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New Insights on Differential and Linear Bounds Using Mixed Integer Linear Programming

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Innovative Security Solutions for Information Technology and Communications (SecITC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12596))

Abstract

Mixed Integer Linear Programming (MILP) is a very common method of modelling differential and linear bounds. The Convex Hull (CH) modelling, introduced by Sun et al. (Eprint 2013/Asiacrypt 2014), is a popular method in this regard, which can convert the conditions corresponding to a small (4-bit) SBox to MILP constraints efficiently. Our analysis shows, there are SBoxes for which the CH modelling can yield incorrect modelling. The problem arises from the observation that although the CH is generated for a certain set of points, there can be points outside this set which also satisfy all the inequalities of the CH. As apparently no variant of the CH modelling can circumvent this problem, we propose a new modelling for differential and linear bounds. Our modelling makes use of every points of interest individually. Additionally, we also explore the possibility of using redundant constraints, such that the run time for an MILP solver can be reduced while keeping the optimal result unchanged. With our experiments on round-reduced GIFT-128, we show it is possible to reduce the run time a few folds using a suitable choice of redundant constraints. We also present the optimal linear bounds for 11- and 12-rounds of GIFT-128, extending from the best-known result of 10-rounds.

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Notes

  1. 1.

    https://www.gurobi.com/.

  2. 2.

    https://www.sagemath.org/.

  3. 3.

    The inequalities are not strict, i.e., of the type \(\le \) or \(\ge \) (but not of the type < or >). The MILP solvers generally cannot handle strict inequalities, hence the inequalities representing CH suits well for forming the constraints of MILP instances.

  4. 4.

    https://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/polyhedron/base.html#sage.geometry.polyhedron.base.Polyhedron_base.equations_list.

  5. 5.

    Note that, this assumption is practical. As the run time for higher rounds take significantly longer than the smaller rounds, generally the solutions for the smaller rounds are available.

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Authors and Affiliations

  1. Nanyang Technological University, Singapore, Singapore

    Anubhab Baksi

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  1. Anubhab Baksi
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Correspondence to Anubhab Baksi .

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Editors and Affiliations

  1. Advanced Technologies Institute, Bucharest, Romania

    Diana Maimut

  2. Advanced Technologies Institute, Bucharest, Romania

    Andrei-George Oprina

  3. Faculté des Sciences et Techniques, University of Limoges, Limoges, France

    Damien Sauveron

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Baksi, A. (2021). New Insights on Differential and Linear Bounds Using Mixed Integer Linear Programming. In: Maimut, D., Oprina, AG., Sauveron, D. (eds) Innovative Security Solutions for Information Technology and Communications. SecITC 2020. Lecture Notes in Computer Science(), vol 12596. Springer, Cham. https://doi.org/10.1007/978-3-030-69255-1_4

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  • DOI: https://doi.org/10.1007/978-3-030-69255-1_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-69254-4

  • Online ISBN: 978-3-030-69255-1

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