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+---
+share: true
+toc: true
+math: true
+categories:
+  - Lecture Notes
+  - Modern Cryptography
+tags:
+  - lecture-note
+  - cryptography
+  - security
+title: 2. PRFs, PRPs and Block Ciphers
+date: 2023-09-12
+github_title: 2023-09-12-prfs-prps-block-ciphers
+image:
+  path: assets/img/posts/Lecture Notes/Modern Cryptography/mc-02-block-cipher.png
+attachment:
+  folder: assets/img/posts/Lecture Notes/Modern Cryptography
+---
+
+## Pseudorandom Functions (PRF)
+
+> **Definition.** A **pseudorandom function** $F$ over $(\mathcal{K}, X, Y)$ is an efficiently computable algorithm $F : \mathcal{K} \times X \rightarrow Y$.
+
+We consider a *keyed function* $F : \mathcal{K} \times X \rightarrow Y$ where $\mathcal{K}$ denotes the key space. For $k \in \mathcal{K}$, $F_k(x) := F(k, x)$ is a function from $X$ to $Y$. Thus each key $k$ induces a distribution on functions $X \rightarrow Y$.
+
+Note that $\left\lvert \mathcal{F}[X, Y] \right\lvert = \left\lvert Y \right\lvert^{\left\lvert X \right\lvert}$, but the number of PRFs is at most $\left\lvert K \right\lvert$. In practice, $\left\lvert K \right\lvert$ is very small compared to $\left\lvert Y \right\lvert^{\left\lvert X \right\lvert}$. So PRFs are chosen from a smaller space, but they should behave in the same way a (truly) random function does.
+
+Let $\mathcal{F}[X, Y]$ denote the set of all functions from $X$ to $Y$. A PRF $F$ would be **secure** if it is **indistinguishable from a random function in $\mathcal{F}[X, Y]$**. This notion is formalized as a security game.
+
+> **Definition.** Let $F$ be a PRF defined over $(\mathcal{K}, X, Y)$. For an adversary $\mathcal{A}$, define two experiments 0 and 1.
+> 
+> **Experiment $b$**.
+> 1. The challenger selects $f \in \mathcal{F}[X, Y]$ as follows
+> 	- If $b = 0$, choose $k \leftarrow \mathcal{K}$ and set $f = F_k = F(k, \cdot)$.
+> 	- If $b = 1$, choose $f \leftarrow \mathcal{F}[X, Y]$.
+> 2. The adversary sends a sequence of queries to the challenger.
+> 	- For $i = 1, \dots, q$, send $x_i \in X$ and receive $y_i = f(x_i) \in Y$.
+> 3. The adversary computes and outputs a bit $b' \in \left\lbrace 0, 1 \right\rbrace$.
+> 
+> Let $W_b$ be the event that $\mathcal{A}$ outputs 1 in experiment $b$. Then the **PRF-advantage** of $\mathcal{A}$ with respect to $F$ is defined as
+> 
+> $$
+> \mathrm{Adv}_{\mathrm{PRF}}^q[\mathcal{A}, F] = \left\lvert \Pr[W_0] - \Pr[W_1] \right\lvert.
+> $$
+> 
+> A PRF $F$ is **secure** if $\mathrm{Adv}_{\mathrm{PRF}}^q[\mathcal{A}, F]$ is negligible for any efficient $\mathcal{A}$.
+
+In experiment $0$ above, the challenger returns $y_i = f(x_i)$. To answer the query $x_i$, the challenger would have to keep a lookup table for a random function $f \in \mathcal{F}[X, Y]$. Since $X$ and $Y$ are very large in practice, it is nearly impossible to create and manage such lookup tables.[^1] As a workaround, we can choose $y_i$ uniformly on each query $x_i$, assuming that $x_i$ wasn't queried before. This is possible since for any different inputs $x_i, x_j \in X$, $f(x_i)$ and $f(x_j)$ are random and independent.
+
+Also, there are two ways that the adversary can query the challenger. The first method is the **adaptive** method, where the adversary queries each $x_i$ one by one. In this method, the adversary can *adaptively* choose the next query $x_{i+1}$ after receiving the result from the challenger.
+
+The second method is querying all $x_i$ at once. We will consider the first method, since the adaptive method assumes greater attack power on the adversary.
+
+### OTP as a PRF
+
+As an example, consider the one-time pad function $F(k, x) = k \oplus x$. This function satisfies the definitions of PRFs, but it is not a secure PRF. Consider an adversary $\mathcal{A}$ that outputs $1$ if and only if $y_1 \oplus y_2 = x_1 \oplus x_2$. In experiment $0$, $\mathcal{A}$ will always output $1$, but in experiment $1$, $\mathcal{A}$ will output $1$ with probability $2^{-n}$. Thus the advantage is $1 - 2^{-n}$, which is not negligible.
+
+### PRFs and PRGs
+
+It is easy to construct PRGs from PRFs. We can simply evaluate $F$ on some distinct inputs. For example, given a seed $s$, we evaluate $F_s$ and obtain
+
+$$
+G(s) = F_s(1) \parallel F_s(2) \parallel \cdots \parallel F_s(n)
+$$
+
+for any $n \in \mathbb{N}$.[^2] In fact, we can show that $G$ is secure PRG if $F$ is a secure PRF.
+
+> **Theorem.** If $F$ is a secure length preserving PRF, then $G$ in the above definition is a secure PRG.
+
+*Proof*. Suppose that $\mathcal{A}$ is an efficient PRG adversary against $G$. We construct an efficient $n$-query PRF adversary $\mathcal{B}$, that queries the challenger at $1, \dots, n$ and receives $f(1), \dots, f(n)$. $\mathcal{B}$ passes $f(1) \parallel \cdots \parallel f(n)$ to $\mathcal{A}$, and $\mathcal{B}$ outputs the output of $\mathcal{A}$. Then we see that $\mathrm{Adv}_{\mathrm{PRG}}[\mathcal{A}, G] = \mathrm{Adv}_{\mathrm{PRF}}[\mathcal{B}, F]$. So if $F$ is secure, then $G$ is secure.
+
+As for the converse, a PRG $G$ gives a PRF $F$ with small input length. If $G : \left\lbrace 0, 1 \right\rbrace^n \rightarrow \left\lbrace 0, 1 \right\rbrace^{n 2^m}$, we can define a PRF $F : \left\lbrace 0, 1 \right\rbrace^n \times \left\lbrace 0, 1 \right\rbrace^m \rightarrow \left\lbrace 0, 1 \right\rbrace^n$ as follows: for a seed $s \in \left\lbrace 0, 1 \right\rbrace^n$, consider $G(s)$ as a $2^m \times n$ table and set $F(s, i)$ as the $i$-th row of $G(s)$.[^3] If $G$ is a secure PRG, then PRF $F$ is also secure.
+
+## Pseudorandom Permutations (PRP)
+
+> **Definition.** A **pseudorandom permutation** $E$ over $(\mathcal{K}, X)$ is an efficiently computable algorithm $E : \mathcal{K} \times X \rightarrow X$ that has an efficient inversion algorithm $D$ such that $D(k, E(k, x)) =x$.
+
+Permutations defined on a set $X$ are bijections on $X$, so a PRP $E(k, \cdot)$ is one-to-one for all $k \in \mathcal{K}$, and also has an inverse function.
+
+Similarly, a PRP $E$ is **secure** if it is **indistinguishable from a random permutation on $X$**. Let $\mathcal{P}[X]$ denote the set of all permutations on $X$.
+
+> **Definition.** Let $E$ be a PRP defined over $(\mathcal{K}, X)$. For an adversary $\mathcal{A}$, define two experiments $0$ and $1$.
+> 
+> **Experiment $b$**.
+> 1. The challenger selects $f \in \mathcal{P}[X]$ as follows
+> 	- If $b = 0$, choose $k \leftarrow \mathcal{K}$ and set $f = E_k = E(k, \cdot)$.
+> 	- If $b = 1$, choose $f \leftarrow \mathcal{P}[X]$.
+> 2. The adversary sends a sequence of queries to the challenger.
+> 	- For $i = 1, \dots, q$, send $x_i \in X$ and receive $y_i = f(x_i) \in Y$.
+> 3. The adversary computes and outputs a bit $b' \in \left\lbrace 0, 1 \right\rbrace$.
+> 
+> Let $W_b$ be the event that $\mathcal{A}$ outputs 1 in experiment $b$. Then the **PRP-advantage** of $\mathcal{A}$ with respect to $E$ is defined as
+> 
+> $$
+> \mathrm{Adv}_{\mathrm{PRP}}^q[\mathcal{A}, E] = \left\lvert \Pr[W_0] - \Pr[W_1] \right\lvert.
+> $$
+> 
+> A PRP $E$ is **secure** if $\mathrm{Adv}_{\mathrm{PRP}}^q[\mathcal{A}, E]$ is negligible for any efficient $\mathcal{A}$.
+
+### PRF Switching Lemma
+
+Suppose that $\left\lvert X \right\lvert$ is sufficiently large. Then for $q$ queries of any adversary $\mathcal{A}$, it is highly probable that $f(x_i)$ are all distinct, regardless of whether $f$ is a PRF or a PRP. Thus we have the following property of PRPs.
+
+> **Lemma.** If $E: \mathcal{K} \times X \rightarrow X$ is a secure PRP and $\left\lvert X \right\lvert$ is sufficiently large, then $E$ is also a secure PRF.
+> 
+> For any $q$-query adversary $\mathcal{A}$,
+> 
+> $$
+> \left\lvert \mathrm{Adv}_{\mathrm{PRF}}[\mathcal{A}, E] - \mathrm{Adv}_{\mathrm{PRP}}[\mathcal{A}, E] \right\lvert \leq \frac{q^2}{2\left\lvert X \right\lvert}.
+> $$
+
+This is a matter of *collisions* of $f(x_i)$, so we use the facts from the birthday problem.
+
+*Proof*. Appendix A.4.
+
+## Block Ciphers
+
+A **block cipher** is actually a different name for PRPs. Since a PRP $E$ is a keyed function, applying $E(k, x)$ is in fact encryption, and applying its inverse is decryption.
+
+![mc-02-block-cipher.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-block-cipher.png)
+
+Block ciphers commonly have the following form.
+- A key $k$ is chosen uniformly from $\left\lbrace 0, 1 \right\rbrace^s$.
+- The key $k$ goes through *key expansion* and generates $k_1, \dots, k_n$. These are called **round keys**, where $n$ is the number of rounds.
+- The plaintext goes through $n$ rounds, where in each round, a round function and the round key is applied to the input.
+- After $n$ rounds, the ciphertext is obtained.
+
+## Data Encryption Standard (DES)
+
+### Brief History
+
+- 1972: NIST calls for a block cipher standard (proposal).
+- 1974: Horst Feistel (IBM) designs *Lucifer*, which evolves into DES.
+- 1976: DES is standardized and widely implemented.
+- 1997: DES is broken by exhaustive search.
+- 2001: DES is replaced by AES.
+
+### Feistel Network
+
+Since block ciphers are PRPs, we have to build an invertible function. Suppose we are given **any** functions $F_1, \dots, F_d : \left\lbrace 0, 1 \right\rbrace^n \rightarrow \left\lbrace 0, 1 \right\rbrace^n$. Can we build an **invertible** function $F : \left\lbrace 0, 1 \right\rbrace^{2n} \rightarrow \left\lbrace 0, 1 \right\rbrace^{2n}$?
+
+![mc-02-feistel-network.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-feistel-network.png)
+
+It turns out the answer is yes. Given an $2n$-bit long input, $L_0$ and $R_0$ denote the left and right halves ($n$ bits) of the input, respectively. Define
+
+$$
+L_i = R_{i-1}, \qquad R_i = F_i(R_{i-1}) \oplus L_{i-1}
+$$
+
+for $i = 1, \dots, d$. Then we can restore $L_0$ and $R_0$ from $L_d$ and $R_d$ by applying the same operations in **reverse order**. It is easy to see that $R_{i-1}$ can be restored from $L_i$. As for $L_{i-1}$, we need a swap. Observe that
+
+$$
+F_i(L_i) \oplus R_i = F_i(R_{i-1}) \oplus (F_i(R_{i-1}) \oplus L_{i-1}) = L_{i-1}.
+$$
+
+Note that we did not require $F_i$ to be invertible. We can build invertible functions from arbitrary functions! These are called **Feistel networks**.
+
+> **Theorem.** (Luby-Rackoff'85) If $F : K \times \left\lbrace 0, 1 \right\rbrace^n \rightarrow \left\lbrace 0, 1 \right\rbrace^n$ is a secure PRF, then the $3$-round Feistel using the functions $F_i=  F(k_i, \cdot)$ is a secure PRP.
+
+In DES, the function $F_i$ is the DES round function.
+
+![mc-02-des-round.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-des-round.png)
+
+The Feistel function takes $32$ bit data and divides it into eight $4$ bit chunks. Each chunk is expanded to $6$ bits using $E$. Now, we have 48 bits of data, so apply XOR with the key for this round. Next, each $6$-bit block is compressed back to $4$ bits using a S-box. Finally, there is a permutation $P$ at the end, resulting in $32$ bit data.
+
+### DES Algorithm
+
+DES uses $56$ bit keys that generate $16$ rounds keys. The diagram below shows that DES has 16-round Feistel networks.
+
+![mc-02-DES.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-DES.png)
+
+The input goes through initial/final permutation, which are inverses of each other. These have no cryptographic significance, and just for engineering.
+
+## Advanced Encryption Standard (AES)
+
+DES is not secure, since key space and block length is too small. Thankfully, we have a replacement called the **advanced encryption standard** (AES).
+
+![mc-02-aes-128.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-aes-128.png)
+
+- DES key only had $56$ bits, so DES was broken in the 1990s
+- NIST standardized AES in 2001, based on Rijndael cipher
+- AES has $3$ different key lengths: $128$, $192$, $256$
+	- Different number of rounds for different key lengths
+	- $10$, $12$, $14$ rounds respectively
+- Input data block is $128$ bits, so viewed as $4\times 4$ table of bytes
+	- This table is called the **current state**.
+
+Each round consists of the following:
+- **SubBytes**: byte substitution, 1 S-box on every byte
+- **ShiftRows**: permutes bytes between groups and columns
+- **MixColumns**: mix columns by using matrix multiplication in $\mathrm{GF}(2^8)$.
+- **AddRoundKey**: XOR with round key
+
+### Modules
+
+#### SubBytes
+
+- A simple substitution of each byte using $16 \times 16$ lookup table.
+- Each byte is split into two $4$ bit *nibbles*
+	- Left half is used as row index
+	- Right half is used as column index
+
+#### ShiftRows
+
+- A circular bytes shift for each row, so it is a permutation
+- $i$-th row is shifted $i$ times to the left. ($i = 0, 1, 2, 3$)
+
+#### MixColumns
+
+- For each column, each byte is replaced by a value
+	- The value depends on all 4 bytes of the column
+- Each column is processed separately
+	- Thus effectively, it is a matrix multiplication (Hill cipher)
+
+#### AddRoundKey
+
+- XOR the input with $128$ bits of the round key
+	- The round key is different for each round
+
+These 4 modules are all invertible!
+
+## Attacks on Block Ciphers
+
+### DES S-box
+
+For DES, the S-box is the non-linear part. If the S-box is linear, then the entire DES cipher would be linear.
+
+Specifically, there would be a fixed binary matrix $B _ 1 \in \mathbb{Z} _ 2^{64 \times 64}$ and $B _ 2 \in \mathbb{Z} _ 2^{64 \times (48 \times 16)}$ such that
+
+$$
+\mathrm{DES}(k, m) = B_1 m \oplus B_2 \mathbf{k}
+$$
+
+where $\mathbf{k} \in \mathbb{Z}_2^{48 \times 16}$ is a vector of round keys.
+
+Then we can attack DES with the same idea as OTP. If $c_i = B_1 m_i \oplus B_2 \mathbf{k}$, then $c_1 \oplus c_2 = B_1 (m_1 \oplus m_2)$.
+
+Choosing the S-boxes at random results in a insecure block cipher, with high probability.
+
+### Strengthening Ciphers Against Exhaustive Search
+
+For DES (and AES-128), it is known that *three pairs of plaintext, ciphertext blocks* are enough to ensure that there is a *unique* key mapping the given plaintext to the ciphertext, with high probability. We assume here that we are given plaintext, ciphertext block pairs.
+
+If we were to find the key by brute force, DES is easy. We can strengthen the DES algorithm by using **nested ciphers**. The tradeoff here is that these are slower than the original DES.
+
+> Define
+> - (**2DES**) $2E: \mathcal{K}^2 \times \mathcal{M} \rightarrow \mathcal{M}$ as $2E((k_1, k_2), m) = E(k_1, E(k_2, m))$.
+> - (**3DES**) $3E: \mathcal{K}^3 \times \mathcal{M} \rightarrow \mathcal{M}$ as $3E((k_1, k_2, k_3), m) = E(k_1, D(k_2, E(k_3, m)))$.[^4]
+
+Then the key space has increased (exponentially). As for 2DES, the key space is now $2^{112}$, so maybe nested ciphers increase the level of security.
+
+#### $2E$ is Insecure: Meet in the Middle
+
+Unfortunately, 2DES is only secure as DES, with the attack strategy called **meet in the middle**. The idea is that if $c = E(k_1, E(k_2, m))$, then $D(k_1, c) = E(k_2, m)$.
+
+![mc-02-2des-mitm.png](../../../assets/img/posts/Lecture%20Notes/Modern%20Cryptography/mc-02-2des-mitm.png)
+
+Since we have the plaintext and the ciphertext, we first build a table of $(k, E(k_2, m))$ over $k_2 \in \mathcal{K}$ and sort by $E(k_2, m)$. Next, we check if $D(k_1, c)$ is in the table for all $k_1 \in \mathcal{K}$.
+
+The complexity of this attack is shown as follows:
+- Space complexity: We must store $\left\lvert \mathcal{K} \right\lvert$ entries with approximately $\log \left\lvert \mathcal{M} \right\lvert$ bits. Thus we need $\left\lvert \mathcal{K} \right\lvert \log \left\lvert \mathcal{M} \right\lvert$ bits in total.
+- Time complexity: We run $E$ for $\left\lvert \mathcal{K} \right\lvert$ times and sorting takes $\left\lvert \mathcal{K} \right\lvert \log \left\lvert \mathcal{K} \right\lvert$. Searching the decrypted messages will also take $\left\lvert \mathcal{K} \right\lvert \log \left\lvert \mathcal{K} \right\lvert$.[^5]
+
+Considering DES, $\left\lvert \mathcal{K} \right\lvert = 2^{56}$ and $\left\lvert \mathcal{M} \right\lvert = 2^{64}$, the search space for the key is far less than $2^{112}$. Moreover, one would need a huge amount of memory to perform this attack, but time and space can be traded off.
+
+3DES can be attacked in a similar manner, so the search space for the key is less than $2^{168}$, around $2^{112}$. It is currently infeasible to brute force this, so 3DES is used in practice.
+
+The above argument can be generalized for any scheme $(E, D)$. Thus, the $2E$ construction is not secure.
+
+#### $EX$ Construction and DESX
+
+There is another method called the **$EX$ construction** for block cipher $(E, D)$ defined over $(\mathcal{K}, \mathcal{X})$. The $EX$ construction uses a new block cipher defined as follows. Intuitively, it can be thought of as applying OTP before and after encryption.
+
+> Let $k_1 \in \mathcal{K}$ and $k_2, k_3 \in \mathcal{X}$.
+> - Encryption: $EX\left((k_1, k_2, k_3), m\right) = E(k_1, m \oplus k_2) \oplus k_3$.
+> - Decryption: $DX((k_1, k_2, k_3), c) = D(k_1, c \oplus k_3) \oplus k_2$.
+
+Then the new cipher $(EX, DX)$ has a key space $\mathcal{K} \times \mathcal{X}^2$, which is much larger than $\mathcal{K}$. Specifically for DESX, the key length would be $56 + 2 \times 64 = 184$ bits.
+
+As a side note, using $E(k_1, m) \oplus k_2$ or $E(k_1, m \oplus k_2)$ does not improve security, since it can be attacked with meet in the middle method. Similarly, DESX also has $184$ bit key space, but actual search space is about $56 + 64 = 120$ bits.
+
+### Attacks on AES
+
+The best known attack on AES is almost a brute force attack.
+- AES-128 requires $2^{126.1}$ evaluations of AES.[^6]
+- AES-192 requires $2^{189.74}$ evaluations of AES.
+- AES-256 requires $2^{254.42}$ evaluations of AES.
+
+These are about $4$ times better than exhaustive search, so they have little impact on the security of AES.
+
+There are also attacks with quantum computers, which uses an algorithm that can recover the key in time $\mathcal{O}(\sqrt{\left\lvert \mathcal{K} \right\lvert})$.[^7] We might need to use AES-256 when quantum computers are widely used.
+
+[^1]: The size of the lookup table is at least $\left\lvert X \right\lvert \log \left\lvert Y \right\lvert$ bits. The table should have at least $\left\lvert X \right\lvert$ rows, and each row must contain $\log \left\lvert Y \right\lvert$ bits of information.
+[^2]: This method is even *parallelizable*!
+[^3]: Note that $m$ has to be $\mathcal{O}(\log n)$ for $F$ to be efficient.
+[^4]: We could also encrypt $3$ times.
+[^5]: The running times of $E$ and $D$ were not considered here.
+[^6]: A. Bogdanov, D. Khovratovich, and C. Rechberger. Biclique cryptanalysis of the full AES. In ASIACRYPT 2011, pages 344–371. 2011.
+[^7]: L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219. ACM, 1996.
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