Paper 2025/1348 Original: eprint.iacr.org/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).