Tight Lower Bounds for Testing Linear Isomorphism

We study lower bounds for testing membership in families of linear/affine-invariant Boolean functions over the hypercube. Motivated by the recent resurgence of attention to the permutation isomorphism problem, we first focus on families that are linearly/affinely isomorphic to some fixed function.

[1]  Shachar Lovett,et al.  New Extension of the Weil Bound for Character Sums with Applications to Coding , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[2]  Robert L. McFarland,et al.  A Family of Difference Sets in Non-cyclic Groups , 1973, J. Comb. Theory A.

[3]  Claude Carlet,et al.  An Alternate Characterization of the Bentness of Binary Functions, with Uniqueness , 1998, Des. Codes Cryptogr..

[4]  Yuichi Yoshida,et al.  Testing Linear-Invariant Function Isomorphism , 2013, ICALP.

[5]  Alan Guo,et al.  New affine-invariant codes from lifting , 2012, ITCS '13.

[6]  Madhu Sudan,et al.  2-Transitivity Is Insufficient for Local Testability , 2008, Computational Complexity Conference.

[7]  Guy Kindler,et al.  Testing juntas , 2002, J. Comput. Syst. Sci..

[8]  Shachar Lovett,et al.  Every locally characterized affine-invariant property is testable , 2013, STOC '13.

[9]  Eli Ben-Sasson,et al.  Sparse affine-invariant linear codes are locally testable , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[10]  Chris Charnes,et al.  The eight variable homogeneous degree three bent functions , 2008, J. Discrete Algorithms.

[11]  Noga Alon,et al.  Testing Boolean Function Isomorphism , 2010, APPROX-RANDOM.

[12]  Natalia N. Tokareva.  On the number of bent functions: lower bounds and hypotheses , 2011, IACR Cryptol. ePrint Arch..

[13]  Dana Ron,et al.  Testing Basic Boolean Formulae , 2002, SIAM J. Discret. Math..

[14]  László Lovász,et al.  Graph limits and parameter testing , 2006, STOC '06.

[15]  Ulrich Dempwolff.  Geometric and design-theoretic aspects of semibent functions II , 2012, Des. Codes Cryptogr..

[16]  Venkatesh Medabalimi.  Property Testing Lower Bounds via Communication Complexity , 2012 .

[17]  Eli Ben-Sasson,et al.  Symmetric LDPC Codes are not Necessarily Locally Testable , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[18]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[19]  Noga Alon,et al.  A combinatorial characterization of the testable graph properties: it's all about regularity , 2006, STOC '06.

[20]  B. Green.  A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[21]  Timo Neumann,et al.  BENT FUNCTIONS , 2006 .

[22]  Claude Carlet,et al.  CCZ-equivalence of bent vectorial functions and related constructions , 2011, Des. Codes Cryptogr..

[23]  Madhu Sudan,et al.  Algebraic property testing: the role of invariance , 2008, Electron. Colloquium Comput. Complex..

[24]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[25]  Noga Alon,et al.  Nearly tight bounds for testing function isomorphism , 2011, SODA '11.

[26]  Ryan O'Donnell,et al.  Lower Bounds for Testing Function Isomorphism , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[27]  Ryan O'Donnell,et al.  Testing Fourier Dimensionality and Sparsity , 2009, ICALP.

[28]  Eli Ben-Sasson,et al.  Limits on the Rate of Locally Testable Affine-Invariant Codes , 2011, APPROX-RANDOM.

[29]  Xiaoming Sun,et al.  Randomized Communication Complexity for Linear Algebra Problems over Finite Fields , 2012, STACS.

[30]  Eli Ben-Sasson,et al.  On Sums of Locally Testable Affine Invariant Properties , 2011, APPROX-RANDOM.

[31]  Madhu Sudan,et al.  Succinct Representation of Codes with Applications to Testing , 2012, SIAM J. Discret. Math..

[32]  Eyal Kushilevitz,et al.  Communication Complexity: Index of Notation , 1996 .

[33]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[34]  Daniel M. Kane,et al.  Tight Bounds for Testing k-Linearity , 2012, APPROX-RANDOM.

[35]  Asaf Shapira,et al.  A unified framework for testing linear-invariant properties , 2015, Random Struct. Algorithms.

[36]  Daniel Král,et al.  A combinatorial proof of the Removal Lemma for Groups , 2008, J. Comb. Theory, Ser. A.

[37]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[38]  Asaf Shapira.  Green's Conjecture and Testing Linear Invariant Properties , 2010, Property Testing.

[39]  Yuichi Yoshida,et al.  Partially Symmetric Functions Are Efficiently Isomorphism Testable , 2015, SIAM J. Comput..

[40]  Eldar Fischer,et al.  Junto-Symmetric Functions, Hypergraph Isomorphism and Crunching , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[41]  Arnab Bhattacharyya,et al.  Lower Bounds for Testing Triangle-freeness in Boolean Functions , 2010, SODA.

[42]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[43]  Claude Carlet.  Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions , 2011, Des. Codes Cryptogr..