We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional'cells' connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square lattice'carpets'. We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies.

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We consider rectangular graph superlattices of sides l 1, l 2 with the wave-function coupling at the junctions either of the type, when they are continuous and the sum of their derivatives is proportional to the common value at the junction with a coupling constant , or the s type with the roles of functions and derivatives reversed; the latter corresponds to the situations where the junctions are realized by complicated geometric scatterers. We show that the band spectra have a hidden fractal structure with respect to the ratio := l 1/l 2. If the latter is an irrational badly approximable by rationals, lattices have no gaps in the weak-coupling case. We show that there is a quantization for the asymptotic critical values of at which new gap series open, and explain it in terms of number-theoretic properties of . We also show how the irregularity is manifested in terms of Fermi-surface dependence on energy, and possible

We show that Multivariate Public Key Cryptosystems (MPKCs) over fields of small odd prime characteristic, say 31, can be highly efficient. Indeed, at the same design security of 2^{80} under the best known attacks, odd-char MPKC is generally faster than prior MPKCs over \GF{2^k}, which are in turn faster than "traditional'' alternatives. This seemingly counter-intuitive feat is accomplished by exploiting the comparative over-abundance of small integer arithmetic resources in commodity hardware, here embodied by SSE2 or more advanced special multimedia instructions on modern x86-compatible CPUs. We explain our implementation techniques and design choices in implementing our chosen MPKC instances modulo small a odd prime. The same techniques are also applicable in modern FPGAs which often contains a large number of multipliers.

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