Public key recovery for EC leaves in P2MR (BIP-360)

The discussion revolves around the integration of P2TRH into P2MR as a specific instance, highlighting the flexibility and potential benefits of such an approach in cryptographic applications. P2TRH can be considered a simpler form of P2MR where it operates as a depth-zero tree featuring a single recoverable elliptic curve (EC) leaf. This setup might include a commitment to a more complex script tree, enhancing its functionality.

P2MR, on the other hand, supports multiple script leaves that can utilize either post-quantum (PQ) or EC signatures, offering a robust framework for various cryptographic operations. The spending mechanism in this structure involves using a single signature pair $(R,s)$, followed by the verification process which includes recovering the public key $P'$ and validating it against the P2MR root hash $H(P')$. An alternative spending method involves revealing an internal key $P$, a script $S$, and a control block $b$. The combination of these elements adjusts $P'$ through the function $P + H(P\ |\ \mathsf{RecomputeMerkleRoot}(H(S), b)) \cdot G$, ensuring the script executes as intended.

The proposal to integrate P2TRH under the P2MR framework could satisfy both groups supporting P2TRv2 and P2MR by maintaining witness size overhead comparable to P2TR and achieving full quantum security without further modifications. This dual approach does not only blend two cryptographic methods but also proposes a unified address format, potentially streamlining implementation and adoption in relevant systems. Such a merger suggests a significant advancement in cryptographic practices, balancing efficiency with enhanced security measures.