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docs/lectures/cryptography/06_finite_fields.md

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+# Finite Field Arithmetic
+
+- A **finite field** is a set containing a finite number of elements
+  - This is sometimes called a *Galois Field*
+- In a Galois field you can:
+  - Add
+  - Subtract
+  - Multiply
+  - Invert (divide)
+- Fields are an extension of *groups* and related to *rings*
+
+### Groups
+
+A group is a set of elements $G$ together with an operation $\circ$ that combines two elements of $G$
+
+> 1. The operation $\circ$ is **closed**
+>    - i.e. for all $a,b \in G$ then $a\circ b=c\in G$
+> 2. The operation is associative
+>    - i.e. $a\circ(b\circ c) = (a\circ b)\circ c$ for all $a,b,c \in G$
+> 3. There is an element $1\in G$ called a **neutral element** such that $a\circ 1 = 1\circ a = a$ for all $a\in G$
+> 4. For each $a \in G$ there exists an element $a^{-1}\in G$ called the **inverse** of $a$ such that $a\circ a^{-1} = a^{-1}\circ a = 1$
+> 5. A group $G$ is **abelian** (commutative) if $a\circ b = b \circ a$ for all $a,b\in G$
+
+##### Example Group
+
+- The set of integers $\mathbb{Z}_m = \{0,1,...m-1\}$ with the operation addition modulo m form a group with the neutral element 0
+- Every element would have an inverse where $a + (-a) = 0$ mod m
+- This group would not form a group with multiplication, as not all elements would have an inverse
+  - We wouldn’t have an inverse, we would need $5\times \frac15=1$ however $\frac15 \notin \mathbb{Z}$ 
+
+### Fields
+
+A field $F$ is a set of elements with the following properties
+
+> 1. All elements of $F$ form an **additive group** with the group operation $+$ and the neutral element 0
+> 2. All elements of $F$ except 0 form a multiplicative group with the group operation $\times$ and the neutral element 1
+> 3. When the two group operations are mixed, the distributivity law holds.
+>    - i.e. for all $a,b,c \in F, a\cdot(b+c) = (a\cdot b) + (a\cdot c)$
+
+##### Example Field
+
+- The set of real numbers $\mathbb{R}$ is a field with neutral element 0 for addition and 1 for multiplication
+- Every real number $a$ has a additive inverse $-a$
+- Every non-zero number $a$ has a multiplicative inverse $\frac{1}{a}$
+
+![1646405235.png](img/1646405235.png)
+
+#### Finite Fields
+
+> A finite field only exists if it has $p^m$ elements
+>
+> Where:
+>
+> - $p$ is a prime 
+> - $m$ is a positive integer
+
+###### Examples
+
+- There is a field with 11 elements: $GF(11)$
+- There is a field with 256 elements: $GF(256)$ or $GF(2^8)$
+- $GF(12)$ is not a finite field $(2^2 \cdot3)$
+
+###### Prime and Extension Fields
+
+When $m=1$ it creates a **prime field**
+
+When $m>1$ it creates an **extension field**
+
+### Prime Fields
+
+- A prime field $GF(p)$ contains the integers $\{0,1,...p-1\}$
+
+![1646405270.png](img/1646405270.png)
+
+- These operations satisfy the properties of fields (*closure*)
+
+##### Inversion in Prime Fields
+
+$a \cdot a^{-1} \equiv 1 \space (mod \space p)$
+
+- A modular inverse exists when $gcd(a,p) = 1$
+- Because $p$ is prime, every number has a multiplicative inverse
+  - $gcd(a,p) = 1, \forall a \neq0 \in GF(p)$
+- $a^{-1}$ can be calculated using the **extended Euclidean algorithm**
+
+#### Extension Fields
+
+- In prime fields, the elements are integers
+- Elements in extension fields $GF(2^m)$ are polynomials of degree $m$
+
+$a_{m-1}x^{m-1}, ..., a_1x  + a_0 = A(x) \in GF(2^m)$
+
+where $a_i \in GF(2) = \{0,1\}$
+
+The coefficients of the polynomial are elements in $GF(2)$ the **sub-field**
+
+##### Example $GF(2^3)$
+
+- The field $GF(2^3)$, sometimes called $GF(8)$ is an extension field containing elements of the form: $A(x) = a_2 x^2 + a_1x^1 + a_0$
+- Its often easier to simply write the coefficients $(a_2, a_1, a_0)$ e.g. 001 or 101
+- $GF(2^3) = \{0, 1, x, x+1, x^2, x^2+1, x^2 + x, x^2 + x + 1\}$ 
+  - $|GF(2^3)| = 8$
+
+#### Arithmetic in $GF(2^3)$
+
+- Adding or subtracting two polynomials happens as expected, but adding the coefficients
+  - $A(x) = x^2 + x + 1$
+  - $B(x) = x^2 + 1$
+  - $A(x) + B(x) = (1+1)x^2 + (1)x + (1+1) = x$
+- mod 2 is simply `xor`
+- Addition and subtraction are identical
+
+#### Multiplication in $GF(2^3)$
+
+- $A(x) = x^2 + x + 1$
+- $B(x) = x^2 + 1$
+- $A(x) \cdot B(x) = (x^2 + x + 1)(x^2 + 1) = x^4 + x^3 + (1+1)x^2 + x + 1$
+- $x^4 + x^3 + x + 1$ however this is **not in the field**
+- The result must be reduced by the result modulo an **irreducible polynomial**
+
+$$
+A(x) \cdot B(x) = x^4 + x^3 + x + 1\space (mod \space x^3 + x + 1)
+$$
+
+- This means we have to do polynomial long division
+
+![img](img/IMG_0230.jpg)
+
+##### Inversion
+
+- Inversion is performed in a similar way to prime fields, we find:
+- $A(x) \cdot A^{-1}(x) \equiv 1 \space (mod \space P(x))$
+  - $A^{-1}(x)$ is calculated using the extended euclidean algorithm
+
+### AES’ Finite Field
+
+- AES uses the extension field $GF(2^8)$ for many of its operations
+- Operations are the same as those in other $GF(2^m)$ fields, using the irreducible polynomial
+
+$$
+  P(x) = x^8 + x^4 + x^3 + x + 1
+$$
+
+- As you might expect, these polynomials are typically represented as single bytes