Construction and Benchmarking Methodology for Crypto-Code Systems on Elliptic and Hyperelliptic Curves

The advent of quantum computing threatens the long-term security of classical public-key cryptosystems: RSA and standard elliptic-curve cryptography are vulnerable to Shor’s algorithm, making quantum-resistant alternatives a priority for modern cryptography. Code-based constructions are among the strongest candidates in post-quantum cryptography. The McEliece and Niederreiter cryptosystems remain competitive after decades of study but suffer from large public keys and reliance on high-order Galois fields, imposing heavy overhead on resource-constrained platforms such as mobile and embedded devices. The symmetric Rao–Nam (RN) construction offers a more compact alternative by operating over smaller fields. This article develops a construction and measurement methodology for crypto-code systems that combine the asymmetric McEliece and Niederreiter schemes with the symmetric Rao–Nam scheme over algebraic-geometric codes derived from elliptic and hyperelliptic curves. The contribution is threefold: an explicit construction procedure for the evaluation and parity-check matrices over small binary fields, a controlled protocol for measuring the per-operation cost of the resulting implementations, and a quantified generic-decoding baseline for the parameter sets used. The parameter sets employed in the experiments are experimental sets selected for measurement and carry no cryptographic security; no deployment-ready post-quantum construction is proposed. It examines systems synthesizing crypto-code constructions with flawed codes to reduce the computational cost of cryptographic transformations by operating over the small field GF(22), under an unchanged code-based security assumption, alongside an alternative approach using hyperelliptic curves, and CCC built for mobile protocols using LDPC codes. These code families differ structurally: shortened codes tighten parameter control and reduce key material, extended codes raise the minimum distance and robustness, and higher-genus hyperelliptic curves provide a richer algebraic structure and a larger set of rational points for comparable field sizes. Whether this translates into stronger security against structural cryptanalysis is not established here and is identified as an open question.