There's a lot of confusion here, mostly bits and pieces of the whole scheme that is Hierarchical Deterministic derivation, and finally two questions that seem to indicate missing some point about it.
The answer to the first question is No. The second question is more interesting :
Let's start from extended keys, specifically BIP32 keys.
Like private keys and public keys, extended keys can be either "private" or "public". I put both in quotes becaus both types of extended keys do contain private information.
At least enough to track key use. This mechanism is used by hardware wallets and "watchonly" software wallets on a PC.
An extended key is just a base58 encoded serialization of a few pieces of data :
[ magic ][ depth ][ parent fingerprint ][ key index ][ chain code ][ key ]
Where key can be either a public key or a private key. Private keys are prepended with a single 0x00 byte, so the length of this blob stays the same.
An extended key is usually derived by "traversing" some path, meaning you would start your derivation at some parent extended key, and consecutively derive child keys with specific indexes until you finally derive the final extended key in the path.
I'll stop using "extended" in this answer. From now on I'll refer to an extended private key as xprv and to an extended public key as xpub, and just "keys" sometimes. Non-extended are just "private key" or "public key".
An xprv or xpub's magic are 4 bytes to indicate the network it belongs to: testnet or mainnet (t or x respectively), and the type of key it is (pub and prv respectively).
The depth is a byte that indecates how deep an xpriv or xpout is in a path, starting from 00 as the depth of the master key, and incremented by one as the derivation of more child keys is done along the path.
Note that up until now, the only difference between xprv and xpub keys that I mentioned is the prv or pub part in the magic.
It also should be clear that an xprv and xpub can be in the same path, and in the same depth. This means that for such pair of xprv and xpub, the [ key ] part of the will have a 32 byte private key (prepended with one 00 byte) in the xprv, and a 33 byte public key which is the public key which which you get from the private key in the xprv.
A parent's fingerprint are the first 4 bytes of the hash160 of the public key of the parent. This means that even if a parent xprv was used to derive a child xprv, it would have the same parent fingerprint as if a parent xpub was used to derive a child xpub.
A parent-child relationship between keys means that they are adjacent in a path.
A path is an n-tuple of indexes, usually in base 10, separated by /. The range of an index can be between zero and and 4294967295 (or 2^32-1), where anything in [0,2147483647] follows non-hardnened derivation, and indexes in [2147483648,4294967295] follow hardened derivation.
You can see that each half of the range of indexes is used for a different method. We can say that there are two ranges. [0,2147483647] for non-hardened keys, and [0h,2147483647h] for hardened keys. The h indicates that the index (we'll call it i) should be treated as i + 2147483648.
You're probably more likely to see the h notation as a caret ' instead, so 1' == 1h, but I don't think it's very pretty so I'll stick with h for now.
An example of what a path looks like is :
m/0h/1/2h/2/1000000000
The m means that the key at this index is a master xprv or master xpub. A small m means that this extended key is a master xprv, and big M a master xpub. Following the previous definitions, you can tell that m is the parent of the key at 0h, and the key at 2h is the child of the key before it at index 1.
To make this easier to follow, we'll annote the different keys in the path with letters {a..e} if we mean that these are xprvs and {A..B} if xpubs.
m / 0h / 1 / 2h / 2 / 1000000000
m a b c d e
A path is usually given with indexes in base10, but in the key itself they are encoded in hex (base16), so a [ key index ] is always 4 bytes with zeros prepended if needed. The depth and index of a master key are both alwasy zero, so 00 and 00000000, and they can get to a maximum of FF and FFFFFFFF respectively.
So m and a are parent and child, and so are d and e. The depth of b is 02 and its index is 00000001, and the depth of c is 03 while its index is 80000002 (80000000 + 2).
The last child key to be derived is e. We can say that we followed a path starting at m, from it we derived the key a at index 0h, then from a we derived the key b at index 1.. and so on. But what does it mean to derive a new key?
The remaining two elements in the extended key format, the parent's [ chain code ] and [ key ] are used together with what would be the child key's index to derive it.
That means that to derive c from b, we'd feed some function with b's chain code and key, and c's index.
A specific example of our b and c would be :
b :
0488ADE4
02
5C1BD648
00000001
2A7857631386BA23DACAC34180DD1983734E444FDBF774041578E9B6ADB37C19
003C6CB8D0F6A264C91EA8B5030FADAA8E538B020F0A387421A12DE9319DC93368
c :
0488ADE4
03
BEF5A2F9
80000002
04466B9CC8E161E966409CA52986C584F07E9DC81F735DB683C3FF6EC7B1503F
00CBCE0D719ECF7431D88E6A89FA1483E02E35092AF60C042B1DF2FF59FA424DCA
The fields are ordered as in the structure above. On both, the magic says xprv, the depth is incremented between the parent and child, the fingerprint at c is the hash160 of the public key that you would get from the private key at b, and b's index is in the first, non-hardened half of the range while c's is the second, hardned half. Finally the chain code and keys of each of the xprvs are encoded.
Deriving the chain code and key for c from b is done with a process called CKDpriv, which means deriving a child xprv from a parent xprv. In this process we used the chain code and key from b, and the index from c.
The important point to make: We only encoded c after deriving its chain code and key from what would be its index.
Any xprv can be used with CKDpriv to derive a child xprv at any index. The specific way CKDpriv will act on the input depends on the child's index being in the hardened range, or the non-hardned range. Basically, a CKDpriv function runs an HMAC-SHA512 on the parent's chain code and key, and the child's index. This hmac function takes two values a key* (not to be confused with our occurences of key, will be refered to as hkey), and text.
The parent's chain code is used as the hkey, while the text is made up of the parent's key in the private key form if the the child's index is in the hardened range, [0h,2147483647h], and in the public key form if the index is in the non-hardened range. It is then concatenated with the child's index.
c's index is in the hardened range, so CKDpriv's hmac-sha512 runs with the inputs:
HMAC-SHA512( 2A7857631386BA23DACAC34180DD1983734E444FDBF774041578E9B6ADB37C19,
003C6CB8D0F6A264C91EA8B5030FADAA8E538B020F0A387421A12DE9319DC9336880000002 )
Which returns a 64 byte hash :
8F6154A0A82D0F68B9E5B586EA66D951DAAA071BEBD390097CC516285C791A6204466B9CC8E161E966409CA52986C584F07E9DC81F735DB683C3FF6EC7B1503F
The 32 bytes on the right half of this hash, 04466B9C...C7B1503F become the child's (c here) chain code, and the 32 bytes on the left are used to "tweak", meaning just "addition mod n" to the parent's key, in this example :
8F6154A0A82D0F68B9E5B586EA66D951DAAA071BEBD390097CC516285C791A62
+
3C6CB8D0F6A264C91EA8B5030FADAA8E538B020F0A387421A12DE9319DC93368
=
CBCE0D719ECF7431D88E6A89FA1483E02E35092AF60C042B1DF2FF59FA424DCA mod n
- I didn't write the
00 prepended bytes in the keys here because this is just adding numbers, but those zero bytes are very important for the hash function, so I purposely included them there.
Now that we've got c's chain code and key (in private key form), we would want to actually encode c for it to be a usable xprv.
To get the fingerprint from b, we need to know the public key of the key from b. Since it's in private key form, we'll have to do multiplication:
CBCE0D719ECF7431D88E6A89FA1483E02E35092AF60C042B1DF2FF59FA424DCA * G
= 03501E454BF00751F24B1B489AA925215D66AF2234E3891C3B21A52BEDB3CD711C
Take the hash160 of this public key, and the returned hash is BEF5A2F9A56A94AAB12459F72AD9CF8CF19C7BBE.
The first four bytes are b's fingerprint : BEF5A2F9.
Encoding the rest of c is easy. Start with the magic xprv since we derived a child xprv, increment the depth of b by one, then the fingerprint.
Next c's index is encoded. We derived index 2h, so this would be 80000002, and then the new chain code and key that we got from CKDpriv.
This is basically what hardened derivation is. The parent's private key and chain code are used to derive the child key at some hardened index.
What if we want to derive d? It's at index 2, so a non-hardened index. This is the second case of CKDpriv.
The difference is in what is used for the text parameter of the HMAC-SHA512 function. Instead of using the parent's key in private key form, we use the public key form, so to derive d at index 2 from c, we first find the public key of c :
CBCE0D719ECF7431D88E6A89FA1483E02E35092AF60C042B1DF2FF59FA424DCA * G
= 0357BFE1E341D01C69FE5654309956CBEA516822FBA8A601743A012A7896EE8DC2
Then continue following the same steps as the above:
HMAC-SHA512( 04466B9CC8E161E966409CA52986C584F07E9DC81F735DB683C3FF6EC7B1503F,
0357BFE1E341D01C69FE5654309956CBEA516822FBA8A601743A012A7896EE8DC200000002 )
tweak chain code
437984D45C4A2F5840C65B3DC6D7274E2859AD25D092DB032C49AA4D006A426B|CFB71883F01676F587D023CC53A35BC7F88F724B1F8C2892AC1275AC822A3EDD
* note that 00 is not prepended to the text, since this is a public key.
437984D45C4A2F5840C65B3DC6D7274E2859AD25D092DB032C49AA4D006A426B
+
CBCE0D719ECF7431D88E6A89FA1483E02E35092AF60C042B1DF2FF59FA424DCA
=
0F479245FB19A38A1954C5C7C0EBAB2F9BDFD96A17563EF28A6A4B1A2A764EF4 mod n
hash160( 0357BFE1E341D01C69FE5654309956CBEA516822FBA8A601743A012A7896EE8DC2 )
finger
print
EE7AB90C|DE56A8C0E2BB086AC49748B8DB9DCE72
The rest is easy, and we can encode :
d :
0488ADE4
04
EE7AB90C
00000002
CFB71883F01676F587D023CC53A35BC7F88F724B1F8C2892AC1275AC822A3EDD
000F479245FB19A38A1954C5C7C0EBAB2F9BDFD96A17563EF28A6A4B1A2A764EF4
The difference between these two methods of deriving child xprvs is subtle but important. It enables CKDpub, which is a function to derive child xpubs from a parent xpub. CKDpub works almost the same as CKDpriv's non-hardened derivation, but it does the derivation using point addition, so rather than adding up integers to make child private keys, we're adding up points to make child public keys.
Notice how in the non-hardened derivation we used the parent's public point for the HMAC-SHA512, we used the tweak as the added value to the parent private key to derive the child private key, specifically, we derived d's private key.
To understand CKDpub, it helps to first know about yet another BIP32 function called Neuter. It's purpose is to convert an xprv to an xpub.
Let's "run" Neuter on our xprv d. We'll call the resulting xpub D. Neuter does two things to an xprv:
1. Replace the magic from 0488ADE4 to 0488B21E (replaces xprv with xpub)
2. Replaces the private key in the key field` with the public point of the same private key
for our xprv d, the public point is:
0F479245FB19A38A1954C5C7C0EBAB2F9BDFD96A17563EF28A6A4B1A2A764EF4 * G
= 02E8445082A72F29B75CA48748A914DF60622A609CACFCE8ED0E35804560741D29
(this is just normal process of private key -> public key)
so the result is:
D:
0488B21E
04
EE7AB90C
00000002
CFB71883F01676F587D023CC53A35BC7F88F724B1F8C2892AC1275AC822A3EDD
02E8445082A72F29B75CA48748A914DF60622A609CACFCE8ED0E35804560741D29
Now d is "neutered", D has the public key encoded, but see how the chain code, depth, fingerprint and index persisted. The xpub D is at the same position in the path as the xprv d. We will be using the chain code and key (public key) for CKDpub, same as CKDpriv with non-hardened derivation, but as for CKDpriv, we derived the child private key using:
tweak + (parent private key) = child private key
for CKDpub we will be using:
tweak*G + (parent public key) = child public key
This works because parent public key is really just (parent private key)*G, and child public key is just (child private key)*G. That is, if we take the CKDpriv tweak equation and multiply all elements by G, we get exactly the CKDpub tweak equation.
CKDpub can only derive child xpub keys in the non-hardened index range. This is because the information present in the parent xpub, specifically the public key in the [ key ], only applies to the non-hardened range.
Where in CKDpriv we could use the private key to know the public key, we can't go the other way. the HMAC-SHA512 round that uses public keys in CKDpriv applies to the non-hardened index range.
Now that we have neutered d to create the xpub D, next in the path is e's with index 1000000000 (or 3B9ACA00), which is in the non-hardened range, so we should be able to derive E the child xpub from D using CKDpub. We start with hmac-sha512 of the parent chain code as hkey and parent key (public key) concatenated with the child E's index :
HMAC-SHA512( CFB71883F01676F587D023CC53A35BC7F88F724B1F8C2892AC1275AC822A3EDD,
02E8445082A72F29B75CA48748A914DF60622A609CACFCE8ED0E35804560741D293B9ACA00 )
tweak chain code
37D3E49D8ECB854CC518BBA096F46795A9707860BF0FC95E5B19278C997098D4|C783E67B921D2BEB8F6B389CC646D7263B4145701DADD2161548A8B078E65E9E
Multiply the tweak by the generator G so we can tweak the parent's public key using point addition :
37D3E49D8ECB854CC518BBA096F46795A9707860BF0FC95E5B19278C997098D4 * G
= 0327E992F68217BC3E88CFFC3FEAB475880145413CBE008DB22B496DF4E1C3F864 <- tweak*G
Add the tweak to the parent point. The result is the child's public key :
0327E992F68217BC3E88CFFC3FEAB475880145413CBE008DB22B496DF4E1C3F864
+
02E8445082A72F29B75CA48748A914DF60622A609CACFCE8ED0E35804560741D29
=
022A471424DA5E657499D1FF51CB43C47481A03B1E77F951FE64CEC9F5A48F7011
Get the paren'ts fingerprint :
hash160(02E8445082A72F29B75CA48748A914DF60622A609CACFCE8ED0E35804560741D29) = D880D7D8....
Finally we can encode E :
0488B21E
05
D880D7D8
3B9ACA00
C783E67B921D2BEB8F6B389CC646D7263B4145701DADD2161548A8B078E65E9E
022A471424DA5E657499D1FF51CB43C47481A03B1E77F951FE64CEC9F5A48F7011
Neutering d to make D then deriving E, we can say that our path now looks like :
m / 0h / 1 / 2h / 2 / 1000000000
m / a / b / c / D / E
Or we can use the N() notation (for Neuter) to show where CKDpub was used, but I think it's less pretty.
m / a / b / c / N(d / e)
So to recap on your question, there are 3 different derivation methods, two using private keys and one using public keys :
CKDpriv to derive a child xprv at a hardened indexCKDpriv to derive a child xprv at a non-hardend indexCKDpub to derive a child xpub at a non-hardened index
