$
+
+$Dec_{1234567890}(brion1234567890)="brion"$
+
+Seems not very secure but valid crypto system.
+
+### Early attempts for crypto system
+
+#### Caesar cipher
+
+$M=\text{finite string of texts}$
+
+$K=[1,26]$
+
+$Enc_k=[(i+K)\% 26\ for\ i \in m]=c$
+
+$Dec_k=[(i+26-K)\% 26\ for\ i \in c]$
+
+```python
+def caesar_cipher_enc(s: str, k:int):
+    return ''.join([chr((ord(i)-ord('a')+k)%26+ord('a')) for i in s])
+
+def caesar_cipher_dec(s: str, k:int):
+    return ''.join([chr((ord(i)-ord('a')+26-k)%26+ord('a')) for i in s])
+```
+
+#### Substitution cipher
+
+$M=\text{finite string of texts}$
+
+$K=\text{set of all bijective linear transformations (for English alphabet},|K|=26!\text{)}$
+
+$Enc_k=[iK\ for\ i \in m]=c$
+
+$Dec_k=[iK^{-1}\ for\ i \in c]$
+
+Fails to frequency analysis
+
+#### Vigenere Cipher
+
+$M=\text{finite string of texts with length }m$
+
+$K=\text{[0,26]}^n$ (assuming English alphabet)
+
+```python
+def viginere_cipher_enc(s: str, k: List[int]):
+    res=''
+    n,m=len(s),len(k)
+    j=0
+    for i in s:
+        res+=caesar_cipher_enc(i,k[j])
+        j=(j+1)%m
+    return res
+
+def viginere_cipher_dec(s: str, k: List[int]):
+    res=''
+    n,m=len(s),len(k)
+    j=0
+    for i in s:
+        res+=caesar_cipher_dec(i,k[j])
+        j=(j+1)%m
+    return res
+```
+
+#### One time pad
+
+Completely random string, sufficiently long.
+
+$M=\text{finite string of texts with length }n$
+
+$K=\text{[0,26]}^n$ (assuming English alphabet)$
+
+$Enc_k=m\oplus k$
+
+$Dec_k=c\oplus k$
+
+```python
+def one_time_pad_enc(s: str, k: List[int]):
+    return ''.join([chr((ord(i)-ord('a')+k[j])%26+ord('a')) for j,i in enumerate(s)])
+
+def one_time_pad_dec(s: str, k: List[int]):
+    return ''.join([chr((ord(i)-ord('a')+26-k[j])%26+ord('a')) for j,i in enumerate(s)])
+```