Flattening Nested 2-of-2 Of a 1-of-1 And a k-of-n

The concept of flattening nested quorum signing groups, specifically a 2-of-2 composed of a single signer and a k-of-n quorum signer into a single-layer quorum signing group, is explored in depth. This process involves adjusting the shares within the group to ensure that the "single signer" has multiple shares in a larger, non-nested k-of-n group. The mathematical foundation for this adjustment includes increasing the total number of participants (n) by one to form the new required quorum size (flattened_k = n + 1) and calculating the new total number of shares (flattened_n = 2 * n - k + 1) to maintain balance and prevent any subgroup from overpowering the single privileged signer.

Several examples are provided to illustrate the practical application of this concept. In one scenario, a 2-of-3 configuration involving Ursula as the single signer and Alice, Bob, and Carol as part of the original quorum is transformed into a 4-of-5 setup. Here, Ursula holds two shares, ensuring she remains an essential part of any new quorum formation. Other examples further demonstrate how varying the initial conditions (e.g., increasing the number of signers in the k-of-n group) affects the distribution of shares and the resulting flattened group's configuration.

The derivation of this method emphasizes the importance of the single signer's role in the flattened structure, ensuring their participation is mandatory for quorum decisions. This approach not only simplifies the quorum signing process but also enhances security by preventing any subset of the original k-of-n group from bypassing the single signer.

The discussion extends to potential applications of this method in real-world scenarios, highlighting its relevance in various cryptographic and blockchain-related contexts. For instance, it mentions the use of such a signing structure in the Lightning Network, where flexibility and high availability are crucial for operations like outgoing HTLCs/PTLCs/MultiPTLCs among different Lightning Service Providers (LSPs). Additionally, it touches on the implications for security improvements in statechain operations and the avoidance of resharing issues inherent in nested k-of-n structures.

Critically, the piece delves into concerns about security vulnerabilities, particularly in the context of hardware wallets and the potential for reverse-engineering. It underscores the limitations of relying solely on technical measures for security, suggesting practical, real-world strategies to verify the integrity of hardware devices. This discussion points to the broader challenges in cryptographic security and the importance of comprehensive approaches that include both technical solutions and pragmatic verification methods.

For further reading on related topics, including advancements in multi-signature protocols and security considerations, you can explore MultiChannel. This resource provides insights into ongoing developments aimed at enhancing Bitcoin payments' global availability, consistency, and partition tolerance through innovative cryptographic techniques.