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docs/lectures/cryptography/01_intro.md

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+# Cryptography
+
+**Cryptology**
+
+> “The science and art of writing and solving codes to hide the meaning of messages.”
+
+**Symmetric**
+
+> “Encryption methods in which both the encryption and decryption algorithms use the same key.”
+
+**Asymmetric**
+
+>“Methods which use separate, but related, private and public keys.”
+
+**Protocols**
+
+> “The application of cryptographic algorithms in secure systems.”
+
+**Cryptanalysis**
+
+> “The science and art of breaking cryptosystems.”
+
+### Modern Cyptography (1970-)
+
+**Fundamentally different** - a scientific and mathematical discipline
+
+**Rigorously tested** - New approaches tested, justified through mathematical proofs and theory
+
+**Extremely powerful** - Ciphers usually take milliseconds to use and lifetimes of the universe to break
+
+**Wider uses** - including message integrity and authenticity
+
+**Civilian use** - everyone benefits from cryptography now
+
+## Ciphers
+
+- Ciphers have been used for thousands of years
+- Usually based around either transposition or substitution
+
+#### Caesar Cipher
+
+- An early substitution cipher, we replace each letter of plain text with a shifted letter $n$ letters away from the letter
+- Therefore our key is an integer $-25\leq n \leq 25$
+
+### Modular Arithmetic
+
+- Modular arithmetic is a system of arithmetic for finite sets of integers
+- Common sets include
+  - $\mathbb{N} = \{1,2,3,...\}$
+  - $\mathbb{Z} = \{..., -3, -2, -1, 0,1,2,3,...\}$
+  - Also $\mathbb{Q}, \mathbb{R}, \mathbb{C}$
+- Cryptography is almost always interested in finite sets
+- This is useful as it avoids overflow errors
+  - When we add or multiply two 1 byte binary digits, the result will always be 1 byte
+
+
+###### Congruence
+
+Let $a, r, m \in \mathbb{Z}$ and $m > 0$
+
+$a \equiv r (mod\space m)$ if $\frac{m}{a-r}$
+
+Check:
+
+$a=12, m=7$
+
+$a\equiv 5 (mod\space 7)$
+
+$\frac{7}{12-5}$ :white_check_mark:
+
+This can be rewritten as: $a = q\cdot m+r$
+
+###### Equivalence Classes
+
+- The sets of all integers **mod 5** form a series of equivalence classes
+- All these numbers act the same in any modluo sum
+
+For example
+
+$74\cdot 62 - 47 (mod \space 5) \equiv 74\%5 \cdot 62\%5 - 47\%5$
+
+Also works with exponentiation
+
+$3^8\space (mod\space 7)$
+
+$3^2 = 3\cdot 3 = 9 \equiv 2\space (mod\space 7)$
+
+$3^4 = 3^2\cdot 3^2 = 2\cdot 2 \equiv 4\space (mod\space 7)$
+
+$3^8 = 3^4 \cdot 3^4 = 4\cdot 4 = 16 \equiv 2 \space (mod\space 7)$
+
+#### Integer Rings
+
+- Modular arithmetic forms what in mathematics we would call a Ring
+
+###### Ring Definition
+
+The integer ring $\mathbb{Z}_m$ consists of:
+
+1. The set $\mathbb{Z}_m = \{0, 1,\ldots m-1\}$ 
+2. Two operations $+$ and $\cdot$ for all $a, b \in \mathbb{Z}_m$ such that:
+   1. $a+b \equiv c \space (mod\space m), (c\in \mathbb{Z})$
+   2. $a\cdot b \equiv d \space (mod\space m), (d\in \mathbb{Z})$
+
+Any time you add or multiply any two numbers in the set, the result is always in the set. We use $\equiv$ instead of $=$ as it could be an intermediatary number e.g. 12 instead of 2.
+
+##### Properties of Rings
+
+- We can add or multiply any two numbers in the ring, and the result is in the ring
+  - It is closed
+- Addition and multiplication are associative 
+  - (a+b)+c = a + (b+c)
+- There is a neutral element 0 for addition
+  - $a + 0 \equiv a\space mod \space m$
+- The additive inverse always exists
+  - $a + (-a) = 0\space mod \space m$
+- There is a neutral element for multiplication
+  - $a\cdot 1 \equiv a\space mod\space m$
+- The multiplicative inverse exists for some but not all elements
+  - $a\cdot a^{-1} \equiv 1 \space mod \space m$
+
+#### Modular Inversion
+
+> In rings, the multiplicative inverse exists for some but not all elements
+
+- Multiplicative inverses allow us to *divide* by a number
+
+$$
+\frac{b}{a} \equiv b \cdot a^{-1} \space (mod \space m)
+$$
+
+- Not all numbers in a ring have an inverse, you can determine whether one exists quite simply:
+
+$$
+gcd(a,m)=1
+$$
+
+Example
+
+$3\cdot 9 \equiv 1 \space (mod\space 26)$
+
+$5\cdot 9 \equiv 19 \space (mod\space 26)$
+
+$19\cdot 3 \equiv 57 \equiv 5\space (mod\space 26)$
+
+Here a=3 and b=5, we can *divide* by 19 to get back to 5.
+
+#### Shift Cipher
+
+We can formalise the shift cipher using modular arithmetic
+
+Let $x, y, k \in \mathbb{Z}_{26}$
+
+$$
+e_k(x) = y \equiv x+k \space (mod \space 26) \\
+d_k(y) = x \equiv y-k \space (mod \space 26)
+$$
+
+##### Frequency Analysis
+
+- The frequency of occurrences of each character are very consistent
+- The longer a cipher text is, the easier this becomes
+
+#### Affine Cipher
+
+We can extend the shift cipher into an affine cipher
+
+Let $x,y,a,b \in \mathbb{Z}_{26}$
+
+$$
+e_k(x) = y \equiv a\cdot x+b\space (mod \space 26)\\
+d_k(y) = x \equiv a^{-1}\cdot(y-b)\space (mod \space m)
+$$
+
+where $k=(a,b)$ and $gcd(a,26)=1$
+
+This is a multiplication and a addition analagous to $y=mx+c$
+
+In a Affine cipher, letters can be themselves
+
+- The keyspace of an affine cipher
+  - a can be 0-25
+  - b can be 0-12
+  - 25*12=300
+- More secure than a caesar cipher
+
+Frequency analysis can still be used, in this case the columns will not only be shifted, but jumbled aswell.
+
+- This is not hard to crack
+
+#### The Vigenere Cipher
+
+- An early stream cipher, the Vigenere cipher is a shift cipher with a running key
+- Unlike caesar cipher, the key is repeated for as long as required.
+- It is the equivalent to multiple interleaved Caesar ciphers
+- Spreads outs occurrances of characters making frequency analysis hard.