Your calculations are right, except there aren't exactly 2^256 private keys -- there are "FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141" (this number is named N in the ETH source code, and is the order of the generator of the elliptic curve secp256k1, from which Ethereum key pairs are generated).
In answer to your question, yes, private keys mapping to the same address will both be able to spend the money in that address on a first come first served basis. They will create the same public key, which the ECDSA verification algorithm is performed against, and so signatures generated by either private key will verify against the same address!
In go-ethereum/accounts/key.go, we have a private key generated from S256 which is secp256k1's curve, meaning they will be less than N be default.
func newKeyFromECDSA(privateKeyECDSA *ecdsa.PrivateKey) *Key {
id := uuid.NewRandom()
key := &Key{
Id: id,
Address: crypto.PubkeyToAddress(privateKeyECDSA.PublicKey),
PrivateKey: privateKeyECDSA,
}
return key
}
func newKey(rand io.Reader) (*Key, error) {
privateKeyECDSA, err := ecdsa.GenerateKey(secp256k1.S256(), rand)
if err != nil {
return nil, err
}
return newKeyFromECDSA(privateKeyECDSA), nil
}
go-ethereum/crypto/secp256k1/secp256 also generates the key pairs in accordance with secp2656k1 albeit in a slightly different way :)
Cool illustrative explanation for what N really is and why 2 keys will be able to spend money from the same account
Imagine private keys are miles driven in your car, and public keys are number of miles on your car's odometer. If the odometer rolls over from 999,999 to 000,000, a person with miles driven = 1,000,001 will have the same 'public key', and hence the same address, as a person with miles driven = 1.
EC group arithmetic is surprisingly similar to this, but rather than 999,999, we have the seemingly arbitrary number given above! Due to this property, even with your different 'private keys', you will be able to create signatures that verify against the same public key, and hence spend ETH from the same address!!
Boring math explanation
Private keys are 256 bit numbers, and to calculate the public key from a private key you multiply by the generator, g, of the elliptic curve group. The generator in use is defined in the parameters of the secp256k1 libraries being used in ETH. It itself is also an elliptic curve point, and as elliptic curves are cyclic, there exists an n such that n.g = 1 (this is called the generator order).
With this equation we can see that if we had a private key k with k>n, we would have k.g = (k-n).g = k'.g, for a k' that is possibly someone else's private key! So we have keys generated at random modulo n, rather than 2^256.
A private key in both bitcoin and ethereum is simply a random 256 bit number (actually a number between 0 and the order of the secp256k1 curve, but that's not really important).
If you can get your raw bitcoin public key, i.e.something like a random hex string of length 64, then it can be used directly as a raw ETH private key in any library or client.
For example, to import it into geth just use
geth account import <(my_privkey)
for parity see How to import a plain private key into Parity?
To simply derive an address from the key in JS, you can use the keythereum library or ethereumjs-util
You can even use openssl, see https://kobl.one/blog/create-full-ethereum-keypair-and-address/
Best Answer
According to this github issue page it turns out the answer is the private key is not generated as part of the node structure but instead the result of HMACing the first 32 bytes of the the entropy. An example below
The actual private key generation occurs in hdkey