Does derivation from public and private keys give the same result in normal key derivation?

Question

I am reading about key derivation in chapter 5 of the book "Mastering Bitcoin" by Andreas, along with this detailed thread and BIP-32. Here are some of my understanding about these two procedures:

k: private key // K: public key // i: index // c: chain code // H: HMAC hashing result // Hleft: the first 32 bits of the hash result. // n: order of Ecliptic Curve. // G: starting point of Ecliptic Curve

Normal Key Derivation

Case 1: parPrivkey -> childPrivkey (and from that, childPubkey)

H = HMAC(cpar, Kpar || ichild) 
=> kchild = (kpar + Hleft) mod n
=> Kchild = G*kchild = G*[ (kpar + Hleft) mod n)]

Case 2: parPubkey -> childPubkey

H = HMAC(cpar, Kpar || ichild)
=> Kchild = G*Hleft + Kpar

Hardened Key Derivation

Case 3: parPrivkey -> childPrivkey (and from that childPubkey)

H = HMAC(cpar, kpar || ichild)
=> kchild = (kpar + Hleft) mod n
=> Kchild = G*kchild = G*[ (kpar + Hleft) mod n]

Given those 3 methods I have some pretty confusion:

1. the difference in the generation equation between cases 1 and 2 are quite subtle such that we only need to multiply kchild = (kpar + Hleft) mod n by G to get that in case 2. Nevertheless, since there is a factor mod n at the end, I couldn't tell whether Kchild of case 1 will relate to that of case 2. If it does not, then what's the point of generating just public key without being able to spend the fund sent to to it?

Thank you very much in advance.

Answer 1

The purpose of unhardened derivation is to allow for cases 1 and 2 to derive the same keys as each other. It works because both actually hash the same data, and that adding two scalars will result in the discrete log for the result of adding two points.

In case 1, we take the left part of the hash as a scalar, and add it to the parent private key (also a scalar). In case 2, we take the left part of the hash as a scalar, multiply it by the generator point G to get a point, and add it to the parent public key (which is the parent private key multiplied by G).