Cyclotomic Constructions of Skew Hadamard Difference Sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A

On quasi-orthogonal cocycles

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.Junta de Andalucía FQM-01

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

Generalized binary arrays from quasi-orthogonal cocycles

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of 2-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.Junta de Andalucía FQM-01