Des. Codes Cryptogr. DOI 10.1007/s10623-017-0418-5 Boolean functions with maximum algebraic immunity: further extensions of the Carlet–Feng construction Konstantinos Limniotis 1,2 · Nicholas Kolokotronis 3 Received: 31 January 2017 / Revised: 7 September 2017 / Accepted: 15 September 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−1 2 ⌋, 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 also attainable. 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 [18]. 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. B Konstantinos Limniotis klimn@di.uoa.gr; klimniotis@dpa.gr Nicholas Kolokotronis nkolok@uop.gr 1 Department of Informatics and Telecommunications, University of Athens, 15785 Athens, Greece 2 Hellenic Data Protection Authority, Kifissias 1-3, 11523 Athens, Greece 3 Department of Informatics and Telecommunications, University of Peloponnese, 22100 Tripolis, Greece 123