63 research outputs found
Effective Construction of a Class of Bent Quadratic Boolean Functions
In this paper, we consider the characterization of the bentness of quadratic
Boolean functions of the form where ,
is even and . For a general , it is difficult to determine
the bentness of these functions. We present the bentness of quadratic Boolean
function for two cases: and , where and are two
distinct primes. Further, we give the enumeration of quadratic bent functions
for the case
A Computational Search for Cubic-Like Bent Functions
Boolean functions are a central topic in computer science. A subset of Boolean functions, Bent Boolean functions, provide optimal resistance to various cryptographical attack vectors, making them an interesting subject for cryptography, as well as many other branches of mathematics and computer science. In this work, we search for cubic Bent Boolean functions using a novel characterization presented by Carlet & Villa in [CV23]. We implement a tool for the search of Bent Boolean functions and cubic-like Bent Boolean functions, allowing for constraints to be set on the form of the ANF of Boolean functions generated by the tool; reducing the search space required for an exhaustive search. The tool guarantees efficient traversal of the search space without redundancies. We use this tool to perform an exhaustive search for cubic-like Bent Boolean functions in dimension 6. This search proves unfeasible for dimension 8 and higher. We further attempt to find novel instances of Bent functions that are not Maioarana-McFarland in dimension 10 but fail to find any interesting results. We conclude that the proposed characterization does not yield a significant enough reduction of the search space to make the classification of cubic Bent Boolean functions of dimensions 8 or higher viable; nor could we use it to produce new instances of cubic Bent Boolean functions in 10 variables.Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN
Decomposing generalized bent and hyperbent functions
In this paper we introduce generalized hyperbent functions from to
, and investigate decompositions of generalized (hyper)bent functions.
We show that generalized (hyper)bent functions from to
consist of components which are generalized (hyper)bent functions from
to for some . For odd , we show
that the Boolean functions associated to a generalized bent function form an
affine space of semibent functions. This complements a recent result for even
, where the associated Boolean functions are bent.Comment: 24 page
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
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