# Secure Cryptographic Key Encapsulation and Recovery Scheme in Noisy Network Conditions
## 2 Background  
Error-correcting codes (ECCs) are widely used in many cryptographic protocols, in particular across unreliable noisy networks to enhance data integrity and confidentiality. Their role is to detect and correct errors in data transmission or storage, thus safeguarding the information from corruption or unauthorized alterations [1; 2; 3]. The authors of [4] extend classical error correction methods to distributed communication networks, introducing network error correction. It leverages network coding principles that require redundant data to correct errors and ensure reliable communication in the presence of noise. Paper [5] addresses the challenges of reliable UAV-to-ground communication and relies on iterative message-passing techniques in the decoding process to enhance error correction and ensure robust data transmission. Paper [6] presents a novel method combining cryptography and error correction to detect and correct transmission errors in wireless communication. The method in [6], termed Joint Channel Coding and Cryptography, uses cryptographic check values for error recovery, achieving a coding gain of 0.72 dB while maintaining security requirements. This approach addresses the inherent error-proneness and security challenges in wireless transmission, providing a significant improvement over traditional convolutional coding methods. The authors in [7] use multi-path capillary routing combined with forward error correction to enhance the reliability of real-time streaming over unreliable networks. By leveraging redundant packets and a dynamic feedback mechanism, they adjust forward error correction overhead as needed to combat packet losses effectively. The authors of [8] present a pilot-free polar-coded transmission scheme that leverages frozen bits for joint channel state information estimation and decoding, significantly improving performance over traditional pilot-assisted transmission. In paper [9], a general framework for error detection in symmetric cryptography ciphers that proposes an operation-centered approach is presented. It enumerates and analyzes arithmetic and logic operations within ciphers to apply suitable error-detecting codes (EDCs) for efficient fault detection. Paper [10] reviews various error control techniques in bio-cryptography, focusing on methods such as score-based thresholding, quantization, and error correction coding. The findings in [10] highlight the challenges of biometric variability and the need for robust error control mechanisms in bio-cryptographic systems to ensure secure and accurate authentication. The authors of [10] also discuss the importance of selecting appropriate error correction methods tailored to specific biometric modalities and their error characteristics. Similarly, the authors of [11] examine the vulnerability of biometric encryption systems, particularly focusing on the security of ECCs in these systems. This study reveals how the error-correcting scheme, which relies on zero insertions, is susceptible to attacks, demonstrating that an attacker could reconstruct the entire key with minimal knowledge of zero locations in the ECC. Fuzzy extractors enable consistent repeatable cryptographic keys from noisy data such as physically unclonable functions (PUFs), corrupted databases, and biometric information. By employing ECCs, fuzzy extractors compensate for the variability inherent in these data sources, ensuring that a user can reliably regenerate the same key despite slight differences in input [12; 13; 14; 15; 16]. A fuzzy extractor that leverages the Learning Parity with Noise (LPN) problem to efficiently correct errors in noisy biometric sources is presented in [17]. The authors of [18] introduce a new cryptographic concept where private keys in asymmetric systems are subject to noise similar to biometric data variability. The authors of [18] propose methods for public key encryption and digital signatures using noisy keys leveraging a computational version of the fuzzy vault scheme for key generation and management.