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

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.

• > whats the point of generating just public key without being able to spend the fund sent to to it? A. deriving the Receiving address without giving ability to spend, as with less secure store website, or a Watch-Only wallet Mar 7 at 14:46