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We study conformal symmetry breaking differential operators which map differential forms on

Database outsourcing has become popular in recent years, although it introduces substantial security and privacy risks. In many applications, users may not want to reveal their data even to a generally trusted database service provider. Several researchers have proposed encryption schemes, such as privacy homomorphisms, that allow service providers to process confidential data sets without learning too much about them. In this paper, the authors discuss serious flaws of these solutions. The authors then present a new definition of security for homomorphic database encryption schemes that avoids these flaws and show that it is difficult to build a privacy homomorphism that complies with this definition. As a practical compromise, the authors present a relaxed variant of the security definition and discuss arising security implications. They present a new method to construct encryption schemes for exact selects and prove that the resulting schemes satisfy this notion.

We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This leads to an explicit description of \\(Q\\)-curvature and gauge companion operators on differential forms.

The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in terms of associated tractor bundles to compute these correction terms. Depending on the signature of the semi Riemannian manifold in question, the obtained conformal powers of the Dirac operator turn out to be formally self-adjoint with respect to the \\(L^2-\\)scalar product, or formally anti-self-adjoint, respectively. Working out this algorithm we present explicit formulas for the conformal third and fifth power of the Dirac operator. Furthermore, we present a new family of conformally covariant differential operators acting on the spin tractor bundle which are induced by conformally covariant differential operators acting on the spinor bundle. Finally, we will give polynomial structures for the first examples of conformal powers in terms of first order differential operators acting on the spinor bundle.