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Algebraic and SAT models for SCA generation

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

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Abstract

In this paper, we compute sequence covering arrays (SCAs), which are arrays, consisting of sequences, such that all subsequences with pairwise different entries of some length are covered, via a novel approach based on commutative algebra and symbolic computation. Hereby, we provide various algebraic models being capable to characterize possibly small sets of permutations collectively containing particular shorter subsequences. These models take the form of multivariate polynomial systems of equations and are then processed via supercomputing by a Gröbner Basis solver in order to compute solutions from them. If the variety is not empty, i.e. the Gröbner basis is non-trivial, then each point in the computed variety can be transformed to a SCA. In our experiments, we observed varying computational performance depending on the chosen model, while all of them exhibited scalability issues. Additionally and for comparison, we give new SAT descriptions modelling SCAs. By employing a SAT solver on our provided SAT models, we are able to provide upper bounds, one of which is best among literature results. Lastly, we adapt our SAT approach to answer a question posed by Yuster (Des Codes Cryptogr 88(3):585–593, 2020). As a result, we find a characterization of the dimensions of all perfect SCAs with coverage multiplicity two of strength three.

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Change history

  • 10 April 2023

    In the original publication, there were some typesetting problems. Now, those errors have been fixed

Notes

  1. We denote the set of positive integers with \(\mathbb {N^{\times }}\).

  2. We denote finite, nonempty sequences in tuple notation, e.g. \(\left( 0,1,2\right)\), highlighting that the order of the appearing elements is of uttermost importance. We also employ this notation to encode a single permutation of pairwise different elements.

  3. Informally, for the scope of this work, isomorphic solutions could be described as those, for which the resulting array structures can be transformed into each other by operations such as permuting the order of the rows in the array or applying a relabeling on the symbols.

  4. Here we use an inclusive interpretation of disjunction.

  5. Instead of \(\mathbb {Q}\) we can work with any ring without non-trivial zero divisors.

  6. Instead of \(\mathbb {Q}\) we can take any commutative ring with unity, which has no non-trivial zero divisors.

  7. If we work with an arbitrary commutative ring with unity, we have invertible elements instead of non-zero elements.

  8. Other integral domains would also be suitable as well, e.g. \(\mathcal {R} ={\text {GF}}(4)\), where the \(\alpha _k\) then have to be mapped to all field elements by a bijection. Notice that \(\alpha _k \mapsto \sum _{j=1}^k 1_{\mathcal {R}}\) is not admissible here.

  9. Note that other models for enforcing coverage do not assume any structure to already exist in the considered matrix.

  10. More generally, we can take any commutative ring with unity instead of \(\mathbb {Q}\).

  11. Instead of \(\mathbb {Q}\) we can work with any commutative ring with unity without non-trivial zero divisors with characteristic equal to zero or at least v.

  12. or in any commutative ring with unity without non-trivial zero divisors of characteristic zero or sufficiently large.

  13. or in any commutative ring with unity without non-trivial zero divisors.

  14. or over any commutative ring with unity without non-trivial zero divisors with characteristic equal to zero or at least \(N+1\).

  15. Recall that depending on the model, different lower bounds for the characteristic of the integral domain over which the model is built have to be met.

  16. We use the exhaustive search only for experimental performance comparison, therefore we considered just one coverage model.

  17. https://github.com/BRiAl/BRiAl.

  18. https://github.com/mzinin/bpb, being a C++ implementation of the involutive algorithm based on Pommaret monomial division (for further information consult the site).

  19. http://minisat.se/MiniSat.html.

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Acknowledgements

SBA Research (SBA-K1) is a COMET Centre within the framework of COMET—Competence Centers for Excellent Technologies Programme and funded by BMK, BMDW, and the federal state of Vienna. The COMET Programme is managed by FFG. This work was performed partly under the following financial assistance award 70NANB21H124 from U.S. Department of Commerce, National Institute of Standards and Technology.

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Authors and Affiliations

  1. MATRIS, SBA Research, 1040, Vienna, Austria

    Marlene Koelbing, Bernhard Garn, Enrico Iurlano & Dimitris E. Simos

  2. CARGO Lab, Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias S. Kotsireas

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  1. Marlene Koelbing
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  2. Bernhard Garn
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  3. Enrico Iurlano
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  4. Ilias S. Kotsireas
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  5. Dimitris E. Simos
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Correspondence to Dimitris E. Simos.

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Koelbing, M., Garn, B., Iurlano, E. et al. Algebraic and SAT models for SCA generation. AAECC (2023). https://doi.org/10.1007/s00200-023-00597-4

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