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From what I understand, you want to share a secret to B in such a way that every single member of B is required to reconstruct the secret. In that case just use a threshold secret sharing scheme where the threshold is the number of players in B. This ensures that no subset of players in B can reconstruct the secret. I’m not convinced you need encryption here ...


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would using S = r + H(A, M) be a secure variant? Actually, it would become trivial to generate a signature for an abitrary message with just the public key. The verification check would be: $$2^h s G = 2^h R + 2^h H(A, M) A$$ where $h$ is the curve cofactor, $G$ is the curve generator, $A$ is the public key, $M$ is the message and $(R, s)$ is the signature. ...


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This answer is assuming you are not removing the private key $a$ from the computation of $S$, and instead actually meant what is said in the title of the question: $S = r + a H(A, M)$ Removing $a$ from the computation would be terrible. The first issue that comes to mind is malleability, on top of collision resistance. The signature process for EdDSA ...


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