923 research outputs found
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
The weight distributions of irreducible cyclic codes of length 2m
AbstractLet m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2m over Fq are determined explicitly
Cyclic Orbit Codes
In network coding a constant dimension code consists of a set of
k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes
which are defined as orbits of a subgroup of the general linear group, acting
on the set of all subspaces of F_q^n. If the acting group is cyclic, the
corresponding orbit codes are called cyclic orbit codes. In this paper we give
a classification of cyclic orbit codes and propose a decoding procedure for a
particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor
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