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**Published in:** Dagstuhl Seminar Proceedings, Volume 9031, Symmetric Cryptography (2009)

MutantXL is an algorithm for solving systems of polynomial equations
that was proposed at SCC 2008 and improved in PQC 2008. This article
gives an overview over the MutantXL algorithm. It also presents
experimental results comparing the behavior of the MutantXL
algorithm to the $F_4$ algorithm on HFE and randomly generated
multivariate systems. In both cases MutantXL is faster and uses less
memory than the Magma's implementation of $F_4$.

Johannes A. Buchmann, Jintai Ding, Mohamed Saied Emam Mohamed, and Wael Said Abd Elmageed Mohamed. MutantXL: Solving Multivariate Polynomial Equations for Cryptanalysis. In Symmetric Cryptography. Dagstuhl Seminar Proceedings, Volume 9031, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{buchmann_et_al:DagSemProc.09031.10, author = {Buchmann, Johannes A. and Ding, Jintai and Mohamed, Mohamed Saied Emam and Mohamed, Wael Said Abd Elmageed}, title = {{MutantXL: Solving Multivariate Polynomial Equations for Cryptanalysis}}, booktitle = {Symmetric Cryptography}, pages = {1--7}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9031}, editor = {Helena Handschuh and Stefan Lucks and Bart Preneel and Phillip Rogaway}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09031.10}, URN = {urn:nbn:de:0030-drops-19456}, doi = {10.4230/DagSemProc.09031.10}, annote = {Keywords: Multivariate systems, MutantXL} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 9031, Symmetric Cryptography (2009)

MutantXL is an algorithm for solving systems of polynomial equations
that was proposed at SCC 2008 and improved in PQC 2008. This article
gives an overview over the MutantXL algorithm. It also presents
experimental results comparing the behavior of the MutantXL
algorithm to the $F_4$ algorithm on HFE and randomly generated
multivariate systems. In both cases MutantXL is faster and uses less
memory than the Magma's implementation of $F_4$.

Johannes A. Buchmann, Jintai Ding, Mohamed Saied Emam Mohamed, and Wael Said Abd Elmageed Mohamed. MutantXL: Solving Multivariate Polynomial Equations for Cryptanalysis. In Symmetric Cryptography. Dagstuhl Seminar Proceedings, Volume 9031, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{buchmann_et_al:DagSemProc.09031.10, author = {Buchmann, Johannes A. and Ding, Jintai and Mohamed, Mohamed Saied Emam and Mohamed, Wael Said Abd Elmageed}, title = {{MutantXL: Solving Multivariate Polynomial Equations for Cryptanalysis}}, booktitle = {Symmetric Cryptography}, pages = {1--7}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2009}, volume = {9031}, editor = {Helena Handschuh and Stefan Lucks and Bart Preneel and Phillip Rogaway}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09031.10}, URN = {urn:nbn:de:0030-drops-19456}, doi = {10.4230/DagSemProc.09031.10}, annote = {Keywords: Multivariate systems, MutantXL} }

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**Published in:** LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

We consider the problem of preprocessing a text T of length n and a dictionary 𝒟 in order to be able to efficiently answer queries CountDistinct(i,j), that is, given i and j return the number of patterns from 𝒟 that occur in the fragment T[i..j]. The dictionary is internal in the sense that each pattern in 𝒟 is given as a fragment of T. This way, the dictionary takes space proportional to the number of patterns d=|𝒟| rather than their total length, which could be Θ(n⋅ d). An 𝒪̃(n+d)-size data structure that answers CountDistinct(i,j) queries 𝒪(log n)-approximately in 𝒪̃(1) time was recently proposed in a work that introduced internal dictionary matching [ISAAC 2019]. Here we present an 𝒪̃(n+d)-size data structure that answers CountDistinct(i,j) queries 2-approximately in 𝒪̃(1) time. Using range queries, for any m, we give an 𝒪̃(min(nd/m,n²/m²)+d)-size data structure that answers CountDistinct(i,j) queries exactly in 𝒪̃(m) time. We also consider the special case when the dictionary consists of all square factors of the string. We design an 𝒪(n log² n)-size data structure that allows us to count distinct squares in a text fragment T[i..j] in 𝒪(log n) time.

Panagiotis Charalampopoulos, Tomasz Kociumaka, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, Juliusz Straszyński, Tomasz Waleń, and Wiktor Zuba. Counting Distinct Patterns in Internal Dictionary Matching. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{charalampopoulos_et_al:LIPIcs.CPM.2020.8, author = {Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Mohamed, Manal and Radoszewski, Jakub and Rytter, Wojciech and Straszy\'{n}ski, Juliusz and Wale\'{n}, Tomasz and Zuba, Wiktor}, title = {{Counting Distinct Patterns in Internal Dictionary Matching}}, booktitle = {31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-149-8}, ISSN = {1868-8969}, year = {2020}, volume = {161}, editor = {G{\o}rtz, Inge Li and Weimann, Oren}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.8}, URN = {urn:nbn:de:0030-drops-121336}, doi = {10.4230/LIPIcs.CPM.2020.8}, annote = {Keywords: dictionary matching, internal pattern matching, squares} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We introduce data structures answering queries concerning the occurrences of patterns from a given dictionary D in fragments of a given string T of length n. The dictionary is internal in the sense that each pattern in D is given as a fragment of T. This way, D takes space proportional to the number of patterns d=|D| rather than their total length, which could be Theta(n * d).
In particular, we consider the following types of queries: reporting and counting all occurrences of patterns from D in a fragment T[i..j] (operations Report(i,j) and Count(i,j) below, as well as operation Exists(i,j) that returns true iff Count(i,j)>0) and reporting distinct patterns from D that occur in T[i..j] (operation ReportDistinct(i,j)). We show how to construct, in O((n+d) log^{O(1)} n) time, a data structure that answers each of these queries in time O(log^{O(1)} n+|output|) - see the table below for specific time and space complexities.
Query | Preprocessing time | Space | Query time
Exists(i,j) | O(n+d) | O(n) | O(1)
Report(i,j) | O(n+d) | O(n+d) | O(1+|output|)
ReportDistinct(i,j) | O(n log n+d) | O(n+d) | O(log n+|output|)
Count(i,j) | O({n log n}/{log log n} + d log^{3/2} n) | O(n+d log n) | O({log^2n}/{log log n})
The case of counting patterns is much more involved and needs a combination of a locally consistent parsing with orthogonal range searching. Reporting distinct patterns, on the other hand, uses the structure of maximal repetitions in strings. Finally, we provide tight - up to subpolynomial factors - upper and lower bounds for the case of a dynamic dictionary.

Panagiotis Charalampopoulos, Tomasz Kociumaka, Manal Mohamed, Jakub Radoszewski, Wojciech Rytter, and Tomasz Waleń. Internal Dictionary Matching. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ISAAC.2019.22, author = {Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Mohamed, Manal and Radoszewski, Jakub and Rytter, Wojciech and Wale\'{n}, Tomasz}, title = {{Internal Dictionary Matching}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {22:1--22:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.22}, URN = {urn:nbn:de:0030-drops-115182}, doi = {10.4230/LIPIcs.ISAAC.2019.22}, annote = {Keywords: string algorithms, dictionary matching, internal pattern matching} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

A border u of a word w is a proper factor of w occurring both as a prefix and as a suffix. The maximal unbordered factor of w is the longest factor of w which does not have a border. Here an O(n log n)-time with high probability (or O(n log n log^2 log n)-time deterministic) algorithm to compute the Longest Unbordered Factor Array of w for general alphabets is presented, where n is the length of w. This array specifies the length of the maximal unbordered factor starting at each position of w. This is a major improvement on the running time of the currently best worst-case algorithm working in O(n^{1.5}) time for integer alphabets [Gawrychowski et al., 2015].

Tomasz Kociumaka, Ritu Kundu, Manal Mohamed, and Solon P. Pissis. Longest Unbordered Factor in Quasilinear Time. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kociumaka_et_al:LIPIcs.ISAAC.2018.70, author = {Kociumaka, Tomasz and Kundu, Ritu and Mohamed, Manal and Pissis, Solon P.}, title = {{Longest Unbordered Factor in Quasilinear Time}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {70:1--70:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.70}, URN = {urn:nbn:de:0030-drops-100184}, doi = {10.4230/LIPIcs.ISAAC.2018.70}, annote = {Keywords: longest unbordered factor, factorisation, period, border, strings} }

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**Published in:** LIPIcs, Volume 88, 17th International Workshop on Algorithms in Bioinformatics (WABI 2017)

The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word w in a given sequence x can be used for classifying w as avoided or overabundant. The definitions used for the expectation and deviation of w in this statistical model were described and biologically justified by Brendel et al. (J Biomol Struct Dyn 1986). We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (Algorithms Mol Biol 2017). In this article, we extend this study by presenting an O(n)-time and O(n)-space algorithm for computing all overabundant words in a sequence x of length n over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree T of x: the number of distinct factors of x whose longest infix is the label of an explicit node of T is no more than 3n-4. We further show that the presented algorithm is time-optimal by proving that O(n) is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms.

Yannis Almirantis, Panagiotis Charalampopoulos, Jia Gao, Costas S. Iliopoulos, Manal Mohamed, Solon P. Pissis, and Dimitris Polychronopoulos. Optimal Computation of Overabundant Words. In 17th International Workshop on Algorithms in Bioinformatics (WABI 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 88, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{almirantis_et_al:LIPIcs.WABI.2017.4, author = {Almirantis, Yannis and Charalampopoulos, Panagiotis and Gao, Jia and Iliopoulos, Costas S. and Mohamed, Manal and Pissis, Solon P. and Polychronopoulos, Dimitris}, title = {{Optimal Computation of Overabundant Words}}, booktitle = {17th International Workshop on Algorithms in Bioinformatics (WABI 2017)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-050-7}, ISSN = {1868-8969}, year = {2017}, volume = {88}, editor = {Schwartz, Russell and Reinert, Knut}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2017.4}, URN = {urn:nbn:de:0030-drops-76468}, doi = {10.4230/LIPIcs.WABI.2017.4}, annote = {Keywords: overabundant words, avoided words, suffix tree, DNA sequence analysis} }

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**Published in:** LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

The paper deals with a load-balancing algorithm for a closed stochastic network with two zones with different demands. The algorithm is motivated by an incentive algorithm for redistribution of cars in a large-scale car-sharing system. The service area is divided into two zones. When cars stay too long in the low-demand zone, users are encouraged to pick them up and return them in the high-demand zone. The zones are divided in cells called stations. The cars are the network customers. The mean-field limit solution of an ODE gives the large scale distribution of the station state in both clusters for this incentive policy in a discrete Markovian framework. An equilibrium point of this ODE is characterized via the invariant measure of a random walk in the quarter-plane. The proportion of empty and saturated stations measures how the system is balanced. Numerical experiments illustrate the impact of the incentive policy. Our study shows that the incentive policy helps when the high-demand zone observes a lack of cars but a saturation must be prevented especially when the high-demand zone is small.

Bianca Marin Moreno, Christine Fricker, Hanene Mohamed, Amaury Philippe, and Martin Trépanier. Mean Field Analysis of an Incentive Algorithm for a Closed Stochastic Network. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{moreno_et_al:LIPIcs.AofA.2022.13, author = {Moreno, Bianca Marin and Fricker, Christine and Mohamed, Hanene and Philippe, Amaury and Tr\'{e}panier, Martin}, title = {{Mean Field Analysis of an Incentive Algorithm for a Closed Stochastic Network}}, booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-230-3}, ISSN = {1868-8969}, year = {2022}, volume = {225}, editor = {Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.13}, URN = {urn:nbn:de:0030-drops-160998}, doi = {10.4230/LIPIcs.AofA.2022.13}, annote = {Keywords: Large scale analysis, mean-field, car-sharing, incentive algorithm, stochastic network, cluster, load balancing, closed Jackson networks, product-form distribution} }

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**Published in:** LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to zero, we derive the first terms of the expansion in this parameter for the stationary probabilities that a queue has 0 to 3 customers. We investigate the error, comparing these expansion results to numerical values obtained by simulations. Then we provide the asymptotics, as the load tends to zero, for the stationary probabilities of the queue length, for a fixed number of queues. It quantifies the difference between policies with this local choice, no choice and the choice between two queues chosen at random.

Plinio S. Dester, Christine Fricker, and Hanene Mohamed. Stationary Distribution Analysis of a Queueing Model with Local Choice. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dester_et_al:LIPIcs.AofA.2018.22, author = {Dester, Plinio S. and Fricker, Christine and Mohamed, Hanene}, title = {{Stationary Distribution Analysis of a Queueing Model with Local Choice}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.22}, URN = {urn:nbn:de:0030-drops-89152}, doi = {10.4230/LIPIcs.AofA.2018.22}, annote = {Keywords: queueing model, local choice, stationary analysis, balance equations, power series expansion} }