BIP32 [1] says: "In order to prevent these from depending solely on the key
itself, we extend both private and public keys first with an extra 256 bits of
entropy. This extension, called the chain code...".

My argument is that the chain code is not needed.
To support such claim, I'll show a schematic of BIP32 operations to be compared
with an alternative proposal and discuss the differences.

I have two main questions:
- Is this claim false?
- Has anyone shared this idea before?

## BIP32 schematic

Let `G` be the secp256k1 generator.
Let `i` be the child index.
Let `(p, P=pG)` and `(p_i, P_i=p_iG)` be the parent and i-th child keypairs
respectively.
Let `c` and `c_i` be the corresponding chain codes.
Let `h1, h2, h3, h4` be hash functions so that the formulae below match the
definitions given in BIP32 [2].
Define private and public child derivation as follow:

    p_i(p, c, i) = (i < 2^31)  p + h1(c, pG, i)
                   (i >= 2^31) p + h2(c, p, i)

    c_i(p, c, i) = (i < 2^31)  h3(c, pG, i)
                   (i >= 2^31) h4(c, p, i)

    P_i(P, c, i) = (i < 2^31)  P + h1(c, P, i)G
                   (i >= 2^31) not possible

    c_i(P, c, i) = (i < 2^31)  h3(c, P, i)
                   (i >= 2^31) not possible

The above formula for unhardened public derivation resembles a pay-to-contract
[3] scheme.

## Alternative proposal

Let `h` be an adequately strong hash function which converts its output to
integer.
Consider the following derivation scheme:

    p_i(p, i) = (i < 2^31)  p + h(pG, i)
                (i >= 2^31) h(p, i)

    P_i(P, i) = (i < 2^31)  P + h(P, i)G
                (i >= 2^31) not possible

Which is basically the above one without the chaincode.

## Considerations

I claim that this has the same properties as BIP32 [4]:
- The problem of finding `p` given `p_i, i` relies on brute-forcing `h` in the
  same way the analogous problem relies on brute-forcing `h2` in BIP32.
- The problem of determining whether `{p_i, i}_i=1..n` are derived from a common
  parent `p` relies on brute-forcing `h` in the same way the analogous problem
  relies on brute-forcing `h2` in BIP32.
- Given `i < 2^31, p_i, P`, an attacker can find `p`. This is analogous to
  BIP32, where the parent extended pubkey is needed (`P, c`). One could argue
  that `c` is never published on the blockchain, while `P` may be. On the other
  hand most wallets either use hardened derivation (so the attack does not work)
  or derive scriptpubkeys from keys at the same depth (so the parent key is
  never published on the blockchain).
  Anyway, if the parent public key is kept as secret as BIP32 extended keys are,
  then the situation is analogous to BIP32's.

_If_ these claims are correct, the proposed derivation scheme has two main
advantages:

1) Shorter backups for public and private derivable keys

Backups are especially relevant for output descriptors. For instance, when using
a NofM multisig, each participant must backup M-1 exteneded public keys and its
extended private key, which can be included in an output descriptor. Using the
proposed derivation reduces the backup size by `~M*32` bytes.

2) User-friendly backup for child keys

Most wallets use user-friendly backups, such as BIP39 [5] mnemonics. They map
16-32 bytes of entropy to 12-24 words. However BIP32 exteneded keys are at least
64(65) bytes (key and chain code), so they cannot be mapped back to a
mnemonic.

A common wallet setup is (`->` one-way derivation, `<->` two-way mapping):

    entropy (16-32 bytes) <-> user-friendly backup
      -> BIP32 extended key (64-65 bytes)
         -> BIP32 extended child keys (64-65 bytes)

With the proposed derivation, it would be possible to have:

    derivable private key (32 bytes) <-> user-friendly backup
      -> derivable public key (33 bytes) <-> user-friendly backup
      -> derivable child keys (32-33 bytes) <-> user-friendly backup

This would allow having mnemonics for subaccount keys.

## References

[1] https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki

[2] h1, h2, h3 and h4 can be defined as follows

    Ip(c, p, i) = (i >= 2^31) HMAC-SHA512(c, 0x00 || ser256(p) || ser32(i))
                  (i < 2^31)  HMAC-SHA512(c, pG || ser32(i))

    IP(c, P, i) = (i >= 2^31) not possible
                  (i < 2^31)  HMAC-SHA512(c, P || ser32(i))

    h1(c, P, i) = parse256(IP(c, P, i)[:32])
    h2(c, p, i) = parse256(Ip(c, p, i)[:32])
    h3(c, P, i) = IP(c, P, i)[32:]
    h4(c, p, i) = Ip(c, p, i)[32:]

[3] https://blockstream.com/sidechains.pdf Appendix A

[4] https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki#security

[5] https://github.com/bitcoin/bips/blob/master/bip-0039.mediawiki


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