Tidy up

docs/lectures/cryptography/15_hash_functions.md

@@ -4,7 +4,7 @@
 
 - Could we simply split up a message and sign parts?
 
-![1649192960.png](img/1649192960.png)\
+![1649192960.png](img/1649192960.png)
 
 A lot of faff for signing large files
 
@@ -16,7 +16,7 @@ A lot of faff for signing large files
 2. Fixed output length
 3. Pre-image resistance (one way)
 4. Second pre-image resistance
-   - If we have a hashed message, we cannot find another message with the same hash
+    - If we have a hashed message, we cannot find another message with the same hash
 5. Collision resistance
 
 #### Pre-image Resistance
@@ -24,7 +24,7 @@ A lot of faff for signing large files
 - Hash functions must be one-way
 - Given a hash of a message $H(x)$ it must be infeasible to calculate $x$
 - Less applicable to digital signatures
-  - Crucial to password storage and key derivation
+    - Crucial to password storage and key derivation
 
 #### Second Pre-image Resistance
 
@@ -37,7 +37,7 @@ A lot of faff for signing large files
 
 ![1649193645.png](img/1649193645.png)
 
-Oscar finds a weak message (one of the messages is known ahead of time), he replaces the message $x_1$ with $x_2$. Now Oscar can send a signed message to Alice
+Oscar finds a weak message (one of the messages is known ahead of time); he replaces the message $x_1$ with $x_2$. Now Oscar can send a signed message to Alice
 
 #### Collision Resistance
 
@@ -80,8 +80,8 @@ P(n)&=(1-\frac{1}{365})\cdot (1-\frac{2}{365})\dots (1-\frac{n-1}{365})
 $$
 
 - The probability of at least one collision is $1 – P(\textrm{no collision})$.
-  - The probability of a collision with only 23 people is ~50%!
-  - For 40 people it’s ~90%
+    - The probability of a collision with only 23 people is ~50%!
+    - For 40 people it’s ~90%
 - The same principle applies to hash functions, the more hashes computed, the more likely a collision becomes
 
 ![1649194470.png](img/1649194470.png)
@@ -93,4 +93,4 @@ $$
 - You will find a collision after approx $\sqrt{(2^n)}=2^{\frac n2}$ random attempts
 - This means that your bit length needs to be double the size of your desired security margin
 - `SHA-256` therefore offers equivalent security to `AES 128`
-- left at `25:55`
+- left at `25:55`