BIP 32: Add simpler explanations where I got confused originally reading this.

bip-0032.mediawiki

@@ -51,10 +51,11 @@ Addition (+) of two coordinate pair is defined as application of the EC group op
 Concatenation (||) is the operation of appending one byte sequence onto another.
 
 As standard conversion functions, we assume:
-* point(p): returns the coordinate pair resulting from EC point multiplication (repeated application of the EC group operation) of the secp256k1 base point with the integer p.
+* point(p): returns the coordinate pair resulting from EC point multiplication (repeated application of the EC group operation) of the secp256k1 base point with the integer p (similar to making public key).
 * ser<sub>32</sub>(i): serialize a 32-bit unsigned integer i as a 4-byte sequence, most significant byte first.
 * ser<sub>256</sub>(p): serializes the integer p as a 32-byte sequence, most significant byte first.
-* ser<sub>P</sub>(P): serializes the coordinate pair P = (x,y) as a byte sequence using SEC1's compressed form: (0x02 or 0x03) || ser<sub>256</sub>(x), where the header byte depends on the parity of the omitted y coordinate.
+* ser<sub>P</sub>(P): serializes the coordinate pair P = (x,y) (aka. the public key) as a byte sequence using [https://github.com/bitcoin-core/secp256k1/blob/master/include/secp256k1.h#L177 SEC1]'s compressed form: (0x02 or 0x03) || ser<sub>256</sub>(x), where the header byte depends on the parity of the omitted y coordinate.
+** [https://github.com/bitcoin-core/secp256k1 Libsecp256k1]'s pubkey_serialize and pubkey_parse functions implement this format.
 * parse<sub>256</sub>(p): interprets a 32-byte sequence as a 256-bit number, most significant byte first.
 
 
@@ -64,6 +65,8 @@ In what follows, we will define a function that derives a number of child keys f
 
 We represent an extended private key as (k, c), with k the normal private key, and c the chain code. An extended public key is represented as (K, c), with K = point(k) and c the chain code.
 
+When deriving child keys, a 'hardened' child key can only be generated using a private key. This provides security advantages and prevents adversarial public key tracing. It is typically used to separate 'accounts' from one another.
+
 Each extended key has 2<sup>31</sup> normal child keys, and 2<sup>31</sup> hardened child keys. Each of these child keys has an index. The normal child keys use indices 0 through 2<sup>31</sup>-1. The hardened child keys use indices 2<sup>31</sup> through 2<sup>32</sup>-1. To ease notation for hardened key indices, a number i<sub>H</sub> represents i+2<sup>31</sup>.
 
 ===Child key derivation (CKD) functions===
@@ -105,6 +108,8 @@ To compute the public child key of a parent private key:
 * CKDpub(N(k<sub>par</sub>, c<sub>par</sub>), i) (works only for non-hardened child keys).
 The fact that they are equivalent is what makes non-hardened keys useful (one can derive child public keys of a given parent key without knowing any private key), and also what distinguishes them from hardened keys. The reason for not always using non-hardened keys (which are more useful) is security; see further for more information.
 
+The non-hardened result is typically used by a server to continually generate receive addresses without the ability to spend funds.
+
 ====Public parent key &rarr; private child key====
 
 This is not possible.