Tidy up

docs/lectures/cryptography/01_intro.md

@@ -10,7 +10,7 @@
 
 **Asymmetric**
 
->“Methods which use separate, but related, private and public keys.”
+> “Methods which use separate, but related, private and public keys.”
 
 **Protocols**
 
@@ -20,7 +20,7 @@
 
 > “The science and art of breaking cryptosystems.”
 
-### Modern Cyptography (1970-)
+### Modern Cryptography (1970-)
 
 **Fundamentally different** - a scientific and mathematical discipline
 
@@ -46,13 +46,12 @@
 
 - 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 $\mathbb{Q}, \mathbb{R}, \mathbb{C}$
+    - $\mathbb{N} = \{1,2,3,...\}$
+    - $\mathbb{Z} = \{..., -3, -2, -1, 0,1,2,3,...\}$
+    - 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
-
+    - When we add or multiply two 1 byte binary digits, the result will always be 1 byte
 
 ###### Congruence
 
@@ -73,7 +72,7 @@ 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
+- All these numbers act the same in any modulo sum
 
 For example
 
@@ -99,25 +98,25 @@ 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. $a\cdot b \equiv d \space (mod\space m), (d\in \mathbb{Z})$
+    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.
+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 intermediate 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
+    - It is closed
 - Addition and multiplication are associative 
-  - (a+b)+c = a + (b+c)
+    - (a+b)+c = a + (b+c)
 - There is a neutral element 0 for addition
-  - $a + 0 \equiv a\space mod \space m$
+    - $a + 0 \equiv a\space mod \space m$
 - The additive inverse always exists
-  - $a + (-a) = 0\space mod \space m$
+    - $a + (-a) = 0\space mod \space m$
 - There is a neutral element for multiplication
-  - $a\cdot 1 \equiv a\space mod\space m$
+    - $a\cdot 1 \equiv a\space mod\space m$
 - The multiplicative inverse exists for some but not all elements
-  - $a\cdot a^{-1} \equiv 1 \space mod \space m$
+    - $a\cdot a^{-1} \equiv 1 \space mod \space m$
 
 #### Modular Inversion
 
@@ -158,7 +157,7 @@ $$
 
 ##### Frequency Analysis
 
-- The frequency of occurrences of each character are very consistent
+- The frequency of occurrences of each character is very consistent
 - The longer a cipher text is, the easier this becomes
 
 #### Affine Cipher
@@ -174,23 +173,23 @@ $$
 
 where $k=(a,b)$ and $gcd(a,26)=1$
 
-This is a multiplication and a addition analagous to $y=mx+c$
+This is a multiplication and an addition analogous to $y=mx+c$
 
-In a Affine cipher, letters can be themselves
+In an 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
+    - 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.
+Frequency analysis can still be used; in this case the columns will not only be shifted, but jumbled as well.
 
 - 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.
+- Unlike the 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.
+- Spreads out occurrences of characters, making frequency analysis hard.