Abstract
Linear modular constraints are a powerful class of constraints that arise naturally in cryptanalysis, checksums, hash functions, and the like. Given their importance, the past few years have witnessed the design of combinatorial solvers with native support for linear modular constraints, and the availability of such solvers has led to the emergence of new applications. While there exist global constraints in cp that consider congruence classes over domain values, linear modular arithmetic constraints have yet to appear in the global constraint catalogue despite their past investigation in the context of model counting for csps. In this work we seek to remedy the situation by advocating the integration of linear modular constraints in state-of-the-art cp solvers.
Contrary to previous belief, we conclude from an empirical investigation that Gauss-Jordan Elimination based techniques can provide an efficient and scalable way to handle linear modular constraints. On the theoretical side, we remark on the pairwise independence offered by hash functions based on linear modular constraints, and then discuss the design of hashing-based model counters for cp, supported by empirical results showing the accuracy and computational savings that can be achieved. We further demonstrate the usefulness of native support for linear modular constraints with applications to checksums and model counting.
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Notes
- 1.
- 2.
Available at https://github.com/PesantGilles/MiniCPBP.
- 3.
In practice we actually implement the compact table filtering algorithm and apply it directly instead of repeatedly posting and retracting Table constraints.
- 4.
For example if p is chosen as the smallest prime number larger than the domain size, as one can always find a prime between d and 2d for any \(d>1\).
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Acknowledgements
We thank the anonymous reviewers for their constructive criticism which helped us improve the original version of the paper. Financial support for this research was provided in part by NSERC Discovery Grant 218028/2017 and by National Research Foundation Singapore under its NRF Fellowship Programme [NRF-NRFFAI1-2019-0004].
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Pesant, G., Meel, K.S., Mohammadalitajrishi, M. (2021). On the Usefulness of Linear Modular Arithmetic in Constraint Programming. In: Stuckey, P.J. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2021. Lecture Notes in Computer Science(), vol 12735. Springer, Cham. https://doi.org/10.1007/978-3-030-78230-6_16
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