Optimal Threshold Signatures in Bitcoin

This raises questions about the optimal threshold setting—balancing security benefits against the risk of self-lockout. The challenge lies in determining the most effective threshold level that minimizes expected loss, which is a sum of losses due to potential attacks and the risk of losing access to one's own funds. Enhancements in security measures or usability can justify higher thresholds by reducing the likelihood of attacks (increasing the cost for attackers) and mitigating the risk of self-lockout, respectively.

The model presented introduces a formal way to calculate the optimal threshold, represented by $\tau^{*}$, based on the differential impact of security parameters ($a$ and $b$) on user and attacker probability functions. A higher threshold is favored when the gap between these parameters is wide, indicating better security or usability conditions. Conversely, a narrow gap necessitates a lower threshold to prevent lockouts without significantly compromising security. The analysis further extends to dynamic scenarios where user and attacker access probabilities evolve over time, affecting optimal threshold adjustments and timelock settings. Two dynamic models are proposed: one considering mutual decay in access probabilities and another accounting for opposite trends—decay in user access versus improvement in attacker's access over time.

The first dynamic model suggests starting with higher thresholds and gradually lowering them while extending timelocks to accommodate decreasing access probabilities. The second model, conversely, indicates that thresholds may need to increase over time if the rate at which an attacker gains access outpaces the rate of user access decay. These models cater to the reality of changing conditions in security and access over time, providing a more nuanced strategy for managing threshold signatures in Bitcoin.

Significant implications of this research include the conceptual framework it provides for understanding and optimizing threshold signatures within the Bitcoin ecosystem, particularly following the Taproot upgrade. It opens avenues for future exploration into economic and security dynamics of threshold schemes, potentially broadening their application beyond human users to include AI agents capable of leveraging Bitcoin Script for complex transactions. Further insights and the source code for simulations discussed can be found at the provided GitHub repository. This work underscores the importance and complexity of choosing optimal thresholds in securing Bitcoin transactions and highlights the potential of Taproot to facilitate diverse economic activities through advanced contract structures.