# Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng construction

Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng... The algebraic immunity of Boolean functions is studied in this paper. More precisely, having the prominent Carlet–Feng construction as a starting point, we propose a new method to construct a large number of functions with maximum algebraic immunity. The new method is based on deriving new properties of minimal codewords of the punctured Reed–Muller code $$\mathrm{RM}^{\star }(\lfloor \frac{n-1}{2}\rfloor ,n)$$ RM ⋆ ( ⌊ n - 1 2 ⌋ , n ) for any n, allowing—if n is odd—for efficiently generating large classes of new functions that cannot be obtained by other known constructions. It is shown that high nonlinearity, as well as good behavior against fast algebraic attacks, is also attainable. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

# Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng construction

, Volume 86 (8) – Oct 10, 2017
22 pages

/lp/springer_journal/boolean-functions-with-maximum-algebraic-immunity-further-extensions-QSEThxi0n0
Publisher
Springer US
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-017-0418-5
Publisher site
See Article on Publisher Site

### Abstract

The algebraic immunity of Boolean functions is studied in this paper. More precisely, having the prominent Carlet–Feng construction as a starting point, we propose a new method to construct a large number of functions with maximum algebraic immunity. The new method is based on deriving new properties of minimal codewords of the punctured Reed–Muller code $$\mathrm{RM}^{\star }(\lfloor \frac{n-1}{2}\rfloor ,n)$$ RM ⋆ ( ⌊ n - 1 2 ⌋ , n ) for any n, allowing—if n is odd—for efficiently generating large classes of new functions that cannot be obtained by other known constructions. It is shown that high nonlinearity, as well as good behavior against fast algebraic attacks, is also attainable.

### Journal

Designs, Codes and CryptographySpringer Journals

Published: Oct 10, 2017

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