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

TL;DR

This paper provides an overview of additive combinatorics, highlighting recent breakthroughs and their significant applications in computer science and cryptography, emphasizing the interdisciplinary nature of the field.

Contribution

It offers a comprehensive exposition of recent advances in additive combinatorics and its applications to computer science and cryptography.

Findings

Recent advances in additive combinatorics

Applications to cryptography and computer science

Interdisciplinary connections with other mathematical fields

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.