Optimal Threshold Signatures in Bitcoin

In the discussion surrounding security models, particularly those involving threshold systems, an important point is raised regarding the relationship between the probabilities of a user breaking into their own system (denoted as $p(\tau)$) and an attacker doing the same (denoted as $q(\tau)$). These probabilities, while conceptually distinct—$p(\tau)$ representing the likelihood of legitimate access by the user and $q(\tau)$ representing the likelihood of unauthorized access by an outsider—are fundamentally linked due to their shared basis in the underlying threshold system. This linkage implies that changes affecting one probability could impact the other, albeit in potentially divergent directions. Despite this interconnectedness, for the sake of simplicity, the current work assumes these probabilities are independent, a decision acknowledged as a simplification that overlooks the potential for a more nuanced model accounting for the interplay between user and attacker behaviors.

Furthermore, the conversation touches upon the notion of conducting a statistical analysis to identify an optimal threshold for signature schemes that balances security concerns with usability for legitimate users. This idea stems from a recognition of the gap in existing literature, where prior studies have primarily concentrated on the design and security aspects of threshold signature schemes without explicitly addressing how to statistically determine such an optimum threshold. The dialogue reveals an understanding that, although various works have examined multisignature and threshold mechanisms from different angles, there seems to be a lack of focused research on leveraging statistical methods to enhance both the security and usability dimensions of these systems. This observation underscores a promising direction for future research aimed at filling this identified gap in the scholarly discourse on threshold systems.