Abstract
The hardness of solving multivariate quadratic (\(\mathcal {MQ}\)) systems is the underlying problem for multivariate-based schemes in the field of post-quantum cryptography. The concrete, practical hardness of this problem needs to be measured by state-of-the-art algorithms and high-performance implementations. We describe, implement, and evaluate an adaption of the Crossbred algorithm by Joux and Vitse from 2017 for solving \(\mathcal {MQ}\) systems over \(\mathbb {F}_{2}\). Our adapted algorithm is highly parallelizable and is suitable for solving \(\mathcal {MQ}\) systems on GPU architectures. Our implementation is able to solve an \(\mathcal {MQ}\) system of 134 equations in 67 variables in 98.39 hours using one single commercial Nvidia GTX 980 graphics card, while the original Joux-Vitse algorithm requires 6200 CPU-hours for the same problem size. We used our implementation to solve all the Fukuoka Type-I MQ challenges for \(n \in \{55, \dots , 74\}\). Based on our implementation, we estimate that the expected computation time for solving an \(\mathcal {MQ}\) system of 80 equations in 84 variables is about one year using a cluster of 3600 GTX 980 graphics cards. These parameters have been proposed for 80-bit security by, e.g., Sakumoto, Shirai, and Hiwatari at Crypto 2011.
This work is based on Kai-Chun Ning’s master thesis under the supervision of Ruben Niederhagen, Tanja Lange, and Daniel J. Bernstein. Date: 2018.01.23. Permanent ID of this document: f9066f7294db4f2b5fbdb3e791fed78e.
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Acknowledgments
We would like to thank Daniel J. Bernstein for granting us access to his Saber GPU clusters at Eindhoven University of Technology and the University of Illinois at Chicago. This research was partially supported by the project MOST105-2923-E-001-003-MY3 of the Ministry of Science and Technology, Taiwan.
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Niederhagen, R., Ning, KC., Yang, BY. (2018). Implementing Joux-Vitse’s Crossbred Algorithm for Solving \({\mathcal M\mathcal Q}\) Systems over \({\mathbb F}_2\) on GPUs. In: Lange, T., Steinwandt, R. (eds) Post-Quantum Cryptography. PQCrypto 2018. Lecture Notes in Computer Science(), vol 10786. Springer, Cham. https://doi.org/10.1007/978-3-319-79063-3_6
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