A Fully Device-Independent Ternary Quantum Key Distribution Protocol Based on the Impossible Colouring Game

We propose a Ternary Fully Device-Independent Quantum Key Distribution (TFDIQKD) protocol based on the two-party Impossible Colouring pseudo-telepathy game, utilizing maximally entangled qutrit states to enable secure key generation between distant parties. The protocol harnesses Bell inequality violations that arise from contextuality in the Kochen-Specker theorem, thereby offering a quantum advantage in a task that is classically impossible and eliminating reliance on assumptions about the internal functioning of quantum devices. A specially designed qutrit quantum circuit is used for state preparation. Security and device independence are rigorously analyzed within a composable framework, employing Bell-inequality violations, smooth min-entropy, von Neumann entropy, and Shannon entropy. The protocol achieves optimal key rates in the ideal case and maintains security under significant noise, with a finite-key analysis that supports its practical viability. Overall, the protocol operates within an adequate security framework and demonstrates an improved key generation rate compared to standard quantum key distribution schemes, highlighting the potential of high-dimensional quantum systems for secure communication.