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+# Lecture 1
+
+> I changed all the element in set to lowercase letters. I don't know why K is capitalized.
+
+Brian Garnett
+
+bcgarnett@wustl.edu
+
+Math Phd... Great!
+
+Proof based course and write proofs.
+
+CSE 433 for practical applications.
+
+OH: Right after class! 4-5 Mon, Urbaur Hall 227
+
+Pass and Shalat
+
+## 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 $\mathcal{M}$ be the set of message that Alice will send to Bob. (The message space) "plaintext"
+
+Let $\mathcal{K}$ be the set of key that will ever be used. (The key space)
+
+$Gen$ be the key generation algorithm.
+
+$k\gets Gen(\mathcal{K})$
+
+$c\gets Enc_k(m)$ denotes cipher encryption.
+
+$m'\gets Dec_k(c')$ $m'$ might be null for incorrect $c'$.
+
+$Pr[K\gets \mathcal{K}:Dec_k(Enc_k(M))=m]=1$ The probability of decryption of encrypted message is original message is 1.
+
+*_in some cases we can allow the probailty not be 1_
+
+## Some examples of crypto system
+
+Let $\mathcal{M}=$ {all five letter strings}.
+
+And $\mathcal{K}=$ {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
+
+$\mathcal{M}=$ finite string of texts
+
+$\mathcal{K}=$ {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
+
+$\mathcal{M}=$ finite string of texts
+
+$\mathcal{K}=$ bijective linear transformations (for English alphabet, $|\mathcal{K}|=26!$)
+
+$Enc_k=[iK\ for\ i \in m]=c$
+
+$Dec_k=[iK^{-1}\ for\ i \in c]$
+
+Fails to frequency analysis
+
+### Vigenere Cipher
+
+$\mathcal{M}=$ finite string of texts
+
+$\mathcal{K}=$ key phrase of a fixed length
+
+```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.