Given a set of consecutive HD private keys, can you generate the next private key without knowing the master private key? I'm wondering whether disclosing a consecutive number of HD private keys derived from a given master key would allow someone to compute what the next private key in the sequence would be before it is revealed?

So given HD-derived private keys n, n+1, n+2, ..., n+m, can one figure out what is the private key n+m+1?
No, this cannot be found out if we assume


  


(which is a desired property of BIP32 and which I'm going to refer to as Φ describing the problem φ) to be true and assume that n+m &lt;= 2^32 - 2

Assume you have an efficient solver ξ : P^m → P for the problem you described with

P := { a ∈ ℕ | 1 &lt;= a &lt;= 0xFFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFE BAAE DCE6 AF48 A03B BFD2 5E8C D036 4140}


being the set of valid secp256k1 ECDSA standard private keys.

Let Ψ = (r_1, r_2, ..., r_{m+1}) be an instance of a problem impossible to solve efficiently if Φ is true.

Solve φ as follows: If and only if ξ(r_1, r_2, ..., r_m) = r_{m+1}, return true, otherwise return false. This contradicts Φ.