mirror of
https://github.com/calofmijuck/blog.git
synced 2025-12-06 22:53:51 +00:00
feat: internet security midterm notes (#128)
* [PUBLISHER] upload files #114 * [PUBLISHER] upload files #115 * PUSH NOTE : 1. OTP, Stream Ciphers and PRGs.md * PUSH ATTACHMENT : mc-01-prg-game.png * PUSH ATTACHMENT : mc-01-ss.png * [PUBLISHER] upload files #116 * PUSH NOTE : 03. Symmetric Key Cryptography (2).md * PUSH ATTACHMENT : is-03-feistel-function.png * PUSH ATTACHMENT : is-03-ecb-encryption.png * 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 * [PUBLISHER] upload files #117 * PUSH NOTE : 01. Security Introduction.md * PUSH ATTACHMENT : is-01-cryptosystem.png * [PUBLISHER] upload files #118 * [PUBLISHER] upload files #119 * [PUBLISHER] upload files #120 * [PUBLISHER] upload files #121 * [PUBLISHER] upload files #122 * [PUBLISHER] upload files #123 * PUSH NOTE : 07. Public Key Cryptography.md (#124) * fix: links fixed * [PUBLISHER] upload files #125 * PUSH NOTE : 08. Public Key Infrastructure.md * PUSH ATTACHMENT : is-08-certificate-validation.png * [PUBLISHER] upload files #126 * PUSH NOTE : 09. Transport Layer Security.md * PUSH ATTACHMENT : is-09-tls-handshake.png * [PUBLISHER] upload files #127 * PUSH NOTE : 08. Public Key Infrastructure.md * PUSH ATTACHMENT : is-08-certificate-validation.png * fix: fixed internal, external links
This commit is contained in:
GitHub
@@ -24,7 +24,7 @@ github_title: 2023-09-11-symmetric-key-cryptography-1
|
||||
- A strong encryption algorithm, which is known to the public.
|
||||
- Kerckhoff's principle!
|
||||
- A secret key known only to sender and receiver.
|
||||
- We assume the **existence of a a secure channel for distributing the key**.
|
||||
- We assume the **existence of a a secure channel for distributing the key**.[^1]
|
||||
- **Correctness requirement**
|
||||
- Let $m$, $k$ denote the message and the key.
|
||||
- For encryption/decryption algorithm $E$ and $D$,
|
||||
@@ -32,7 +32,7 @@ github_title: 2023-09-11-symmetric-key-cryptography-1
|
||||
|
||||
## Cryptographic Attacks
|
||||
|
||||
In increasing order of increasing power of the attacker,
|
||||
In increasing order of the power of the attacker,
|
||||
|
||||
- **Ciphertext only attacks**: the attacker has ciphertexts, and tries to obtain information.
|
||||
- **Known plaintext attack**: the attacker has a collection of plaintext/ciphertext pairs.
|
||||
@@ -44,8 +44,10 @@ In increasing order of increasing power of the attacker,
|
||||
The following two properties should hold for a secure cipher.
|
||||
- **Diffusion** hides the relationship between the ciphertext and the plaintext.
|
||||
- It should be hard to obtain the plaintext from the ciphertext.
|
||||
- Changing a single bit of the plaintext affects several bits of the ciphertext, and vice versa.
|
||||
- **Confusion** hides the relationship between the ciphertext and the key.
|
||||
- It should be hard to obtain the key from the ciphertext.
|
||||
- Each bit of the ciphertext should depend on several parts of the key.
|
||||
|
||||
## Primitives
|
||||
|
||||
@@ -66,8 +68,9 @@ In **substitution cipher**, encryption is done by replacing units of plaintext w
|
||||
- In Caesar cipher, $a = 1$ and $b = 3$.
|
||||
- Encryption: $E(x) = ax + b \pmod m$.
|
||||
- Decryption: $D(x) = a^{-1}(x - b) \pmod m$.
|
||||
- There are $12$ possible values for $a$, and $26$ possible values for $b$.
|
||||
- If we use the $26$ alphabets, there are $12$ possible values for $a$, and $26$ possible values for $b$.
|
||||
- $a^{-1}$ does not exist for all $m$.
|
||||
- We need that $\gcd(a, m) = 1$. The number of possible $a$ values is $\phi(m)$.
|
||||
- This scheme is not secure either, since we can try all possibilities and check if the message makes sense.
|
||||
|
||||
#### Monoalphabetic Substitution Cipher
|
||||
@@ -79,17 +82,17 @@ In **substitution cipher**, encryption is done by replacing units of plaintext w
|
||||
- Decryption is done by replacing each letter $x$ by $\pi^{-1}(x)$.
|
||||
- This scheme is still not secure, since we can try all possibilities on a *modern* computer.
|
||||
|
||||
To attack this scheme, we use frequency analysis. Calculate the frequency of each letter and compare it with the actual distribution of English letters. Also, we could use bigrams (2-letters)
|
||||
To attack this scheme, we use frequency analysis. Calculate the frequency of each letter and compare it with the actual distribution of English letters. We could also use *bigrams* (2-letters) for calculating the frequency.
|
||||
|
||||
#### Vigenère Cipher
|
||||
|
||||
- A polyalphabetic substitution
|
||||
- Given a key length $m$, take key $k = (k_1, k_2, \dots, k_m)$.
|
||||
- For the $i$-th letter $x$, set $j = i \pmod m$.
|
||||
- For the $i$-th letter $x$, set $j = i \bmod m$.
|
||||
- Encryption is done by replacing $x$ by $x + k_{j}$.
|
||||
- Decryption is done by replacing $x$ by $x - k_j$.
|
||||
|
||||
To attack this scheme, find the key length by *index of coincidence*. Then use frequency analysis.
|
||||
To attack this scheme, find the key length by [*index of coincidence*](https://en.wikipedia.org/wiki/Index_of_coincidence). Then use frequency analysis.
|
||||
|
||||
#### Hill Cipher
|
||||
|
||||
@@ -113,6 +116,48 @@ This scheme is vulnerable to known plaintext attack, since the equation can be s
|
||||
- To encrypt, reorder the columns by the chosen permutation.
|
||||
- Then the ciphertext is taken by taking letters in column major order.
|
||||
|
||||
##### Example
|
||||
|
||||
Suppose we encrypt the following text:
|
||||
|
||||
$$
|
||||
\texttt{CRYPTOGRAPHY INTERNET SECURITY}
|
||||
$$
|
||||
|
||||
Choose a key $\sigma = (1, 4, 5, 2, 3, 6)$. Then
|
||||
|
||||
$$
|
||||
\begin{matrix} \\
|
||||
4 & 3 & 6 & 5 & 2 & 1 \\ \hline
|
||||
\texttt{C} & \texttt{R} & \texttt{Y} & \texttt{P} & \texttt{T} & \texttt{O} \\
|
||||
\texttt{G} & \texttt{R} & \texttt{A} & \texttt{P} & \texttt{H} & \texttt{Y} \\
|
||||
\texttt{I} & \texttt{N} & \texttt{T} & \texttt{E} & \texttt{R} & \texttt{N} \\
|
||||
\texttt{E} & \texttt{T} & \texttt{S} & \texttt{E} & \texttt{C} & \texttt{U} \\
|
||||
\texttt{R} & \texttt{I} & \texttt{T} & \texttt{Y}
|
||||
\end{matrix}
|
||||
$$
|
||||
|
||||
Now reorder the columns,
|
||||
|
||||
$$
|
||||
\begin{matrix} \\
|
||||
1 & 2 & 3 & 4 & 5 & 6 \\ \hline
|
||||
\texttt{O} & \texttt{T} & \texttt{R} & \texttt{C} & \texttt{P} & \texttt{Y} \\
|
||||
\texttt{Y} & \texttt{H} & \texttt{R} & \texttt{G} & \texttt{P} & \texttt{A} \\
|
||||
\texttt{N} & \texttt{R} & \texttt{N} & \texttt{I} & \texttt{E} & \texttt{T} \\
|
||||
\texttt{U} & \texttt{C} & \texttt{T} & \texttt{E} & \texttt{E} & \texttt{S} \\
|
||||
&& \texttt{I} & \texttt{R} & \texttt{Y} & \texttt{T}
|
||||
\end{matrix}
|
||||
$$
|
||||
|
||||
The ciphertext is
|
||||
|
||||
$$
|
||||
\texttt{OYNU THRC RRNTI CGIER PPEEY YATST}.
|
||||
$$
|
||||
|
||||
The decryption process is the reverse of this operation. It seems to be breakable by inspecting the $i$-th letter of each block and reordering the letters to check if any reordering makes sense.
|
||||
|
||||
### Exclusive OR (XOR)
|
||||
|
||||
- A bitwise operation $x \oplus y = x + y \pmod 2$.
|
||||
@@ -130,8 +175,8 @@ This scheme is vulnerable to known plaintext attack, since the equation can be s
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\mathrm{Pr}[C = 0] &= \mathrm{Pr}[M = 0 \land K = 0] + \mathrm{Pr}[M = 1 \land K = 1] \\ &= \mathrm{Pr}[M = 0] \cdot \mathrm{Pr}[K = 0] + \mathrm{Pr}[M = 1] \cdot \mathrm{Pr}[K = 1] \\
|
||||
&= \frac{1}{2}\left(\mathrm{Pr}[M = 0] + \mathrm{Pr}[M = 1]\right) \\
|
||||
\Pr[C = 0] &= \Pr[M = 0 \land K = 0] + \Pr[M = 1 \land K = 1] \\ &= \Pr[M = 0] \cdot \Pr[K = 0] + \Pr[M = 1] \cdot \Pr[K = 1] \\
|
||||
&= \frac{1}{2}\left(\Pr[M = 0] + \Pr[M = 1]\right) \\
|
||||
&= \frac{1}{2}.
|
||||
\end{align*}
|
||||
$$
|
||||
@@ -140,20 +185,20 @@ The case for $C = 1$ is similar.
|
||||
|
||||
### One-Time Pad (OTP)
|
||||
|
||||
Omitted.
|
||||
[1. OTP, Stream Ciphers and PRGs > One-Time Pad (OTP)](../../modern-cryptography/2023-09-07-otp-stream-cipher-prgs#one-time-pad-otp)
|
||||
|
||||
## Perfect Secrecy
|
||||
|
||||
> **Definition.** Let $(E, D)$ be a cipher defined over $(\mathcal{K}, \mathcal{M}, \mathcal{C})$. We assume that $\lvert \mathcal{K} \rvert = \lvert \mathcal{M} \rvert = \lvert \mathcal{C} \rvert$. The cipher is **perfectly secure** if for all $m \in \mathcal{M}$ and $c \in \mathcal{C}$,
|
||||
>
|
||||
> $$
|
||||
> \mathrm{Pr}[\mathcal{M} = m \mid \mathcal{C} = c] = \mathrm{Pr}[\mathcal{M} = m].
|
||||
> \Pr[\mathcal{M} = m \mid \mathcal{C} = c] = \Pr[\mathcal{M} = m].
|
||||
> $$
|
||||
>
|
||||
> Or equivalently, for all $m_0, m_1 \in \mathcal{M}$, $c \in \mathcal{C}$,
|
||||
>
|
||||
> $$
|
||||
> \mathrm{Pr}[E(k, m _ 0) = c] = \mathrm{Pr}[E(k, m _ 1) = c]
|
||||
> \Pr[E(k, m _ 0) = c] = \Pr[E(k, m _ 1) = c]
|
||||
> $$
|
||||
>
|
||||
> where $k$ is chosen uniformly in $\mathcal{K}$.
|
||||
@@ -163,7 +208,7 @@ In other words, the adversary learns nothing from the ciphertext.
|
||||
With this definition, we can show that **OTP is perfectly secure**. For all $m \in \mathcal{M}$ and $c \in \mathcal{C}$,
|
||||
|
||||
$$
|
||||
\mathrm{Pr}[E(k, m) = c] = \frac{1}{\lvert \mathcal{K} \rvert}
|
||||
\Pr[E(k, m) = c] = \frac{1}{\lvert \mathcal{K} \rvert}
|
||||
$$
|
||||
|
||||
since for each $m$ and $c$, $k$ is determined uniquely.
|
||||
@@ -278,3 +323,5 @@ Given a bit string (defined in the specification), the sender performs long divi
|
||||
- $c \oplus (x \parallel \mathrm{CRC}(x)) = k_s \oplus (m\oplus x \parallel \mathrm{CRC}(m\oplus x))$
|
||||
- The receiver will decrypt and get $(m\oplus x \parallel \mathrm{CRC}(m\oplus x))$.
|
||||
- CRC check by the receiver will succeed.
|
||||
|
||||
[^1]: This assumption will be removed when we learn public key cryptography.
|
||||
|
||||
Reference in New Issue
Block a user