Revisiting the security of quantum bit commitment schemes

Abstract

Bit commitment is one of the cornerstones of both classical and quantum cryptography. In the classical world, security analysis is straightforward because both hiding and binding each admit a single natural interpretation. In the quantum world, however, security notions can be defined in different ways that are not necessarily equivalent. Over three decades, several works have introduced numerous binding notions whose precise statements and mu- tual implications remain scattered across the literature. This thesis clarifies that ambiguity. We study quantum binding notions within the framework of canonical purified quantum bit commitment (QBC) schemes, a class that captures both the theoretical generality and the practical mechanisms of recent constructions. In the statistical setting, we examine six binding notions: strong-binding, honest-binding, sum-binding, collapse-binding, extractor- binding, and arbitrary-binding. We prove that all these definitions except strong-binding are equivalent for the class of canonical purified schemes. In the computational setting, we establish that honest-binding, sum-binding, and collapse-binding are equivalent for the same class of schemes. Moreover, we introduce a quantitative framework based on the work of Ref. [26] to quantify the tradeoff between hiding and binding properties, and we use it to provide a geometric proof of the impossibility theorem of Mayers [21] and Lo & Chau [20], for canonical purified schemes. We next revisit the quantum notion of hiding, which is far less ambiguous than binding. Finally, we survey some technical tools that can be used to construct secure QBC schemes.