Tidy up

docs/lectures/cryptography/09_public_key_maths.md

@@ -11,15 +11,15 @@ $$
 
 #### Euclidean Algorithm
 
-- The euclidean algorithm calculates the greatest common divisor of two numbers $gcd(r_0, r_1)$
-  - This is the largest number that divides both $r_0$ and $r_1$
+- The Euclidean algorithm calculates the greatest common divisor of two numbers $gcd(r_0, r_1)$
+    - This is the largest number that divides both $r_0$ and $r_1$
 - If $gcd(x,y)=1$ then $x$ and $y$ are **coprime** (sometimes called relatively prime)
 - The Euclidean algorithm is based around the fact:
-  - $gcd(r_0, r_1) = gcd(r_1, r_0 - r_1)$
+    - $gcd(r_0, r_1) = gcd(r_1, r_0 - r_1)$
 
 ![1646755531.png](img/1646755531.png)
 
-- Computing $(x-y)\cdot gcd(r_0, r_1)$ is easier as its a smaller number
+- Computing $(x-y)\cdot gcd(r_0, r_1)$ is easier as it's a smaller number
 - Doing this repeatedly is slow, we can use $gcd(r_0,r_1) = gcd(r_1, r_0\space mod \space r_1)$
 
 ![1646755650.png](img/1646755650.png)
@@ -37,16 +37,16 @@ $r_0=q\cdot r_1 + r_2 \\57=4\cdot 12 + 9\\ r_1=q\cdot r_2 + r_3 \\ 12=1\cdot 9 +
 
 ![1646756075.png](img/1646756075.png)
 
-#### Bezout’s Identity
+#### Bézout’s Identity
 
-- Bezout’s identity tells us that the greatest common divisor of two numbers can be expressed as the sum of multiples of these numbers
+- Bézout’s identity tells us that the greatest common divisor of two numbers can be expressed as the sum of multiples of these numbers
 - $gcd(r_0,r_1) = s\cdot r_0 + t\cdot r_1$
-  - e.g. $gcd(99,20)=-1\cdot 99+5\cdot 20=1$
-  - $gcd(141,50)=11\cdot 141+-31\cdot 50=1$
+    - e.g. $gcd(99,20)=-1\cdot 99+5\cdot 20=1$
+    - $gcd(141,50)=11\cdot 141+-31\cdot 50=1$
 
 ##### Extended Euclidean Algorithm
 
-- The extended euclidean algorithm calculates the $gcd(r_0,r_1)$ as normal, and in addition calculates $s$ and $t$.
+- The extended Euclidean algorithm calculates the $gcd(r_0,r_1)$ as normal, and in addition calculates $s$ and $t$.
 
 | Euclidean Algorithm                | Extended Euclidean Algorithm                                 |
 | ---------------------------------- | ------------------------------------------------------------ |
@@ -81,4 +81,3 @@ $$
 $$
 
 Where $t$ is our multiplicative inverse
-