NDLI: Bent vectorial functions and linear codes from o-polynomials

Content Provider	Springer Nature Link
Author	Mesnager, Sihem
Copyright Year	2014
Abstract	The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and $$q$$ -ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. We show that the hyperovals of $$PG_{2}(2^m)$$ from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of $$2^r$$ -ary ( $$r$$ being a divisor of $$m$$ ) simplex linear codes (whose duals are the perfect $$2^r$$ -ary Hamming codes) over an extension field $${\mathbb F}_{2^{r}}$$ of $${\mathbb F}_{2^{}}$$ . The following diagram gives an indication of the main topics and interconnections arising in this paper.
Ending Page	116
Page Count	18
Starting Page	99
File Format	PDF
ISSN	09251022
e-ISSN	15737586
Journal	Designs, Codes and Cryptography
Issue Number	1
Volume Number	77
Language	English
Publisher	Springer US
Publisher Date	2014-06-25
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Information and Communication, Circuits o-polynomials Bent functions Data Encryption Linear codes, general Applications to coding theory and cryptography Coding theory Data Structures, Cryptology and Information Theory Finite geometry Symmetric cryptography Discrete Mathematics in Computer Science Linear codes Coding and Information Theory Cryptography Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science Applications

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