Des. Codes Cryptogr. (2018) 86:1685–1706
Boolean functions with maximum algebraic immunity:
further extensions of the Carlet–Feng construction
· Nicholas Kolokotronis
Received: 31 January 2017 / Revised: 7 September 2017 / Accepted: 15 September 2017 /
Published online: 10 October 2017
© Springer Science+Business Media, LLC 2017
Abstract The algebraic immunity of Boolean functions is studied in this paper. More pre-
cisely, 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 RM
, n) for any n, allowing—if n is odd—for efﬁciently gener-
ating 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
Keywords Algebraic attack · Algebraic immunity · Annihilators · Boolean functions ·
Cryptography · Reed–Muller codes
Mathematics Subject Classiﬁcation 94A60 · 06E30
Part of this work has been presented at the BalkanCryptSec 2015, Koper, Slovenia, 3–4 September 2015
. The results on functions with odd number of variables have been extended, providing a wider class of
functions (i.e., Theorem 6,Alg.modifyCFand Propositions 10, 11 and 12 are new), whereas new
sub-sections have been added with results on functions with even number of variables (Sects. 3.2 and 4.2).
Moreover, the results of Sect. 3 have been extended to cover the even case too.
Communicated by C. Carlet.
Department of Informatics and Telecommunications, University of Athens, 15785 Athens, Greece
Hellenic Data Protection Authority, Kiﬁssias 1-3, 11523 Athens, Greece
Department of Informatics and Telecommunications, University of Peloponnese,
22100 Tripolis, Greece