# 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.
