Lower bound on the size of the largest metrically regular subset of the boolean cube

Oblaukhov, Alexey

doi:10.17223/2226308x/11/4

Cited by 2 publications

(4 citation statements)

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“…Note: Sets A and B satisfying condition from Step 1 of the problem (for an arbitrary constant not necessarily greater than 26) are called strongly metrically regular and are studied in [9].…”

Section: Solutionmentioning

confidence: 99%

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The Fifth International Students’ Olympiad in cryptography—NSUCRYPTO: Problems and their solutions

Gorodilova¹,

Agievich²,

Carlet

et al. 2019

Cryptologia

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Problems and their solutions of the Fifth International Students' Olympiad in cryptography NSUCRYPTO'2018 are presented. We consider problems related to attacks on ciphers and hash functions, Boolean functions, quantum circuits, Enigma, etc. We discuss several open problems on orthogonal arrays, Sylvester matrices and disjunct matrices. The problem of existing an invertible Sylvester matrix whose inverse is again a Sylvester matrix was completely solved during the Olympiad.

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Section: Solutionmentioning

confidence: 99%

“…The Enigma machine is a symmetric cipher famous for Figure 4 shows how a simplified Enigma machine works. The key components are the set of input switches (2) -which are reduced to 4 in the example but could have been 26 for the Latin alphabet -an input plugboard (3,7,8), three rotors (5), the reflector (6) and the output board (9).…”

Section: Formulationmentioning

confidence: 99%

The Fifth International Students’ Olympiad in cryptography—NSUCRYPTO: Problems and their solutions

Gorodilova¹,

Agievich²,

Carlet

et al. 2019

Cryptologia

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show abstract

“…if A is a metrically regular set, then its metric complement A is also a metrically regular set and both of them have the same covering radius. For some simple examples of metric complements and metrically regular sets, refer to [8,9,10].…”

Section: Definitions and Examplesmentioning

confidence: 99%

“…Metric regularity of several classes of partition set functions is studied in [13], while the work [4] touches upon metric properties of self-dual bent functions. Metric regularity has been actively investigated by the author: metric complements of linear subspaces of the Boolean cube are studied in the paper [8], while the works [9] and [10] are studying possible sizes of the largest and smallest metrically regular set.…”

Section: Introductionmentioning

confidence: 99%

On metric regularity of Reed-Muller codes

Oblaukhov¹

2019

Preprint

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In this work we study metric properties of the well-known family of binary Reed-Muller codes. Let A be an arbitrary subset of the Boolean cube, and A be the metric complement of A -the set of all vectors of the Boolean cube at the maximal possible distance from A. If the metric complement of A coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets appeared when studying bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this work we describe metric complements and establish metric regularity of the codes RM(0, m) and RM(k, m) for k m − 3. Additionally, metric regularity of the RM(1, 5) code is proved. Combined with previous results by Tokareva N. (2012) concerning duality of affine and bent functions, this proves metric regularity of most Reed-Muller codes with known covering radius. It is conjectured that all Reed-Muller codes are metrically regular.

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Lower bound on the size of the largest metrically regular subset of the boolean cube

Cited by 2 publications

References 1 publication

The Fifth International Students’ Olympiad in cryptography—NSUCRYPTO: Problems and their solutions

The Fifth International Students’ Olympiad in cryptography—NSUCRYPTO: Problems and their solutions

On metric regularity of Reed-Muller codes

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