MPC for MPC: Secure Computation on a Massively Parallel Computing Architecture

Massively Parallel Computation (MPC) is a model of computation widely believed to best capture realistic parallel computing architectures such as large-scale MapReduce and Hadoop clusters. Motivated by the fact that many data analytics tasks performed on these platforms involve sensitive user data, we initiate the theoretical exploration of how to leverage MPC architectures to enable efficient, privacy-preserving computation over massive data. Clearly if a computation task does not lend itself to an efficient implementation on MPC even without security, then we cannot hope to compute it efficiently on MPC with security. We show, on the other hand, that any task that can be efficiently computed on MPC can also be securely computed with comparable efficiency. Specifically, we show the following results: - any MPC algorithm can be compiled to a communication-oblivious counterpart while asymptotically preserving its round and space complexity, where communication-obliviousness ensures that any network intermediary observing the communication patterns learn no information about the secret inputs; - assuming the existence of Fully Homomorphic Encryption with a suitable notion of compactness and other standard cryptographic assumptions, any MPC algorithm can be compiled to a secure counterpart that defends against an adversary who controls not only intermediate network routers but additionally up to 1/3 - ? fraction of machines (for an arbitrarily small constant ?) - moreover, this compilation preserves the round complexity tightly, and preserves the space complexity upto a multiplicative security parameter related blowup. As an initial exploration of this important direction, our work suggests new definitions and proposes novel protocols that blend algorithmic and cryptographic techniques

Extreme Lagrangian acceleration in confined turbulent flow

A Lagrangian study of two-dimensional turbulence for two different geometries, a periodic and a confined circular geometry, is presented to investigate the influence of solid boundaries on the Lagrangian dynamics. It is found that the Lagrangian acceleration is even more intermittent in the confined domain than in the periodic domain. The flatness of the Lagrangian acceleration as a function of the radius shows that the influence of the wall on the Lagrangian dynamics becomes negligible in the center of the domain and it also reveals that the wall is responsible for the increased intermittency. The transition in the Lagrangian statistics between this region, not directly influenced by the walls, and a critical radius which defines a Lagrangian boundary layer, is shown to be very sharp with a sudden increase of the acceleration flatness from about 5 to about 20

Reversal-field memory in magnetic hysteresis

We report results demonstrating a singularity in the hysteresis of magnetic materials, the reversal-field memory effect. This effect creates a nonanalyticity in the magnetization curves at a particular point related to the history of the sample. The microscopic origin of the effect is associated with a local spin-reversal symmetry of the underlying Hamiltonian. We show that the presence or absence of reversal-field memory distinguishes two widely studied models of spin glasses (random magnets).Comment: 3 pages, 5 figures. Proceedings of "2002 MMM Conferece", Tampa, F

Bifurcations and Complete Chaos for the Diamagnetic Kepler Problem

We describe the structure of bifurcations in the unbounded classical Diamagnetic Kepler problem. We conjecture that this system does not have any stable orbits and that the non-wandering set is described by a complete trinary symbolic dynamics for scaled energies larger then $\epsilon_c=0.328782\ldots$.Comment: 15 pages PostScript uuencoded with figure

Alternative method to find orbits in chaotic systems

We present here a new method which applies well ordered symbolic dynamics to find unstable periodic and non-periodic orbits in a chaotic system. The method is simple and efficient and has been successfully applied to a number of different systems such as the H\'enon map, disk billiards, stadium billiard, wedge billiard, diamagnetic Kepler problem, colinear Helium atom and systems with attracting potentials. The method seems to be better than earlier applied methods.Comment: 5 pages, uuencoded compressed tar PostScript fil

Magnetization reversal in Kagome artificial spin ice studied by first-order reversal curves

Magnetization reversal of interconnected Kagome artificial spin ice was studied by the first-order reversal curve (FORC) technique based on the magneto-optical Kerr effect and magnetoresistance measurements. The magnetization reversal exhibits a distinct six-fold symmetry with the external field orientation. When the field is parallel to one of the nano-bar branches, the domain nucleation/propagation and annihilation processes sensitively depend on the field cycling history and the maximum field applied. When the field is nearly perpendicular to one of the branches, the FORC measurement reveals the magnetic interaction between the Dirac strings and orthogonal branches during the magnetization reversal process. Our results demonstrate that the FORC approach provides a comprehensive framework for understanding the magnetic interaction in the magnetization reversal processes of spin-frustrated systems

Reversal-Field Memory in the Hysteresis of Spin Glasses

We report a novel singularity in the hysteresis of spin glasses, the reversal-field memory effect, which creates a non-analyticity in the magnetization curves at a particular point related to the history of the sample. The origin of the effect is due to the existence of a macroscopic number of "symmetric clusters" of spins associated with a local spin-reversal symmetry of the Hamiltonian. We use First Order Reversal Curve (FORC) diagrams to characterize the effect and compare to experimental results on thin magnetic films. We contrast our results on spin glasses to random magnets and show that the FORC technique is an effective "magnetic fingerprinting" tool.Comment: 4 pages, 6 figure