From Subset-Sum to Decoding: Improved Classical and Quantum Algorithms via Ternary Representation Technique

The subset-sum problem, a foundational NP-hard problem in theoretical computer science, serves as a critical building block for cryptographic constructions. This work introduces novel classical and quantum heuristic algorithms for the random subset-sum problem at density $d=1$, where exactly one solution is expected. Classically, we propose the first algorithm based on a ternary tree representation structure, inspired by recent advances in lattice-based cryptanalysis. Through numerical optimization, our method achieves a time complexity of $\tilde{\mathcal{O}}\left(2^{0.2400n}\right)$ and space complexity of $\tilde{\mathcal{O}}\left(2^{0.2221n}\right)$, improving upon the previous best classical heuristic result of $\tilde{\mathcal{O}}\left(2^{0.2830n}\right)$. In the quantum setting, we develop a corresponding algorithm by integrating the classical ternary representation technique with a quantum walk search framework. The optimized quantum algorithm attains a time and space complexity of $\tilde{\mathcal{O}}\left(2^{0.1843n}\right)$, surpassing the prior state-of-the-art quantum heuristic of $\tilde{\mathcal{O}}\left(2^{0.2182n}\right)$. Furthermore, we apply our algorithms to information set decoding in code-based cryptography. For half-distance decoding, our classical algorithm improves the time complexity to $\tilde{\mathcal{O}}\left(2^{0.0453n}\right)$, surpassing the previous best of $\tilde{\mathcal{O}}\left(2^{0.0465n}\right)$. For full-distance decoding, we achieve a quantum complexity of $\tilde{\mathcal{O}}\left(2^{0.058326n}\right)$, advancing beyond the prior best quantum result of $\tilde{\mathcal{O}}\left(2^{0.058696n}\right)$. These findings demonstrate the broad applicability and efficiency of our ternary representation technique across both classical and quantum computational models.