Additive combinatorics with a view towards computer science and cryptography: An exposition

Details Additive combinatorics with a view towards computer science and cryptography: An exposition Additive combinatorics with a view towards computer science and cryptography: An exposition

2011-08-18

arxiv.org/abs/1108.3790v5

Mathematics

Combinatorics

31 pages. Some other useful comments were received and incorporated in the paper

Scientific paper

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 newly 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! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating 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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