From a4d6cf19052d9b786279acb34f1ad196a679220e Mon Sep 17 00:00:00 2001 From: Sungchan Yi Date: Sun, 24 Sep 2023 17:41:01 +0900 Subject: [PATCH] [PUBLISHER] upload files #91 * PUSH NOTE : 03. Symmetric Key Cryptography (2).md * PUSH ATTACHMENT : is-03-cbc-encryption.png * PUSH ATTACHMENT : is-03-cfb-encryption.png * PUSH ATTACHMENT : is-03-ofb-encryption.png * PUSH ATTACHMENT : is-03-ctr-encryption.png --- ...2023-09-18-symmetric-key-cryptography-2.md | 318 ++++++++++++++++++ .../is-03-cbc-encryption.png | Bin 0 -> 46130 bytes .../is-03-cfb-encryption.png | Bin 0 -> 40114 bytes .../is-03-ctr-encryption.png | Bin 0 -> 41891 bytes .../is-03-ofb-encryption.png | Bin 0 -> 38565 bytes 5 files changed, 318 insertions(+) create mode 100644 _posts/Lecture Notes/Internet Security/2023-09-18-symmetric-key-cryptography-2.md create mode 100644 assets/img/posts/Lecture Notes/Internet Security/is-03-cbc-encryption.png create mode 100644 assets/img/posts/Lecture Notes/Internet Security/is-03-cfb-encryption.png create mode 100644 assets/img/posts/Lecture Notes/Internet Security/is-03-ctr-encryption.png create mode 100644 assets/img/posts/Lecture Notes/Internet Security/is-03-ofb-encryption.png diff --git a/_posts/Lecture Notes/Internet Security/2023-09-18-symmetric-key-cryptography-2.md b/_posts/Lecture Notes/Internet Security/2023-09-18-symmetric-key-cryptography-2.md new file mode 100644 index 0000000..8380769 --- /dev/null +++ b/_posts/Lecture Notes/Internet Security/2023-09-18-symmetric-key-cryptography-2.md @@ -0,0 +1,318 @@ +--- +share: true +toc: true +math: true +categories: + - Lecture Notes + - Internet Security +tags: + - lecture-note + - security + - cryptography +title: 03. Symmetric Key Cryptography (2) +date: 2023-09-18 +github_title: 2023-09-18-symmetric-key-cryptography-2 +image: + path: /assets/img/posts/Lecture Notes/Internet Security/is-03-ecb-encryption.png +attachment: + folder: assets/img/posts/Lecture Notes/Internet Security +--- + +## Block Cipher Overview + +- We need confusion and diffusion + - Confusion: relationship between ciphertext and key is complex + - Diffusion: relationship between message and ciphertext is complex +- Series of **substitutions** and **permutations** can achieve confusion and diffusion + +### Modules + +- **S-box**: a substitution module + - Usually for confusion + - $m \times n$ lookup box is needed, since it should be invertible +- **P-box**: a permutation module + - Usually for diffusion + - Compared to the number of input bits, + - *Expansion* if the number of output bits is larger + - *Compression* if the number of output bits is smaller + - *Straight* if the number of output bits is equal + +## Data Encryption Standard (DES) + +- Standardized in 1979. +- Block size is $64$ bits ($8$ bytes) +- $64$ bits input $\rightarrow$ $64$ bits output +- Key is $56$ bits, but there are $8$ bits representing parity, so total of $64$ bits + - Every $8$th bit is a parity bit + +### Encryption + +(Diagram) +- From the $56$-bit key, generate $16$ different $48$ bit keys $k_1, \dots, k_{16}$. +1. Initially, input goes through the P-box. +2. The output goes through $16$ rounds, and in the round $i$, key $k_i$ is used. +3. After $16$ rounds, split the output into two $32$ bit halves and swap them. +4. The output goes through the inverse of the P-box from Step 1. + +Let $L_{i-1} \parallel R_{i-1}$ be the output of round $i-1$, where $L_{i-1}$ and $R_{i-1}$ are $32$ bit halves. Also let $f$ be the Mangler function. + +In each round $i$, +- $L_i = R_{i - 1}$ +- $R_i = L_{i-1} \oplus f(k_i, R_{i-1})$ + +#### Mangler Function + +*Is the Mangler function invertible?* + +### Questions + +- Why does the input go through the P-box and its inverse at the end? + - Not for security, but for efficient hardware design. +- Why do we swap each $32$ bit halves? + - Not for security, but for engineering purposes, see below. +- Is DES invertible? + - Yes, message should be decrypted. + +But a Mangler function is *not invertible*, since it sends $4$ bits to $6$ bits during the evaluation process. Then how is decryption possible? + +### Decryption + +Let $f$ be the Mangler function. We can define each round as a function $F$, + +$$ +F(L_i \parallel R_i) = R_i \parallel L_i \oplus f(R_i). +$$ + +Consider a function $G$, defined as + +$$ +G(L_i \parallel R_i) = R_i \oplus f(L_i) \parallel L_i. +$$ + +Then, we see that + +$$ +\begin{align*} +G(F(L_i \parallel R_i)) &= G(R_i \parallel L_i \oplus f(R_i)) \\ +&= (L_i \oplus f(R_i)) \oplus f(R_i) \parallel R_i \\ +&= L_i \parallel R_i. +\end{align*} +$$ + +Thus $F$ and $G$ are inverses of each other, thus $f$ doesn't have to be invertible. This is called the **Feistel cipher**. + +Also, note that + +$$ +G(L_i \parallel R_i) = F(L_i \oplus f(R_i) \parallel R_i), +$$ + +so evaluating the decryption round is actually equivalent to running the encryption round with upper/lower $32$ bit halves swapped. Hence the reason for swapping each $32$ bit halves. + +## Advanced Encryption Standard (AES) + +- 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 $\mathbf{GF}(2^8)$. +- **AddRoundKey**: XOR with round key + +The first and last rounds are a little different. +- Before the first round, AddRoundKey is done. +- The last round does not have MixColumns. + +The objectives of AES: +- Build resistance against known attacks +- Code must be compact, and should run fast on many CPUs +- Design must be simple + +### 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! + +### Questions + +- Why is there a AddRoundKey at the beginning? +- Why is the last round different? + +Both are for engineering purposes, to make the encryption and decryption process the same. (Check!) + +## Modes of Operations + +AES, DES use fixed block size for encryption. How do we encrypt longer messages? For long messages, there are many different ways to process each block of the message. This is called the **mode of operation**. We will look at 5 different modes of operations. + +### Electronic Codebook Mode (ECB) + +![is-03-cbc-encryption.png](../../../assets/img/posts/Lecture%20Notes/Internet%20Security/is-03-cbc-encryption.png) + +- Codebook is a mapping table. +- For the $i$-th plaintext block, we use key $k$ to encrypt and obtain the $i$-th ciphertext block. + - **Uses the same key for all blocks** +- Adjacent blocks are independent of each other. +- Advantages + - Good when run in parallel +- Limitations + - Repetitions in messages (if aligned with the block) may lead to repetitions in the ciphertext + - Susceptible to *cut-and-paste attacks* +- Mainly used to send a few blocks data + +#### Cut-and-Paste Attack + +Since the same key is used for all blocks, once a mapping from plaintext to ciphertext is known, a sequence of ciphertext blocks can be easily manipulated. The assumption here is that the encryption keys do not change frequently. So the attacker can *cut* some block from a ciphertext and *paste* it to manipulate the data. This is a chosen ciphertext attack. + +### Cipher Block Chaining Mode (CBC) + +![is-03-cbc-encryption.png](../../../assets/img/posts/Lecture%20Notes/Internet%20Security/is-03-cbc-encryption.png) + +- Two identical messages produce to different ciphertexts. + - This prevents chosen plaintext attacks +- Blocks are linked together in the encryption process + - **Each previous cipher block is chained with current block** + - Initialization vector is used +- Encryption + - Let $c_0$ be the initialization vector. + - $c_i = E(k, p_i \oplus c_{i - 1})$, where $p_i$ is the $i$-th plaintext block. + - The ciphertext is $(c_0, c_1, \dots)$. +- Decryption + - The first block $c_0$ contains the initialization vector. + - $p_i = c_{i - 1} \oplus D(k, c_i)$. + - The plaintext is $(p_1, p_2, \dots)$. +- Used for bulk data encryption, authentication +- Advantages + - Parallelism in decryption. + - Chosen plaintext attacks can be mitigated through randomized IV. +- Limitations + - Encryption is not parallelizable. Each ciphertext block depends on *all* previous blocks. +- Side note: CBC can be used to check message integrity. (MAC) + +#### Error Propagation in CBC + +- If there is a 1-bit error in the *plaintext*, then that error will affect that block and all the other blocks afterwards. + - This error doesn't occur frequently since we are in the same system. +- If there is a 1-bit error in the *ciphertext*, then that error will affect only two blocks. + - This error can happen in transit through the network. + - CBC mode is self-recovering + +#### Initialization Vector in CBC + +- If the IV is the same, then the encryption of the same plaintext is the same. + - Thus IVs should be random. +- IV are not required to be secret, but + - No IVs should be reused under the same key + - IV changes should be unpredictable +- On IV reuse, same message will generate the same ciphertext if key isn't changed +- If IV is predictable, CBC is vulnerable to chosen plaintext attacks + +### Cipher Feedback Mode (CFB) + +![is-03-cfb-encryption.png](../../../assets/img/posts/Lecture%20Notes/Internet%20Security/is-03-cfb-encryption.png) + +- The message is treated as a stream of bits; similar to stream cipher +- **Result of the encryption is fed to the next stage.** + - Standard allows any number of bits to be fed to the next stage + - It is most efficient to use all $64$ bits (CFB-64) +- Initialization vector is used. + - Same requirements on the IV as CBC mode. + - Should be randomized, and should not be predictable. +- Encryption + - Let $c_0$ be the initialization vector. + - $c_i = p_i \oplus E(k, c_{i - 1})$, where $p_i$ is the $i$-th plaintext block. + - The ciphertext is $(c_0, c_1, \dots)$. +- Decryption + - The first block $c_0$ contains the initialization vector. + - $p_i = c_i \oplus E(k, c_{i - 1})$. The same module is used for decryption! + - The plaintext is $(p_1, p_2, \dots)$. +- Advantages + - Appropriate when data arrives in bits/bytes (similar to stream cipher) + - Only encryption module is needed. + - Decryption can be run in parallel. +- Limitations + - Encryption is not parallelizable. + +#### Error Propagation in CFB + +- CFB mode is self-recovering. +- 1 bit error in the ciphertext corrupts some number of blocks. + - Bit errors in the ciphertext will cause bit errors at the same position. + - Since this ciphertext is fed to the next block, the error is propagated +- Some implementations (like CFB-8) use shift registers, so errors will be propagated as long as the erroneous bit is in the shift register. + - If the error is removed from the shift register, it automatically recovers. + +### Output Feedback Mode (OFB) + +![is-03-ofb-encryption.png](../../../assets/img/posts/Lecture%20Notes/Internet%20Security/is-03-ofb-encryption.png) + +- Very similar to stream cipher. +- Initialization vector is used as a seed to generate the key stream. +- Actual encryption and decryption only consists of XOR, so it is fast. +- Blocks are independent of each other + - Encryption/decryption are both parallelizable after key stream is calculated. + - Key stream generation cannot be parallelized. +- Encryption + - Let $s_0$ be the initialization vector. + - $s_i = E(k, s_{i - 1})$ where $s_i$ is the $i$-th key stream. + - $c_i = p_i \oplus s_i$. + - The ciphertext is $(s_0, c_1, \dots)$. +- Decryption + - The first block $s_0$ contains the initialization vector. + - $s_i = E(k, s_{i - 1})$. The same module is used for decryption. + - $p_i = c_i \oplus s_i$. + - The plaintext is $(p_1, p_2, \dots)$. +- Note: IV and successive encryptions act as an OTP generator. +- Advantages + - There is no error propagation. 1 bit error in ciphertext only affects 1 bit in the plaintext. + - Key streams can be generated in advance. + - Fast when parallelized. + - Only encryption module is needed. +- Limitations + - Key streams should not have repetitions. + - We would have $c_i \oplus c_{i+1} = p_i \oplus p_{i + 1}$. + - Size of each $s_i$ should be large enough. + - If attacker knows the plaintext and ciphertext, plaintext can be modified. + - Same as in OTP. + +### Counter Mode (CTR) + +![is-03-ctr-encryption.png](../../../assets/img/posts/Lecture%20Notes/Internet%20Security/is-03-ctr-encryption.png) + +- Without chaining, we use a counter (typically incremented by 1). + - Counter starts from the initialization vector. + - Highly parallelizable. + - Can decrypt from any arbitrary position. +- Counter should not be repeated for the same key. + +--- + +Images are from [Wikipedia](https://en.wikipedia.org/wiki/Block_cipher_mode_of_operation). diff --git a/assets/img/posts/Lecture Notes/Internet Security/is-03-cbc-encryption.png b/assets/img/posts/Lecture Notes/Internet Security/is-03-cbc-encryption.png new file mode 100644 index 0000000000000000000000000000000000000000..99f48d7d15ee7978eefae2c142b8b0d2b4398843 GIT binary patch literal 46130 zcmeFZXH-+|);0NJjR#WzL^Yk|Kx^L&8=HTJr?QCbDrbIw+ z=hZ7?tA|X+v~n2@RbZ}x$R8g*z$wneldIhfzCA@RpE*>yPYNcvRVyZRuTNz;^W2`p z<~j4r(JE4XDx6K4#aVvhWp^prh!uKQt@jQT$#*n#to9alooub)c2!eSy=zPFBxW&l zD|Bi4@#gfUgnB{C*VEiYf(##p7QX+g>RkNtMcU_6$k3ILAt7*Fia-hvA>&+o8+v1l ztz&xhpzaXowDwU?jqv%@v$4`UJsiIr2+rufR$7IF6i>RyMjsrJeSx@Tvi-XHxrV@$ zKAUp8axg9N)^i&bLB8UbXJ00slPqx_Sb#_#Z%@R@h!>XebM&foOO&MRaDH!zJPE5+ znR5Ly5vdzrB^p9IbD{nqRlxi8&uFR~>8W?J?sDwh0)1Q+~BbE2Aw4DkIV)6=ne7h7}FCX?JZ)FAWnA|(sK>@NYK^1wq* zmU3-pf#g?mR!LU%ci>6tXGgC{H3 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