Real $$\tau $$-Conjecture for Sum-of-Squares: A Unified Approach to Lower Bound and Derandomization

A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

SIAM Journal on Computing ◽

10.1137/17m1138236 ◽

2019 ◽

Vol 48 (2) ◽

pp. 687-735 ◽

Cited By ~ 1

Author(s):

Boaz Barak ◽

Samuel Hopkins ◽

Jonathan Kelner ◽

Pravesh K. Kothari ◽

Ankur Moitra ◽

...

Keyword(s):

Lower Bound ◽

Sum Of Squares ◽

Planted Clique ◽

Clique Problem

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A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) ◽

10.1109/focs.2016.53 ◽

2016 ◽

Cited By ~ 27

Author(s):

Boaz Barak ◽

Samuel B. Hopkins ◽

Jonathan Kelner ◽

Pravesh Kothari ◽

Ankur Moitra ◽

...

Keyword(s):

Lower Bound ◽

Sum Of Squares ◽

Planted Clique ◽

Clique Problem

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Perturbed Identity Matrices Have High Rank: Proof and Applications

Combinatorics Probability Computing ◽

10.1017/s0963548307008917 ◽

2009 ◽

Vol 18 (1-2) ◽

pp. 3-15 ◽

Cited By ~ 34

Author(s):

NOGA ALON

Keyword(s):

Lower Bound ◽

Set Theory ◽

Coding Theory ◽

High Rank ◽

Real Matrix ◽

Unified Approach ◽

Absolute Value ◽

Finite Set ◽

Finite Set Theory

We describe a lower bound for the rank of any real matrix in which all diagonal entries are significantly larger in absolute value than all other entries, and discuss several applications of this result to the study of problems in Geometry, Coding Theory, Extremal Finite Set Theory and Probability. This is partly a survey, containing a unified approach for proving various known results, but it contains several new results as well.

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Validating Clusters with the Lower Bound for Sum-of-Squares Error

Psychometrika ◽

10.1007/s11336-003-1272-1 ◽

2007 ◽

Vol 72 (1) ◽

pp. 93-106 ◽

Cited By ~ 6

Author(s):

Douglas Steinley

Keyword(s):

Lower Bound ◽

Sum Of Squares

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ONE SUFFICIENT AND NECESSARY CONDITION ON BALANCED BOOLEAN FUNCTIONS WITH σf = 22n + 2n+3(n ≥ 3)

International Journal of Foundations of Computer Science ◽

10.1142/s0129054114500178 ◽

2014 ◽

Vol 25 (03) ◽

pp. 343-353 ◽

Cited By ~ 2

Author(s):

YU ZHOU ◽

LIN WANG ◽

WEIQIONG WANG ◽

XINFENG DONG ◽

XIAONI DU

Keyword(s):

Lower Bound ◽

Boolean Function ◽

Lower Bounds ◽

Boolean Functions ◽

Sum Of Squares ◽

Necessary Condition ◽

Resilient Function ◽

The Absolute ◽

Global Avalanche Characteristics ◽

Sufficient And Necessary Condition

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

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On the Modified Transparency Order of n , m -Functions

Security and Communication Networks ◽

10.1155/2021/6640099 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Yu Zhou ◽

Yongzhuang Wei ◽

Hailong Zhang ◽

Wenzheng Zhang

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Power Analysis ◽

Sum Of Squares ◽

Walsh Transform ◽

Algebraic Immunity ◽

Hamming Weight ◽

Absolute Value ◽

Transparency Order ◽

Weight Model

The concept of transparency order is introduced to measure the resistance of n , m -functions against multi-bit differential power analysis in the Hamming weight model, including the original transparency order (denoted by TO ), redefined transparency order (denoted by RTO ), and modified transparency order (denoted by MTO ). In this paper, we firstly give a relationship between MTO and RTO and show that RTO is less than or equal to MTO for any n , m -functions. We also give a tight upper bound and a tight lower bound on MTO for balanced n , m -functions. Secondly, some relationships between MTO and the maximal absolute value of the Walsh transform (or the sum-of-squares indicator, algebraic immunity, and the nonlinearity of its coordinates) for n , m -functions are obtained, respectively. Finally, we give MTO and RTO for (4,4) S-boxes which are commonly used in the design of lightweight block ciphers, respectively.

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