Root-Hadamard transforms and complementary sequences
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Root-Hadamard transforms and complementary sequences ˘ a˘ 4 Luis A. Medina1 · Matthew G. Parker2 · Constanza Riera3 · Pantelimon Stanic Received: 22 July 2019 / Accepted: 1 June 2020 / © This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2020
Abstract In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k -Hadamard, consta-Hadamard and all H N -transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f . Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set. Keywords Golay pairs · Boolean functions · Correlations · Generalized root-transforms · Complementary sets Mathematics Subject Classification (2010) 06E30 · 31B83 · 94A55 · 94C10
Dedicated to the memory of our friend and co-author, Francis N. Castro. This article belongs to the Topical Collection: Boolean Functions and Their Applications IV Guest Editors: Lilya Budaghyan and Tor Helleseth Pantelimon St˘anic˘a
[email protected] Luis A. Medina [email protected] Matthew G. Parker [email protected] Constanza Riera [email protected] 1
Department of Mathematics, University of Puerto Rico, San Juan, PR, 00925, USA
2
Department of Informatics, University of Bergen, Bergen, Norway
3
Department of Computer Science, Electrical Engineering, Mathematical Sciences, Western Norway University of Applied Sciences, 5020, Bergen, Norway
4
Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943–521, USA
Cryptography and Communications
1 (Generalized) Boolean functions Let Fn2 be the vector space of the n-tuples over F2 , and, for an integer q, let Zq be the ring of integers modulo q. By ‘+’ and ‘−’ we respectively denote addition and subtraction in Fn2 . A function F : Fn2 → F2 , n > 0, is called a Boolean function in n variables, whose set will be denoted by Bn . A Boolean function can be regarded as a multivariate polynomial over F2 , called the algebraic normal form (ANF) f (x1 , . . . , xn ) = a0 + ai xi + aij xi xj + · · · + a12...n x1 x2 . . . xn , 1≤i
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