Fast GPU Implementation of Dumer’s Algorithm Solving the Syndrome Decoding Problem

Abstract

The syndrome decoding problem (SDP) is a security basis for code-based cryptographic algorithms such as the McEliece cryptosystem, which has been selected as a finalist for the third round of the post-quantum cryptography (PQC) standardization project organized by the National Institute of Standards and Technology (NIST). Information set decoding (ISD) is a general term for algorithms that efficiently solve SDPs based on combinatorial enumeration. In this paper, we focused on Dumer’s algorithm, which is relatively lightweight among the ISD algorithms; and proposed a multi-parallel Dumer’s algorithm optimized for computing devices that perform massively parallel processing. We implemented our algorithm on a GPU and tested it by solving some SDP instances in the decoding challenge hosted by Inria. We report that our algorithm first solved a 1161-dimensional SDP in the Goppa-McEliece setting in approximately 380 hours, which has a computational complexity of 2⁵². Our results contribute to a more detailed security analysis for code-based cryptosystems.