Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts

We compute minimal bases of solutions for a general interpolation problem, which encompasses Hermite-Pad\'e approximation and constrained multivariate interpolation, and has applications in coding theory and security. This problem asks to find univariate polynomial relations between $m$ vectors of size $\sigma$; these relations should have small degree with respect to an input degree shift. For an arbitrary shift, we propose an algorithm for the computation of an interpolation basis in shifted Popov normal form with a cost of $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ field operations, where $\omega$ is the exponent of matrix multiplication and the notation $\mathcal{O}\tilde{~}(\cdot)$ indicates that logarithmic terms are omitted. Earlier works, in the case of Hermite-Pad\'e approximation and in the general interpolation case, compute non-normalized bases. Since for arbitrary shifts such bases may have size $\Theta(m^2 \sigma)$, the cost bound $\mathcal{O}\tilde{~}(m^{\omega-1} \sigma)$ was feasible only with restrictive assumptions on the shift that ensure small output sizes. The question of handling arbitrary shifts with the same complexity bound was left open. To obtain the target cost for any shift, we strengthen the properties of the output bases, and of those obtained during the course of the algorithm: all the bases are computed in shifted Popov form, whose size is always $\mathcal{O}(m \sigma)$. Then, we design a divide-and-conquer scheme. We recursively reduce the initial interpolation problem to sub-problems with more convenient shifts by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms

q-Functional Wick's theorems for particles with exotic statistics

In the paper we begin a description of functional methods of quantum field theory for systems of interacting q-particles. These particles obey exotic statistics and are the q-generalization of the colored particles which appear in many problems of condensed matter physics, magnetism and quantum optics. Motivated by the general ideas of standard field theory we prove the q-functional analogues of Hori's formulation of Wick's theorems for the different ordered q-particle creation and annihilation operators. The formulae have the same formal expressions as fermionic and bosonic ones but differ by a nature of fields. This allows us to derive the perturbation series for the theory and develop analogues of standard quantum field theory constructions in q-functional form.Comment: 15 pages, LaTeX, submitted to J.Phys.

Computation and Homotopical Applications of Induced Crossed Modules

We explain how the computation of induced crossed modules allows the computation of certain homotopy 2-types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.Comment: 15 pages, xypic, latex2

On the representations and Z2\mathbb{Z}_2-equivariant normal form for solenoidal Hopf-zero singularities

In this paper, we deal with the solenoidal conservative Lie algebra associated to the classical normal form of Hopf-zero singular system. We concentrate on the study of some representations and $\mathbb{Z}_2$-equivariant normal form for such singular differential equations. First, we list some of the representations that this Lie algebra admits. The vector fields from this Lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one-degree of freedom Hamiltonian system and the third one depends upon the first two variables. This representation is governed by the associated Poisson algebra to one sub-family of this Lie algebra. Euler's form, vector potential, and Clebsch representation are other representations of this Lie algebra that we list here. We also study the non-potential property of vector fields with Hopf-zero singularity from this Lie algebra. Finally, we examine the unique normal form with non-zero cubic terms of this family in the presence of the symmetry group $\mathbb{Z}_2$ . The theoretical results of normal form theory are illustrated with the modified Chua's oscillator

Partial-indistinguishability obfuscation using braids

An obfuscator is an algorithm that translates circuits into functionally-equivalent similarly-sized circuits that are hard to understand. Efficient obfuscators would have many applications in cryptography. Until recently, theoretical progress has mainly been limited to no-go results. Recent works have proposed the first efficient obfuscation algorithms for classical logic circuits, based on a notion of indistinguishability against polynomial-time adversaries. In this work, we propose a new notion of obfuscation, which we call partial-indistinguishability. This notion is based on computationally universal groups with efficiently computable normal forms, and appears to be incomparable with existing definitions. We describe universal gate sets for both classical and quantum computation, in which our definition of obfuscation can be met by polynomial-time algorithms. We also discuss some potential applications to testing quantum computers. We stress that the cryptographic security of these obfuscators, especially when composed with translation from other gate sets, remains an open question.Comment: 21 pages,Proceedings of TQC 201