Shorter FMD public keys - is it possible?

[vveiln]

@kike

The suggested key derivation. We start with m master keypairs and use a cryptographic hash function \mathsf{H}:\mathbb{G}^m\times\mathbb{I}\rightarrow\mathbb{Z}_q. Derive the i-th child keypair as:

1. h_i := \mathsf{H}(X_1,\ldots,X_m,i)
2. s_i := h_i + \sum_{k\leq m}x_i\mod q
3. P_i := s_i\cdot G

I’m probably missing something, but if we use public key \sum{X_i} (ignoring the rest of the components here) and it allows to recover \sum{x_i}, isn’t it equivalent to having a single public key X = \sum{X_i} and recovering x (because the key shares are always used together, so we don’t actually need to know individual shares)? Or are we using the private/public keys in different combinations and not always together?