Paper 2025/1348

The CRO Trilemma : a formal incompatibility between Confidentiality, Reliability and legal Opposability in Post-Quantum proof systems

Abstract

This work establishes the CRO Trilemma: no post-quantum proof system can simultaneously satisfy the following three properties beyond negligible failure probability: (i) Confidentiality, quantified by Priv > 1 - negl(lambda); (ii) Reliability, with Rel > 1 - negl(lambda); and (iii) Legal Opposability, measured by contextual entropy H_Opp ≈ log |V_J|, where V_J denotes the validation space of a polynomial-time institutional verifier J. The result formalizes the Invisible Authenticity Paradox through contextual entropy H(C) and quantum interpretability loss eta_q(C). Under standard quantum assumptions, we prove the following impossibility bound against QPT adversaries: H_Opp <= (Priv * Rel) / (epsilon_eff * 2^H(C)) + eta_q(C) + negl(lambda) A composable framework is introduced, including a composable model with verifier J (Algorithm 1), and an optimal entropy decomposition theorem (Theorem 2.3). Theoretical predictions are supported by empirical evidence indicating consistent violation of the bound (Gamma_CRO > 0.8) across NIST PQC finalists (e.g., Dilithium) and structured ZKPs (e.g., STARKs, Groth16).

Note: This manuscript introduces the CRO Trilemma, a formal impossibility result demonstrating that no post-quantum proof system can simultaneously achieve confidentiality, reliability, and legal opposability beyond negligible bounds. The trilemma is quantified using contextual entropy and quantum interpretability loss, establishing a new class of constraints on composable cryptographic systems. A prior submission by the same author, titled “ZK-NR: A Layered Cryptographic Architecture for Explainable Non-Repudiation” (ePrint 2025/1138), presents a layered protocol that operates within the feasible space delineated by this trilemma. While ZK-NR offers a concrete protocol-level approach, the present work provides the underlying theoretical boundary that governs such constructions.