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Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

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Number Theory and Related Fields

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 43))

Abstract

Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define—perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremendous advances over the last few years and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply cross-fertilize each other, holding great promise for all of them! In this exposition, we attempt to provide an overview of some breakthroughs in this field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.

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Acknowledgements

The author would like to thank Igor Shparlinski for many invaluable comments and for his inspiration throughout the preparation of this survey and his unending encouragement. I also thank Antal Balog, Emmanuel Breuillard, Ernie Croot, Harald Helfgott, Sergei Konyagin, Liangpan Li, Helger Lipmaa, Devanshu Pandey, Alain Plagne, László Pyber, Jeffrey Shallit, Emanuele Viola, and Van Vu for useful comments on this manuscript and/or sending some papers to me. Finally, I am grateful to the anonymous referees for their suggestions to improve this paper.

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  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Khodakhast Bibak

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  1. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Jonathan M. Borwein

  2. , Department of Computing, Macquarie University, Herring Road, North Ryde, 2109, New South Wales, Australia

    Igor Shparlinski

  3. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Wadim Zudilin

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Bibak, K. (2013). Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_4

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