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+# CSE4303 Introduction to Computer Security (Exam Review)
+
+## Details
+
+Time and location
+
+– In class exam – Thursday, 3/5 at 11:30 AM
+– What is allowed:
+    - One 8.5" X 11" paper of notes, single-sided only, typed or hand-written
+
+Topics covered:
+
+– Security fundamentals
+– TCP/IP network stack
+– Crypto fundamentals
+– Symmetric key cryptography
+– Hash functions
+– Asymmetric key cryptography
+
+## Security fundamentals
+
+### Defining security
+
+- Understand principles of security analysis
+  - The security of a system, application, or protocol is always relative to
+    - A set of desired properties
+    - An adversary with specific capabilities ("threat model")
+
+### Key security concepts
+
+C.I.A. triad:
+
+- Integrity: Prevent unauthorized modification of data, and/or detect if modification occurred.
+  - ARP poisoning (ARP spoofing)
+  - Authentication codes
+- Confidentiality: Prevent unauthorized parties from learning the contents of data (in transit or at rest).
+  - Packet sniffing / eavesdropping
+  - Data encryption
+- Availability: Ensure systems and data are accessible to authorized users when needed.
+  - Denial-of-Service (DoS) / Distributed DoS (DDoS)
+  - Rate limiting + traffic filtering (often with DDoS protection/CDN)
+
+Other security goals:
+
+- Authenticity: identity of an entity (issuer of info/message) is verified
+- Anonymity: identity of an entity remains unknown
+- Non-repudiation: messages can't be denied or taken back (e.g. online transaction commitments)
+
+### Modeling attacks
+
+Common components:
+
+- System being attacked (usually a model, with assumptions and abstractions)
+- Threat model 
+- Attack surface: what can be attacked
+  - Open ports and exposed services
+  - Public APIs and their parameters
+  - Web endpoints, forms, cookies
+  - File system permissions
+  - Hardware interfaces (USB, JTAG)
+  - User roles and privilege boundaries
+- Attack vector: how the attacker attacks
+  - SQL injection via POST /login
+  - Phishing to steal credentials, then SSH login
+  - Buffer overflow in a network daemon
+  - Cross-site scripting through a comment field
+  - Supply-chain poisoning of a dependency
+- Vulnerability: what the attacker can do
+- Exploit: how the attacker exploits the vulnerability
+- Damage: what the attacker can do
+- Mitigation: mitigate vulnerability
+- Defense: close vulnerability gap
+
+Importance of correct modeling
+
+- Attack-surface awareness guides defenses
+  - E.g. pre-Covid-19 vs. post-Covid attack surface of company servers
+- Match resources to expected threat actors
+  - "Script kiddie": individual or group running off-the-shelf attacks
+    - Caveat: off-the-shelf attacks can still be quite powerful! Metasploit, Shodan, dark web market.
+  - "Insider attack": employee with access to internal machines/networks
+  - "Advanced Persistent Threat (APT)": nation-state level resources and patience
+  - All these threats have different motivations, require different defenses/responses!
+- Reevaluate often
+  - Threat capabilities change over time
+
+### TCP/IP network stack
+
+Local and interdomain routing
+
+- TCP/IP for routing and messaging
+- BGP for routing announcements
+
+Domain Name System
+
+- Find IP address from symbolic name (cse.wustl.edu)
+
+#### Layer Summary
+
+Application: the actual sending message
+Transport (TCP, UDP): segment
+Network (IP): packet
+Data Link (Ethernet): frame
+
+### Types of Addresses in Internet
+
+- Media Access Control (MAC) addresses in the network access layer
+  - Associated w/ network interface card (NIC)
+  - 00-50-56-C0-00-01
+- IP addresses for the network layer
+  - IPv4 (32 bit) vs IPv6 (128 bit)
+  - 128.1.1.3 vs fe80::fc38:6673:f04d:b37b%4
+- IP addresses + ports for the transport layer
+  - E.g., 10.0.0.2:8080
+- Domain names for the application/human layer
+  - E.g., www.wustl.edu
+
+#### Routing and Translation of Addresses
+
+(All of them are attack surfaces)
+
+- Translation between IP addresses and MAC addresses
+  - Address Resolution Protocol (ARP) for IPv4
+  - Neighbor Discovery Protocol (NDP) for IPv6
+- Routing with IP addresses
+  - TCP, UDP for connections, IP for routing packets
+  - Border Gateway Protocol for routing table updates
+- Translation between IP addresses and domain names
+  - Domain Name System (DNS)
+
+### Summary for security
+
+- Confidentiality
+  - Packet sniffing
+- Integrity
+  - ARP poisoning
+- Availability
+  - Denial of service attacks
+- Common
+  - Address translation poisoning attacks (DNS, ARP)
+  - Packet spoofing
+- Core protocols not designed for security
+  - Eavesdropping, packet injection, route stealing, DNS poisoning
+  - Patched over time to prevent basic attacks
+- More secure variants exist:
+  - IP $\to$ IPsec (IPsec is )
+  - DNS $\to$ DNSsec
+  - BGP $\to$ sBGP
+
+## Crypto fundamentals
+
+- Well-defined statement about difficulty of compromising a system
+  - ...with clear implicit or explicit assumptions about:
+    - Parameters of the system
+    - Threat model
+    - Attack surfaces
+- Example: "A one-time pad cipher is secure against any cryptanalysis, including a brute-force attack, assuming:
+  - the key is the same length as the plaintext,
+  - the key is truly random, and
+  - the key is never re-used."
+
+### Common roles in cryptography
+
+Alice and Bob: Sender and receiver
+
+Eve: Adversary that can see but not create any packets
+
+Mallory: Man in the middle, can create and modify packets
+
+The message M is called the **plaintext**.
+
+Alice will convert plaintext M to an encrypted form using an
+encryption algorithm E that outputs a **ciphertext** C for M.
+
+#### Cryptography goals
+
+Confidentiality:
+
+- Mallory and Eve cannot recover original message from ciphertext
+
+Integrity:
+
+- Mallory cannot modify message from Alice to Bob without detection by Bob
+
+#### Threat models
+
+- Attacker may have (with increasing power):
+  - a) collection of ciphertexts (ciphertext-only attack)
+  - b) collection of plaintext/ciphertext pairs (known plaintext attack: KPA)
+  - c) collection of plaintext/ciphertext pairs for plaintexts selected by the attacker (chosen plaintext attack: CPA)
+  - d) collection of plaintext/ciphertext pairs for ciphertexts selected by the attacker (chosen ciphertext attack: CCA/CCA2)
+
+### Symmetric key cryptography
+
+#### Classical cryptography
+
+Techniques: substitution and transposition
+
+- Substitution: 1:1 mapping of alphabet onto itself
+- Transposition: permutation of elements (i.e. rearrange letters)
+
+- Caesar cipher: rotate each letter by k positions (k is fixed)
+- Vigenère cipher: If length of key is known, split letters into groups based on index within key and do frequency analysis within groups
+
+> The three steps in cryptography:
+>
+> - Precisely specify threat model
+> - Propose a construction
+> - Prove that breaking construction under threat mode will solve an underlying hard problem
+
+#### Perfect secrecy
+
+Ciphertext attack reveal no "info" about plaintext under ciphertext only attack
+
+Def: A cipher $(E, D)$ over $(K, M, C)$ has perfect secrecy if
+
+- $\forall m_0, m_1 \in M$ $(|m_0| = |m_1|)$ and $\forall c \in C$,
+  - $\Pr[E(k, m_0) = c] = \Pr[E(k, m_1) = c]$ where $k \leftarrow K$
+
+#### XOR One-time pad (perfect secrecy)
+
+Assumptions:
+
+- Key is as long as message
+- Key is random
+- Key is never re-used
+
+In practice, relax this assumption gets "Stream ciphers"
+
+### Stream cipher
+
+- Use pseudorandom generator as keystream for xore encryption (security is guaranteed by pseudorandom generator)
+
+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
+
+#### Semantic security
+
+- $(E, D)$ has semantic secrecy if $\forall m_0, m_1 \in M$ $(|m_0| = |m_1|)$,
+  - $\{E(k, m_0)\} \approx_p \{E(k, m_1)\}$ where $k \leftarrow K$
+- ...and the adversary exhibits $m_0, m_1 \in M$ explicitly
+
+The advantage of adversary is defined as the probability of distinguishing $E(k, m_0)$ from $E(k, m_1)$.
+
+#### Weakness for stream ciphers
+
+- Week pseudorandom generator
+- Key re-use
+- Predicable effect of modifying ciphertext or decrypted plaintext.
+
+### Block cipher
+
+View cipher as a Pseudo-Random Permutation (PRP)
+
+#### Pseudorandom permutation
+
+- 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 one-to-one mapping from message space to message space
+
+#### Security of block ciphers
+
+Intuition: a PRP is secure if: a random function in $Perms[X]$ is indistinguishable from a random function in $SF$ (real random permutation function)
+
+The adversarial game is to let adversary decide $x$, then we choose random key $k$ and give $E(k,x)$ and real random permutation $Perm(X)$ to let adversary decide which is which.
+
+#### Block cipher constructions: Feistel network
+
+Forward network:
+
+![Feistel network](https://notenextra.trance-0.com/CSE4303/Feistel_network.png)
+
+- Forward (round $i$): given $(L_{i-1}, R_{i-1}) \in \{0,1\}^n \times \{0,1\}^n$,
+  - $L_i = R_{i-1}$
+  - $R_i = L_{i-1} \oplus f_i(R_{i-1})$
+
+- Proof (construct the inverse):
+  - Suppose we are given the output of round $i$, namely $(L_i, R_i)$.
+  - Recover the previous right half immediately:
+    - $R_{i-1} = L_i$
+  - Then recover the previous left half by undoing the XOR:
+    - $L_{i-1} = R_i \oplus f_i(R_{i-1}) = R_i \oplus f_i(L_i)$
+  - Therefore each round map is invertible, with inverse transformation:
+    - $R_{i-1} = L_i$
+    - $L_{i-1} = f_i(L_i) \oplus R_i$
+  - Applying this inverse for $i=d,d-1,\ldots,1$ recovers $(L_0,R_0)$ from $(L_d,R_d)$, so the whole Feistel network $F$ is invertible.
+
+- Notation sketch (each wire is $n$ bits):
+  - Input: $(L_0, R_0)$
+  - Rounds:
+    - $L_1 = R_0,\ \ R_1 = L_0 \oplus f_1(R_0)$
+    - $L_2 = R_1,\ \ R_2 = L_1 \oplus f_2(R_1)$
+    - $\cdots$
+    - $L_d = R_{d-1},\ \ R_d = L_{d-1} \oplus f_d(R_{d-1})$
+  - Output: $(L_d, R_d)$
+
+#### Block ciphers: block modes: ECB
+
+New attacker model for multi-use keys (e.g. multiple blocks): CPA (Chosen Plaintext)-capable, not just CT-only
+
+- Attacker sees many PT/CT pairs for same key
+- Conservative model: attacker submits arbitrary PT (hence "C"PA)
+- Cipher goal: maintain semantic security against CPA
+
+#### CPA indistinguishability game
+
+- Updated adversarial game for a CPA attacker:
+  - Let $E = (E, D)$ be a cipher defined over $(K, M, C)$. For $b \in \{0,1\}$ define $\operatorname{EXP}(b)$ as:
+
+- Experiment $\operatorname{EXP}(b)$:
+  - Challenger samples $k \leftarrow K$.
+  - For each query $i = 1,\ldots,q$:
+    - Adversary outputs messages $m_{i,0}, m_{i,1} \in M$ such that $|m_{i,0}| = |m_{i,1}|$.
+    - Challenger returns $c_i \leftarrow E(k, m_{i,b})$.
+
+- Encryption-oracle access (CPA):
+  - If the adversary wants $c = E(k, m)$, it queries with $m_{j,0} = m_{j,1} = m$ (so the response is $E(k,m)$ regardless of $b$).
+
+#### Semantic security under CPA
+
+- Def: $E$ is semantically secure under CPA if for all "efficient" adversaries $A$,
+  - $\operatorname{Adv}^{\operatorname{CPA}}[A,E] = \left|\Pr[\operatorname{EXP}(0)=1] - \Pr[\operatorname{EXP}(1)=1]\right|$
+  - is negligible.
+
+### Summary for symmetric encrption
+
+1. Stream ciphers
+   - Rely on secure PRG
+   - No key re-use
+   - Fast, low-mem, less robust
+2. Block ciphers
+   - Rely on secure PRP
+   - Allow key re-use (usually only across blocks, not sessions)
+   - Provide authenticated encryption in some modes (e.g. GCM)
+   - Slower, higher-mem, more robust
+   - Used in practice for most crypto tasks (including secure network channels)
+
+## Hash functions
+
+### Hash function security properties
+
+- Given a function $h:X \to Y$, we say that $h$ is:
+
+- 1. Preimage resistant (one-way) if:
+  - given $y \in Y$ it is computationally infeasible to find a value $x \in X$ s.t. $h(x) = y$
+
+- 2. 2nd preimage resistant (weak collision resistant) if:
+  - given a specific $x \in X$ it is computationally infeasible to find a value $x' \in X$ s.t. $x' \ne x$ and $h(x') = h(x)$
+
+- 3. Collision resistant (strong collision resistant) if:
+  - it is computationally infeasible to find any two distinct values $x', x \in X$ s.t. $h(x') = h(x)$
+
+### Collision resistance: adversarial definition
+
+- Let $H: M \to T$ be a hash function ($|M| \gg |T|$).
+- A function $H$ is collision resistant if for all (explicit) "efficient" algorithms $A$,
+  - $\operatorname{Adv}^{\operatorname{CR}}[A,H] = Pr[$A outputs a collision for $H$ $]$
+  - is negligible
+
+### Hash function integrity applications
+
+1. Delayed knowledge verification
+2. Password storage
+3. Trusted timestamping / blockchains
+4. Integrity check on software
+
+#### File integrity with secure read-only space
+
+- When user downloads package, can verify that contents are valid
+- $H$ collision resistant $\Rightarrow$ attacker cannot modify package without detection
+- No encryption needed (public verifiability) if publisher has secure read-only space (e.g. trusted website, social media account)
+
+#### Symmetric-crypto message authentication
+
+- Context: Assume no secure RO space (insecure channel only)
+  - Need means of message authentication
+- Idea: add tag to message
+- System: Message Authentication Code (MAC)
+- Def: a MAC $I=(S,V)$ defined over $(K,M,T)$ is a pair of algorithms:
+  - $S(k,m)$ outputs $t \in T$  // "Sign"
+  - $V(k,m,t)$ outputs `yes' or `no'  // "Verify"
+
+- Symmetric-crypto message authentication:
+  - Alice and Bob share secret key $k$
+  - Generate tag: $\text{tag} \leftarrow S(k,m)$
+  - Verify tag: $V(k,m,\text{tag}) = \texttt{yes}?$
+
+#### MAC security model
+
+- For a MAC $I=(S,V)$ and adversary $A$, define a MAC game as:
+- Def: $I=(S,V)$ is a secure MAC if for all "efficient" $A$,
+  - $\operatorname{Adv}^{\operatorname{MAC}}[A,I] = \Pr[\text{Chal. outputs }1]$
+  - is negligible
+
+- MAC game (sketch):
+  - Challenger samples $k \leftarrow K$
+  - Adversary makes queries $m_1,\ldots,m_q \in M$
+    - For each $i$, challenger returns $t_i \leftarrow S(k,m_i)$
+  - Adversary outputs a candidate forgery $(m,t)$
+  - Challenger outputs $b=1$ if:
+    - $V(k,m,t)=\texttt{yes}$ and
+    - $(m,t) \notin \{(m_1,t_1),\ldots,(m_q,t_q)\}$
+  - Otherwise challenger outputs $b=0$
+
+- MAC security example: secure PRF not sufficient
+  - Suppose $F: K \times X \to Y$ is a secure PRF with $Y=\{0,1\}^{10}$.
+  - Is the derived MAC $I_F$ a secure MAC system?
+    - No: tags are too short, anyone can guess the tag for any message
+
+#### MACs from PRFs: sufficient security condition
+
+- Thm: If $F: K \times X \to Y$ is a secure PRF and $1/|Y|$ is negligible (i.e. $|Y|$ is large), then $I_F$ is a secure MAC.
+- In particular, for every efficient MAC adversary $A$ attacking $I_F$, there exists an efficient PRF adversary $B$ attacking $F$ such that:
+  - $\operatorname{Adv}^{\operatorname{MAC}}[A, I_F] \le \operatorname{Adv}^{\operatorname{PRF}}[B, F] + 1/|Y|$
+- Therefore $I_F$ is secure as long as $|Y|$ is large, e.g. $|Y| = 2^{80}$.
+
+#### MACs from collision resistance
+
+- Let $I=(S,V)$ be a MAC for short messages over $(K,M,T)$ (e.g. AES).
+- Let $H: M_{\text{big}} \to M$.
+- Def: $I_{\text{big}}=(S_{\text{big}},V_{\text{big}})$ over $(K,M_{\text{big}},T)$ as:
+  - $S_{\text{big}}(k,m) = S(k, H(m))$
+  - $V_{\text{big}}(k,m,t) = V(k, H(m), t)$
+- Thm: If $I$ is a secure MAC and $H$ is collision resistant, then $I_{\text{big}}$ is a secure MAC.
+- Example: $S(k,m) = \operatorname{AES2\text{-}block\text{-}cbc}(k, \operatorname{SHA\text{-}256}(m))$ is a secure MAC.
+
+#### Using HMACs for confidentiality + integrity
+
+- Confidentiality:
+  - Semantic security under a CPA
+  - Encryption secure against eavesdropping only
+- Integrity:
+  - Existential unforgeability under a CPA
+  - CBC-MAC, HMAC
+  - Hash functions
+- Confidentiality + integrity:
+  - CCA security
+  - Secure against tampering
+  - Method: Authenticated Encryption (AE)
+  - Encryption + MAC, in correct form
+
+#### Authenticated Encryption: security defs
+
+- An authenticated encryption system $(E,D)$ is a cipher where:
+  - $E: K \times M \times N \to C$
+  - $D: K \times C \times N \to M \cup$ cipher text rejected
+- Security: the system must provide
+  - semantic security under a CPA attack, and
+  - ciphertext integrity: attacker cannot create new ciphertexts that decrypt properly
+
+#### Ciphertext integrity
+
+- Let $(E,D)$ be a cipher with message space $M$.
+- Def: $(E,D)$ has ciphertext integrity if for all "efficient" $A$,
+  - $\operatorname{Adv}^{\operatorname{CI}}[A,E] = \Pr[\text{Chal. outputs }1]$
+  - is negligible
+
+- Security model: ciphertext integrity (sketch):
+  - Challenger samples $k \leftarrow K$
+  - Adversary makes encryption queries $m_1,\ldots,m_q \in M$
+    - For each $i$, challenger returns $c_i \leftarrow E(k,m_i)$
+  - Adversary outputs a ciphertext $c$
+  - Challenger outputs $b=1$ if:
+    - $D(k,c) \ne \bot$ and
+    - $c \notin \{c_1,\ldots,c_q\}$
+  - Otherwise challenger outputs $b=0$
+
+#### Authenticated encryption implies CCA security
+
+- Thm: Let $(E,D)$ be a cipher that provides AE. Then $(E,D)$ is CCA secure.
+- In particular, for any $q$-query efficient adversary $A$, there exist efficient $B_1,B_2$ such that:
+  - $\operatorname{Adv}^{\operatorname{CCA}}[A,E] \le 2q \cdot \operatorname{Adv}^{\operatorname{CI}}[B_1,E] + \operatorname{Adv}^{\operatorname{CPA}}[B_2,E]$
+- Interpretation: CCA advantage is $\le O(\text{CT-integrity advantage}) + \text{CPA advantage}$.
+
+- AE implication: authenticity
+  - Attacker cannot fool Bob into thinking a message was sent from Alice
+  - If attacker cannot create a valid ciphertext $c \notin \{c_1,\ldots,c_q\}$, then whenever $D(k,c) \ne \bot$ Bob knows the message is from someone who knows $k$ (but it could be a replay)
+
+- DS construction example: signing a certificate
+
+### Comparison: integrity/authentication approaches
+
+- 1) Collision resistant hashing: need a read-only public space
+  - Allows public verification if the hash is published in a small read-only public space
+- 2) MACs: must compute a new MAC for every client/user
+  - Must manage a long-term secret key per user to verify MACs (depending on application)
+  - Typically useful when one party signs, one verifies
+- 3) Digital signatures: must manage a long-term secret key
+  - E.g. vendor's signature on software is shipped with software
+  - Allows software to be downloaded from an untrusted distribution site
+  - Public-key verification/rejection works, provided public key distribution is trustworthy
+  - Typically useful when one party signs, many verify
+
+## Asymmetric key cryptography
+
+### Asymmetric crypto overview
+
+- Parties: sender, recipient, attacker (eavesdropping)
+- Goal: sender encrypts a plaintext to a ciphertext using a public key; recipient decrypts using a private key.
+
+#### Public-key encryption system
+
+- Def: a public-key encryption system is a triple of algorithms $(G, E, D)$:
+  - $G()$: randomized algorithm that outputs a key pair $(pk, sk)$
+  - $E(pk, m)$: randomized algorithm that takes $m \in M$ and outputs $c \in C$
+  - $D(sk, c)$: deterministic algorithm that takes $c \in C$ and outputs $m \in M$ or $\bot$
+- Consistency: for all $(pk, sk)$ output by $G$, for all $m \in M$,
+  - $D(sk, E(pk, m)) = m$
+
+#### Trapdoor function
+
+- Def: a trapdoor function $X \to Y$ is a triple of efficient algorithms $(G, F, F^{-1})$:
+  - $G()$: randomized algorithm that outputs a key pair $(pk, sk)$
+  - $F(pk, \cdot)$: deterministic algorithm that defines a function $X \to Y$
+  - $F^{-1}(sk, \cdot)$: defines a function $Y \to X$ that inverts $F(pk, \cdot)$
+- More precisely: for all $(pk, sk)$ output by $G$, for all $x \in X$,
+  - $F^{-1}(sk, F(pk, x)) = x$
+
+#### Symmetric vs. asymmetric security: attacker models
+
+- Symmetric ciphers: two security notions for a passive attacker
+  - One-time security (stream ciphers: ciphertext-only)
+  - Many-time security (block ciphers: CPA)
+  - One-time security $\Rightarrow$ many-time security
+  - Example: ECB mode is one-time secure but not many-time secure
+- Public-key encryption: single notion for a passive attacker
+  - Attacker can encrypt by themselves using the public key
+  - Therefore one-time security $\Rightarrow$ many-time security (CPA)
+  - Implication: public-key encryption must be randomized
+  - Analogous to secure block modes for block ciphers
+
+### Semantic security of asymmetric crypto (IND-CPA)
+
+#### IND-CPA game for public-key encryption
+
+- For $b \in \{0,1\}$ define experiments $\operatorname{EXP}(0)$ and $\operatorname{EXP}(1)$:
+
+- Experiment $\operatorname{EXP}(b)$:
+  - Challenger runs $(pk, sk) \leftarrow G()$
+  - Challenger sends $pk$ to adversary $A$
+  - Adversary outputs $m_0, m_1 \in M$ such that $|m_0| = |m_1|$
+  - Challenger returns $c \leftarrow E(pk, m_b)$
+  - Adversary outputs a bit $b' \in \{0,1\}$ (often modeled as outputting 1 if it "guesses $b=1$")
+
+#### Semantic security (IND-CPA)
+
+- Def: $E = (G, E, D)$ is semantically secure (a.k.a. IND-CPA) if for all efficient adversaries $A$,
+  - $\operatorname{Adv}^{\operatorname{SS}}[A, E] = \left|\Pr[\operatorname{EXP}(0)=1] - \Pr[\operatorname{EXP}(1)=1]\right|$
+  - is negligible
+- Note: inherently multiple-round because the attacker can always encrypt on their own using $pk$ (CPA power is "built in").
+
+### RSA cryptosystem: overview
+
+- Setup:
+  - $n = pq$, with $p$ and $q$ primes
+  - Choose $e$ relatively prime to $\phi(n) = (p-1)(q-1)$
+  - Choose $d$ as the inverse of $e$ in $\mathbb{Z}_{\phi(n)}$
+- Keys:
+  - Public key: $K_E = (n, e)$
+  - Private key: $K_D = d$
+- Encryption:
+  - Plaintext $M \in \mathbb{Z}_n$
+  - $C = M^e \bmod n$
+- Decryption:
+  - $M = C^d \bmod n$
+
+- Example:
+  - Setup:
+    - $p = 7$, $q = 17$
+    - $n = 7 \cdot 17 = 119$
+    - $\phi(n) = 6 \cdot 16 = 96$
+    - $e = 5$
+    - $d = 77$
+  - Keys:
+    - public key: $(119, 5)$
+    - private key: $77$
+  - Encryption:
+    - $M = 19$
+    - $C = 19^5 \bmod 119 = 66$
+  - Decryption:
+    - $M = 66^{77} \bmod 119 = 19$
+
+- Security intuition:
+  - To invert RSA without $d$, attacker must compute $x$ from $c = x^e \pmod n$.
+  - Best known approach:
+    - Step 1: factor $n$ (hard)
+    - Step 2: compute $e$-th roots modulo $p$ and $q$ (easy once factored)
+  - Notes (as commonly stated in lectures):
+    - 1024-bit RSA is within reach; 2048-bit is recommended usage
+
+### Diffie-Hellman key exchange (informal)
+
+- Fix a large prime $p$ (e.g., 2000 bits)
+- Fix an integer $g \in \{1,\ldots,p\}$
+
+- Protocol:
+  - Alice chooses random $a \in \{1,\ldots,p-1\}$ and sends $A = g^a \bmod p$
+  - Bob chooses random $b \in \{1,\ldots,p-1\}$ and sends $B = g^b \bmod p$
+  - Shared key:
+    - Alice computes $k_{AB} = B^a \bmod p = g^{ab} \bmod p$
+    - Bob computes $k_{AB} = A^b \bmod p = g^{ab} \bmod p$
+
+- Hardness assumptions:
+  - Discrete log problem: given $p, g, y = g^x \bmod p$, find $x$
+  - Diffie-Hellman function: $\operatorname{DH}_g(g^a, g^b) = g^{ab} \bmod p$
+
+#### Diffie-Hellman: security notes
+
+- As described, the protocol is insecure against active attacks:
+  - A man-in-the-middle (MiTM) can insert themselves and create 2 separate secure sessions
+- Fix idea: need a way to bind identity to a public key
+  - In practice: web of trust (e.g., GPG) or Public Key Infrastructure (PKI)
+
+### Implementing trapdoor functions securely
+
+- Never encrypt by applying $F$ directly to plaintext:
+  - Deterministic: cannot be semantically secure
+  - Many attacks exist for concrete TDFs
+  - Same plaintext blocks yield same ciphertext blocks
+
+- Naive (insecure) sketch:
+  - $E(pk, m)$: output $c \leftarrow F(pk, m)$
+  - $D(sk, c)$: output $F^{-1}(sk, c)$
+
+### Public-key encryption from TDFs
+
+- Components:
+  - $(G, F, F^{-1})$: secure TDF $X \to Y$
+  - $(E_s, D_s)$: symmetric authenticated encryption over $(K, M, C)$
+  - $H: X \to K$: a hash function
+
+- Construction of $(G, E, D)$ (with $G$ same as in the TDF):
+  - $E(pk, m)$:
+    - sample $x \leftarrow X$, compute $y \leftarrow F(pk, x)$
+    - derive $k \leftarrow H(x)$, compute $c \leftarrow E_s(k, m)$
+    - output $(y, c)$
+  - $D(sk, (y, c))$:
+    - compute $x \leftarrow F^{-1}(sk, y)$
+    - derive $k \leftarrow H(x)$, compute $m \leftarrow D_s(k, c)$
+    - output $m$
+
+- Visual intuition:
+  - header: $y = F(pk, x)$
+  - body: $c = E_s(H(x), m)$
+
+- Security theorem (lecture-style statement):
+  - If $(G, F, F^{-1})$ is a secure TDF, $(E_s, D_s)$ provides authenticated encryption, and $H$ is modeled as a random oracle, then $(G, E, D)$ is CCA-secure in the random oracle model (often denoted CCA-RO).
+  - Extension exists to reach full CCA (outside the RO idealization).
+
+### Wrapup: symmetric vs. asymmetric systems
+
+- Symmetric: faster, but key distribution is hard
+- Asymmetric: slower, but key distribution/management is easier
+- Application: secure web sessions (e.g., online shopping)
+  - Use symmetric-key encrypted sessions for bulk traffic
+  - Exchange symmetric keys using an asymmetric scheme
+  - Authenticate public keys (PKI or web of trust)
+
+### Key exchange: summary
+
+- Symmetric-key encryption challenges:
+  - Key storage: one per user pair, $O(n^2)$ total for $n$ users
+  - Key exchange: how to do it over a non-secure channel?
+
+- Possible solutions:
+
+- 1) Trusted Third Party (TTP)
+  - All users establish separate secret keys with the TTP
+  - TTP helps manage user-user keys (storage and secure channel)
+  - Applicability:
+    - Works for local domains
+    - Popular implementation: Kerberos for Single Sign On (SSO)
+  - Challenges:
+    - Scale: central authentication server is not suitable for the entire Internet
+    - Latency: requires online response from central server for every user-user session
+
+- 2) Public/private keys with certificates
+  - All users have a single stable public key (helps with key storage and exchange)
+  - Users exchange per-session symmetric keys via a secure channel using public/private keys
+  - Trusting public keys: binding is validated by a third-party authority (Certificate Authority, CA)
+    - Why better than TTP? CAs can validate statically by issuing certificates, then be uninvolved
+  - CA/certificate process covered in a future lecture
+
+## Appendix for additional algorithms and methods
+
+### Feistel network (used by several items below)
+
+A **Feistel network** splits a block into left/right halves and iterates rounds of the form $(L_{i+1},R_{i+1})=(R_i, L_i\oplus F(R_i,K_i))$, so decryption reuses the same structure with subkeys in reverse order.
+
+Feistel-based here: **DES, 3DES, CAMELLIA, SEED, GOST 28147-89 (and thus GOST89MAC uses a Feistel block cipher internally).**
+
+### Key exchange and authentication selectors (not symmetric encryption, not MAC)
+
+These describe *how keys are negotiated- and/or *how the peer is authenticated*, not whether payload is a block/stream cipher.
+
+#### RSA / DH / ECDH families
+
+- **kRSA, RSA** — (key exchange) the premaster secret is sent encrypted under the server’s RSA public key (classic TLS RSA KX).
+- **aRSA, aECDSA, aDSS, aGOST, aGOST01** — (authentication) the server identity is proven via a certificate signature scheme (RSA / ECDSA / DSA / GOST).
+- **kDHr, kDHd, kDH** — (key exchange) *static- DH key agreement using DH certificates (obsolete/removed in newer OpenSSL).
+- **kDHE, kEDH, DH / DHE, EDH / ECDHE, EECDH / kEECDH, kECDHE, ECDH** — (key exchange) *ephemeral- (EC)DH derives a fresh shared secret each handshake; "authenticated" variants bind it to a cert/signature.
+- **aDH** — (authentication selector) indicates DH-authenticated suites (DH certs; also removed in newer OpenSSL).
+
+#### PSK family
+
+- **PSK** — (keying model) uses a pre-shared secret as the authentication/secret basis.
+- **kPSK, kECDHEPSK, kDHEPSK, kRSAPSK** — (key exchange) PSK combined with (EC)DHE or RSA to derive/transport session keys.
+- **aPSK** — (authentication) PSK itself authenticates endpoints (except RSA_PSK where cert auth may be involved).
+
+---
+
+### Symmetric encryption / AEAD (this is where "block vs stream" applies)
+
+#### AES family
+
+- **AES128 / AES256 / AES** — **encryption/decryption**; **block cipher**; core algorithm: AES is an SPN (substitution–permutation network) of repeated SubBytes/ShiftRows/MixColumns/AddRoundKey rounds.
+- **AES-GCM** — **both encryption + message authentication (AEAD)**; **both** (AES block cipher used in counter mode + auth); core algorithm: encrypt with AES-CTR and authenticate with GHASH over ciphertext/AAD to produce a tag.
+- **AES-ECB**: Functionality is encryption/decryption (confidentiality only) using a block cipher mode; core algorithm encrypts each 128-bit plaintext block independently under the same key, which deterministically leaks patterns because equal plaintext blocks map to equal ciphertext blocks.
+- **AES-CBC**: Functionality is encryption/decryption (confidentiality only) using a block cipher mode; core algorithm XORs each plaintext block with the previous ciphertext block (starting from a fresh unpredictable IV) before AES-encrypting, which hides repetitions but requires correct IV handling and padding for non-multiple-of-block messages.
+- **AES-OFB** — **encryption**; both (stream-like); repeatedly AES-encrypts an internal state to generate a keystream and XORs it with plaintext, where the state evolves independently of the plaintext/ciphertext.
+- **AESCCM / AESCCM8** — **both encryption + message authentication (AEAD)**; **both**; core algorithm: compute CBC-MAC then encrypt with CTR mode, with 16-byte vs 8-byte tag length variants.
+
+#### ARIA family
+
+- **ARIA128 / ARIA256 / ARIA** — **encryption/decryption**; **block cipher**; core algorithm: ARIA is an SPN-style block cipher with byte-wise substitutions and diffusion layers across rounds.
+
+#### CAMELLIA family
+
+- **CAMELLIA128 / CAMELLIA256 / CAMELLIA** — **encryption/decryption**; **block cipher**; core algorithm: Camellia is a **Feistel network** with round functions plus extra FL/FL$^{-1}$ layers for nonlinearity and diffusion. *(Feistel: yes)*
+
+#### ChaCha20
+
+- **CHACHA20** — **encryption/decryption**; **stream cipher**; core algorithm: ChaCha20 generates a keystream via repeated ARX (add-rotate-xor) quarter-rounds on a 512-bit state and XORs it with plaintext.
+
+#### DES / 3DES
+
+- **DES** — **encryption/decryption**; **block cipher**; core algorithm: DES is a 16-round **Feistel network** using expansion, S-boxes, and permutations. *(Feistel: yes)*
+- **3DES** — **encryption/decryption**; **block cipher**; core algorithm: applies DES three times (EDE or EEE) to increase effective security while retaining the **Feistel** DES core. *(Feistel: yes)*
+
+#### RC4
+
+- **RC4** — **encryption/decryption**; **stream cipher**; core algorithm: maintains a 256-byte permutation and produces a keystream byte-by-byte that is XORed with plaintext.
+
+#### RC2 / IDEA / SEED
+
+- **RC2** — **encryption/decryption**; **block cipher**; core algorithm: mixes key-dependent operations (adds, XORs, rotates) across rounds with "mix" and "mash" steps (not Feistel).
+- **IDEA** — **encryption/decryption**; **block cipher**; core algorithm: combines modular addition, modular multiplication, and XOR in a Lai–Massey-like structure to achieve diffusion/nonlinearity (not Feistel).
+- **SEED** — **encryption/decryption**; **block cipher**; core algorithm: a 16-round **Feistel network** with nonlinear S-box-based round functions. *(Feistel: yes)*
+
+---
+
+### Hash / MAC / digest selectors (message authentication side)
+
+These are not "ciphers" but are used for integrity/authentication (often as HMAC, PRF, signatures).
+
+- **MD5** — **message authentication component** (typically via HMAC, historically); **cipher method: N/A**; core algorithm: iterated Merkle–Damgård hash compressing 512-bit blocks into a 128-bit digest (now considered broken for collision resistance).
+- **SHA1, SHA** — **message authentication component** (typically HMAC-SHA1 historically); **N/A**; core algorithm: Merkle–Damgård hash producing 160-bit output via 80-step compression (collisions known).
+- **SHA256 / SHA384** — **message authentication component** (HMAC / TLS PRF / signatures); **N/A**; core algorithm: SHA-2 family Merkle–Damgård hashes with different word sizes/output lengths (256-bit vs 384-bit).
+- **GOST94** — **message authentication component** (HMAC based on GOST R 34.11-94); **N/A**; core algorithm: builds an HMAC tag by hashing inner/outer padded key with the message using the GOST hash.
+- **GOST89MAC** — **message authentication**; **block-cipher-based MAC (so "block" internally)**; core algorithm: computes a MAC using the GOST 28147-89 block cipher in a MAC mode (cipher-based chaining). *(Feistel internally via GOST 28147-89)*
+
+> Latest version of cheatsheet distilled from this note.

content/CSE4303/_meta.js

@@ -3,6 +3,7 @@ export default {
     "---":{
         type: 'separator'
     },
+    CSE4303_E1: "Exam review",
     CSE4303_L1: "Introduction to Computer Security (Lecture 1)",
     CSE4303_L2: "Introduction to Computer Security (Lecture 2)",
     CSE4303_L3: "Introduction to Computer Security (Lecture 3)",

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -78,6 +78,27 @@ An $m$-dimensional **manifold** is a topological space  $X$ that is
 2. Second countable: With a countable basis
 3. Local euclidean: Each point of $x$ of $X$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^m$.
 
+<details>
+<summary>Example of space that is not a manifold but satisfies part of the definition</summary>
+
+Non-hausdorff:
+
+Consider the set with two origin $\mathbb{R}\setminus\{0\}$. with $\{p,q\}$, and the topology defined over all the open intervals that don't contain the origin, with set of the form $(-a,0)\cup \{p\}\cup (0,a)$ for $a\in \mathbb{R}$ and $(-a,0)\cup \{q\}\cup (0,a)$.
+
+---
+
+Non-second-countable:
+
+Consider the long line $\mathbb{R}\times [0,1)$
+
+---
+
+Non-local-euclidean:
+
+Any 1-dimensional CW complex (graph) that has a vertex with 3 or more edges connected to it will be Hausdorff and second-countable, but not locally Euclidean at those vertices.
+
+</details>
+
 #### Whitney's Embedding Theorem
 
 If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$ for some positive integer $N$.
@@ -97,6 +118,12 @@ Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair o
 
 Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
 
+#### Definition of paracompact space
+
+Locally finite: $\forall x\in X$, $\exists$ open $x\in U$ such that $U$ only intersects finitely many open sets in $\mathcal{B}$.
+
+A space $X$ is paracompact if every open cover $A$ of $X$ has a **locally finite** refinement $\mathcal{B}$ of $A$ that covers $X$.
+
 ### Homotopy
 
 #### Definition of homotopy equivalent spaces
@@ -128,7 +155,6 @@ Two pathes $f$ and $f'$ are path homotopic if
 
 The $\simeq$, $\simeq_p$ are both equivalence relations.
 
-
 #### Definition for product of paths
 
 Given $f$ a path in $X$ from $x_0$ to $x_1$ and $g$ a path in $X$ from $x_1$ to $x_2$.