Courant Institute of Mathematical Sciences

About: Courant Institute of Mathematical Sciences is a education organization based out in New York, New York, United States. It is known for research contribution in the topics: Nonlinear system & Boundary value problem. The organization has 2414 authors who have published 7759 publications receiving 439773 citations. The organization is also known as: CIMS & New York University Department of Mathematics.

Recovering Short Generators of Principal Ideals in Cyclotomic Rings.

Abstract: A handful of recent cryptographic proposals rely on the conjectured hardness of the following problem in the ring of integers of a cyclotomic number field: given a basis of a principal ideal that is guaranteed to have a “rather short” generator, find such a generator. Recently, Bernstein and Campbell-Groves-Shepherd sketched potential attacks against this problem; most notably, the latter authors claimed a polynomial-time quantum algorithm. (Alternatively, replacing the quantum component with an algorithm of Biasse and Fieker would yield a classical subexponential-time algorithm.) A key claim of Campbell et al. is that one step of their algorithm—namely, decoding the log-unit lattice of the ring to recover a short generator from an arbitrary one—is classically efficient (whereas the standard approach on general lattices takes exponential time). However, very few convincing details were provided to substantiate this claim. In this work, we clarify the situation by giving a rigorous proof that the log-unit lattice is indeed efficiently decodable, for any cyclotomic of prime-power index. Combining this with the quantum algorithm from a recent work of Biasse and Song confirms the main claim of Campbell et al. Our proof consists of two main technical contributions: the first is a geometrical analysis, using tools from analytic number theory, of the standard generators of the group of cyclotomic units. The second shows that for a wide class of typical distributions of the short generator, a standard lattice-decoding algorithm can recover it, given any generator. By extending our geometrical analysis, as a second main contribution we obtain an efficient algorithm that, given any generator of a principal ideal (in a prime-power cyclotomic), finds a 2 √ -approximate shortest vector in the ideal. Combining this with the result of Biasse and Song yields a quantum polynomial-time algorithm for the 2 √ -approximate Shortest Vector Problem on principal ideal lattices. ∗Cryptology Group, CWI, Amsterdam, The Netherlands & Mathematical Institute, Leiden University, The Netherlands. Email: cramer@cwi.nl, cramer@math.leidenuniv.nl †Cryptology Group, CWI, Amsterdam, The Netherlands. Supported by an NWO Free Competition Grant. Email: ducas@cwi.nl ‡Department of Computer Science and Engineering, University of Michigan. Much of this work was done while the author was at the Georgia Institute of Technology. This material is based upon work supported by the National Science Foundation under CAREER Award CCF-1054495, by DARPA under agreement number FA8750-11-C-0096, and by the Alfred P. Sloan Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, DARPA or the U.S. Government, or the Sloan Foundation. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. §Courant Institute of Mathematical Sciences, New York University. Supported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF) under Grant No. CCF-1320188.

Minimum-Relative-Entropy Calibration of Asset-Pricing Models

Abstract: We present an algorithm for calibrating asset-pricing models to the prices of benchmark securities. The algorithm computes the probability that minimizes the relative entropy with respect to a prior distribution and satisfies a finite number of moment constraints. These constraints arise from fitting the model to the prices of benchmark prices are studied in detail. We find that the sensitivities can be interpreted as regression coefficients of the payoffs of contingent claims on the set of payoffs of the benchmark instruments. We show that the algorithm has a unique solution which is stable, i.e. it depends smoothly on the input prices. The sensitivities of the values of contingent claims with respect to varriations in the benchmark instruments, in the risk-neutral measure. We also show that the minimum-relative-entropy algorithm is a special case of a general class of algorithms for calibrating models based on stochastic control and convex optimization. As an illustration, we use minimum-relative-entropy to construct a smooth curve of instantaneous forward rates from US LIBOR swap/FRA data and to study the corresponding sensitivities of fixed-income securities to variations in input prices.

Ground State of Liquid Helium-4 and Helium-3

Abstract: A variational calculation of the ground-state energy of liquid helium-3 and liquid helium-4 is made using, respectively, Jastrow- and Slater-Jastrow-type trial wave functions. In the boson (${\mathrm{He}}^{4}$) case, the quantum average, analogous to a statistical average over a fictitious canonical ensemble, is computed by a molecular-dynamics method for a system of 864 atoms. The following quantities are obtained: ground-state energy: -5.95\ifmmode^\circ\else\textdegree\fi{}K/atom (experiment: -7.14\ifmmode^\circ\else\textdegree\fi{}K/atom); equilibrium density (0.020\ifmmode\pm\else\textpm\fi{}0.002) atoms/${\mathrm{\AA{}}}^{3}$ (experiment: 0.022 atoms/${\mathrm{\AA{}}}^{3}$); liquid-structure factor; fraction of particles condensed in the zero-momentum state: 0.105\ifmmode\pm\else\textpm\fi{}0.005. This is in good agreement with previous computations. In the fermion (${\mathrm{He}}^{3}$) case, the energy expectation value is calculated approximately by use, up to second order, of a cluster expansion of the effect of the antisymmetrization, developed by Wu and Feenberg. The ground-state energy obtained is -1.35\ifmmode^\circ\else\textdegree\fi{}K/atom (experiment: -2.52\ifmmode^\circ\else\textdegree\fi{}K/atom). The liquid-structure factor of liquid ${\mathrm{He}}^{3}$, for which no experimental result is yet available, is calculated in this approximation.

On recognition of 3-D objects from 2-D images

Abstract: Techniques are described for model-based recognition of 3-D objects from unknown viewpoints using single gray-scale images. The method is especially useful for recognition of scenes with overlapping and partially occluded objects. An efficient matching algorithm, which assumes affine approximation to the perspective viewing transformation, is proposed. The algorithm has an off line model preprocessing phase and a recognition phase to reduce matching complexity. The algorithm was successfully tested in recognition of flat industrial objects appearing in composite occluded scenes. >