Is it possible to prove that a given signature is a part of an aggregated BLS signature? Specifically, given:
- $m_1...m_n$ are $n$ distinct messages
- $p_1...p_n$ are private keys with $P_1...P_n$ being the corresponding public keys
- $S_1...S_n$ are BLS signatures such that $S_i = sig(m_i, p_i)$
- $A = S_1 + S_2 + ... + S_n$ is the aggravated BLS signature
Is it possible to prove that a signature $S_i$ is contained within $A$ using only publicly available data: $A$, $S_i$, $P_i$, and $m_i$?