Tidy up

docs/lectures/cryptography/13_elgamal.md

@@ -1,8 +1,8 @@
 # Elgamal Encryption
 
-#### Extending Diffie-Hellmen to Encryption
+#### Extending Diffie-Hellman to Encryption
 
-We could do is multiply the plain text by the key generated
+What we could do is multiply the plain text by the key generated
 
 $y\equiv x\cdot k_{ab}\mod p \rightarrow x\equiv y\cdot k_{ab}^{-1}$
 
@@ -28,35 +28,35 @@ $y\equiv x\cdot k_{ab}\mod p \rightarrow x\equiv y\cdot k_{ab}^{-1}$
 
 1. Choose $a\in \{1,2,...,p-1\}$
 2. Compute ephemeral key
-   - $k_E\equiv g^a\mod p$
-   - Remember ephemeral means the key is generated every time communication happens
+    - $k_E\equiv g^a\mod p$
+    - Remember ephemeral means the key is generated every time communication happens
 3. Compute masking key
-   - $k_M\equiv B^a\mod p$
+    - $k_M\equiv B^a\mod p$
 4. Encrypt message $x\in\mathbb{Z}^*_p$
-   - $y\equiv x\cdot k_M\mod p$
+    - $y\equiv x\cdot k_M\mod p$
 5. Send $(k_E,y)$
 
 #### Elgamal Decryption
 
 1. Compute masking key
-   - $k_M\equiv k_E^b\mod p$
+    - $k_M\equiv k_E^b\mod p$
 2. Decrypt message
-   - $x\equiv y\cdot k_M^{-1}\mod p$
+    - $x\equiv y\cdot k_M^{-1}\mod p$
 
 ### Computational Efficiency
 
-To calculate bobs private key we use one exponentiation
+To calculate Bob's private key we use one exponentiation
 
-Alice has to do two binary exponentiation to send a message to bob
+Alice has to do two binary exponentiations to send a message to Bob
 
 ![1648752755.png](img/1648752755.png)
 
-- Both the exponentiations during encryption can be pre-computed during down time
+- Both the exponentiations during encryption can be pre-computed during downtime
 - We can also improve on the decryption step using Fermat’s little theorem
 - Fermat’s Little Theorem: $a^{p-1}\equiv 1\mod p$
-  1. Compute $k_M=k_E^b\mod 67$
-  2. Compute $k_M^{-1}$
-  3. Decrypt $y=y\cdot k_M^{-1}\mod p$
+    1. Compute $k_M=k_E^b\mod 67$
+    2. Compute $k_M^{-1}$
+    3. Decrypt $y=y\cdot k_M^{-1}\mod p$
 
 #### Practicalities
 
@@ -119,17 +119,17 @@ Recall: $a^{p-1}\equiv 1\mod p$ for some $m$
 - Computed in a subgroup of prime order q, which is usually 160 bits
 - This means the signature (r, s) is 320 bits
 - Hashing is enforced by the algorithm, and a hash function must match the key size
-  - e.g. SHA-1 for 160-bit q, SHA-256 for 256 bit q
+    - e.g. SHA-1 for 160-bit q, SHA-256 for 256 bit q
 - Index calculus does not apply to the sub-group, so 160 bit DSA has a security of 80 bits
-  - In practice larger keys would be required now
+    - In practice larger keys would be required now
 
 #### ECDSA
 
--  Identical to DSA, ECDSA operates on an elliptic curve over $\mathbb{Z}_p$ with the signature calculated over a subgroup of prime order $\#q$
-  - More efficient, does not require modulus of thousands of bits
+- Identical to DSA, ECDSA operates on an elliptic curve over $\mathbb{Z}_p$ with the signature calculated over a subgroup of prime order $\#q$
+    - More efficient, does not require modulus of thousands of bits
 - Security level is based on generic attacks against EC
-  - i.e $\sqrt{|\#q|}$
+    - i.e. $\sqrt{|\#q|}$
 - Deterministic generation of $k$ is often used for safety (RFC 6979)
-  - This is where the ephemeral key isn’t random, it’s based off the hash of the message
-  - This is because reusing the ephemeral key is bad news
-- Other variants like EdDSA using Edwards curves (Ed25519 / Ed448) exist
+    - This is where the ephemeral key isn’t random, it’s based off the hash of the message
+    - This is because reusing the ephemeral key is bad news
+- Other variants like EdDSA using Edwards curves (Ed25519 / Ed448) exist