Expansion in Finite Simple Groups of Lie
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Tao, Terence

doi:10.1090/gsm/164

Cited by 68 publications

(24 citation statements)

References 100 publications

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“…It is also easy to check that their homomorphic images are also sets of small doubling. To see this, first note that such a box B satisfies a stronger property than (2), in that there exists a set X satisfying |X| = 2 d such that…”

Section: First Examplesmentioning

confidence: 99%

A Brief Introduction to Approximate Groups

Tointon

2020

EMS Newsl.

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Section: First Examplesmentioning

confidence: 99%

A Brief Introduction to Approximate Groups

Tointon

2020

EMS Newsl.

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“…The spectral expansion of these constructions can be analyzed via group representation theory, which is more involved than Fourier analysis over abelian groups, because it deals with matrix-valued (rather than scalar-valued) functions. See the surveys by Hoory, Linial, and Wigderson [207] and Lubotzky [276], the lecture notes of Tao [386], and the notes for Section 4 for more on this and other approaches to analyzing the expansion of Cayley and Schreier graphs.…”

Section: Algebraic Pseudorandomnessmentioning

confidence: 99%

Chapter Notes and References

2020

Emergency Relief Operations

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This is a survey of pseudorandomness, the theory of efficiently generating objects that "look random" despite being constructed using little or no randomness. This theory has significance for a number of areas in computer science and mathematics, including computational complexity, algorithms, cryptography, combinatorics, communications, and additive number theory. Our treatment places particular emphasis on the intimate connections that have been discovered between a variety of fundamental "pseudorandom objects" that at first seem very different in nature: expander graphs, randomness extractors, list-decodable error-correcting codes, samplers, and pseudorandom generators. The structure of the presentation is meant to be suitable for teaching in a graduate-level course, with exercises accompanying each section.Cryptography: Randomization is central to cryptography. Indeed, cryptography is concerned with protecting secrets, and how can something be secret if it is deterministically fixed? For example, we assume that cryptographic keys are chosen at random (e.g., uniformly from the set of n-bit strings). In addition to the keys, it is known that often the cryptographic algorithms themselves (e.g., for encryption) must be randomized to achieve satisfactory notions of security (e.g., that no partial information about the message is leaked).Combinatorial Constructions: Randomness is often used to prove the existence of combinatorial objects with a desired property. Specifically, if one can show that a randomly chosen object has the property with nonzero probability, then it follows that such an object must, in fact, exist. A famous example due to Erdős is the existence of Ramsey graphs: A randomly chosen n-vertex graph has no clique or independent set of size 2 log n with high probability. We will see several other applications of this "Probabilistic Method" in this survey, such as with two important objects mentioned below: expander graphs and errorcorrecting codes.Though these applications of randomness are interesting and rich topics of study in their own right, they are not the focus of this survey. Rather, we ask the following: Main Question: Can we reduce or even eliminate the use of randomness in these settings?We have several motivations for doing this.

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“…O grafo de Ramanujan é um grafo de expansão. Grafos de expansão são grafos esparsos e possuem propriedades muito importantes, como baixo diâmetro, alta conectividade e alto número cromático [Tao 2015]. Para saber mais sobre grafos de expansão e suas aplicações consulte [Hoory et al 2006].…”

Section: Grafo De Ramanujanunclassified

Construção de S-Boxes com Valores Ótimos de Não Linearidade Baseada em uma Relação entre o Multigrafo de Ramanujan e a Matriz da Transformação Aﬁm

Belleza¹,

Borges²

2019

Anais Do XIX Simpósio Brasileiro De Segurança Da Informação E De Sistemas Computacionais (SBSeg 2019)

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Uma S-Box (caixa de substituição) deve ter pelo menos valores ótimos para não linearidade (N L), uniformidade diferencial e grau algébrico. Segundo a literatura, uma S-Box criptograﬁcamente forte deve ter N L > 100. O AES (Advanced Encryption Standard) usa uma matriz binária não singular S para construir sua S-Box. Muitos trabalhos escolhem S em aproximadamente 262 matrizes não singulares ou constroem S-Box aleatoriamente, sem garantir N L > 100. Neste trabalho, identiﬁcamos que S pode ser estudada como uma matriz de adjacência (A(G)) de um multigrafo de Ramanujan e veriﬁcamos esta relação com outras A(G) do tipo rotacionais. Dessa forma, reduzimos a busca por S para a ordem de 1011 e construímos S-Boxes com N L > 100.

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Expansion in Finite Simple Groups of Lie Type

Abstract: We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper [BGGT].

Cited by 68 publications

References 100 publications

A Brief Introduction to Approximate Groups

A Brief Introduction to Approximate Groups

Chapter Notes and References

Construção de S-Boxes com Valores Ótimos de Não Linearidade Baseada em uma Relação entre o Multigrafo de Ramanujan e a Matriz da Transformação Aﬁm

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