Quantum Circuit Design for the Lee-Brickell Based Information Set Decoding

In the race for quantum-safe cryptography, fostered by the ongoing National Institute of Standards and Technology (NIST) post-quantum standardization process, it is crucial to assess the security of the emerging schemes. In this work, we propose a fully quantum algorithm to accelerate the Lee-Brickell’s Information Set Decoding (ISD)—one of the main cryptanalytic techniques used for assessing the security of code-based schemes—on binary error correcting codes. Our solution relies on a careful scheduling of the quantum gates included in the circuit design, coupled with a strategy that applies multiple times the oracle-reflection, from a Grover-like search, within a single Grover iteration. Compared with the state-of-the-art alternatives, our solution shows a reduction of the circuit depth ranging between \(2^{3}\) and \(2^{26}\) , when considering the parameters sets for code-based cryptosystems advanced to the fourth round of the NIST process. Denoting as t and \(t - p\) the two sets of bit flips tackled by the Lee-Brickell’s strategy, as an additional noteworthy fact we show that our solution exhibits 1 as the best value for p instead of 2 as it is the case for the classic ISD, for all concrete parameter sets considered.