A Fast Method for Finding Separable Goppa Polynomials Used in Post-Quantum McEliece-Based Cryptography

This paper introduces a simple and efficient method for generating Goppa polynomials used in post-quantum cryptography based on any variant of the McEliece algorithm. The approach demonstrates that such polynomials can be constructed more rapidly by multiplying several low-degree polynomials that satisfy specific properties. It is also proven that employing these polynomials does not compromise the code's error-correcting capability or overall security. The proposed method is especially advantageous when high-order Goppa polynomials are required. As a proof of concept, we present an application for user identification that combines cryptography and iris biometrics. In this system, encrypted versions of iris templates are securely stored. Using the homomorphic property of McEliece, recognition can be performed within the encrypted domain, ensuring that biometric data remains confidential throughout the entire process.