7 research outputs found
On metric regularity of Reed-Muller codes
In this work we study metric properties of the well-known family of binary
Reed-Muller codes. Let be an arbitrary subset of the Boolean cube, and
be the metric complement of -- the set of all vectors of the
Boolean cube at the maximal possible distance from . If the metric
complement of coincides with , then the set is called a
{\it metrically regular set}. The problem of investigating metrically regular
sets appeared when studying {\it 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 the metric regularity of the codes
and for .
Additionally, the metric regularity of the codes and
is proved. Combined with previous results by Tokareva N.
(2012) concerning duality of affine and bent functions, this establishes the
metric regularity of most Reed-Muller codes with known covering radius. It is
conjectured that all Reed-Muller codes are metrically regular.Comment: 29 page
Classification of the Codewords of Weights 16 and 18 of the Reed-Muller Code RM(n-3,n)
International audienc
On the Sixth International Olympiad in Cryptography NSUCRYPTO
NSUCRYPTO is the unique cryptographic Olympiad containing scientific
mathematical problems for professionals, school and university students from
any country. Its aim is to involve young researchers in solving curious and
tough scientific problems of modern cryptography. From the very beginning, the
concept of the Olympiad was not to focus on solving olympic tasks but on
including unsolved research problems at the intersection of mathematics and
cryptography. The Olympiad history starts in 2014. In 2019, it was held for the
sixth time. In this paper, problems and their solutions of the Sixth
International Olympiad in cryptography NSUCRYPTO'2019 are presented. We
consider problems related to attacks on ciphers and hash functions, protocols,
Boolean functions, Dickson polynomials, prime numbers, rotor machines, etc. We
discuss several open problems on mathematical countermeasures to side-channel
attacks, APN involutions, S-boxes, etc. The problem of finding a collision for
the hash function Curl27 was partially solved during the Olympiad