# Proof of membership in an aggregated BLS signature

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 $$m_i$$?