Extension to Real Number Representations over Elliptic Curves: Precision-Controlled Approximation and Fully Homomorphic Encryption with Advanced Bootstrapping and Noise Management

Building upon our previous work on rational number representations over elliptic curves via equivalence relations, we extend the framework to encompass real number computations through precision-controlled rational approximations. We present four fundamental contributions: (1) a systematic two-layer architecture clearly separating external real number interfaces from internal rational computation engines, (2) advanced bootstrapping techniques for noise reduction and computational capacity extension, (3) a comprehensive noise management framework with concrete bounds and mitigation strategies, and (4) a complete elliptic curve-based fully homomorphic encryption (EC-FHE) scheme that supports arbitrary-precision real arithmetic while maintaining cryptographic security. Our construction enables privacy-preserving computations on real-valued data with formal guarantees on both computational correctness and cryptographic security, opening new applications in secure scientific computing, privacy-preserving machine learning, and confidential financial modeling.