Optimal query complexity for estimating the trace of a matrix

Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form $\frac{1}{k}\sum_{i=1}^k x_i^T A x_i$ for $x_i\in \mathbb{R}^n$ being i.i.d. for some special distribution. Our main results are summarized as follows. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (1) We show that any estimator requires $\Omega(1/\epsilon)$ queries to have a guarantee of variance at most $\epsilon$. (2) We show that any estimator requires $\Omega(\frac{1}{\epsilon^2}\log \frac{1}{\delta})$ queries to achieve a $(1\pm\epsilon)$-multiplicative approximation guarantee with probability at least $1 - \delta$. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.Comment: full version of the paper in ICALP 201

License protection with a tamper-resistant token

Content protection mechanisms are intended to enforce the usage rights on the content. These usage rights are carried by a license. Sometimes, a license even carries the key that is used to unlock the protected content. Unfortunately, license protection is difficult, yet it is important for digital rights management (DRM). Not many license protection schemes are available, and most if not all are proprietary. In this paper, we present a license protection scheme, which exploits tamper-resistant cryptographic hardware. The confidentiality and integrity of the license or parts thereof can be assured with our protection scheme. In addition, the keys to unlock the protected content are always protected and stored securely as part of the license. We verify secrecy and authentication aspects of one of our protocols. We implement the scheme in a prototype to assess the performance.\ud This project is funded by Telematica Instituut, The Netherlands

A Formal Study of the Privacy Concerns in Biometric-Based Remote Authentication Schemes

With their increasing popularity in cryptosystems, biometrics have attracted more and more attention from the information security community. However, how to handle the relevant privacy concerns remains to be troublesome. In this paper, we propose a novel security model to formalize the privacy concerns in biometric-based remote authentication schemes. Our security model covers a number of practical privacy concerns such as identity privacy and transaction anonymity, which have not been formally considered in the literature. In addition, we propose a general biometric-based remote authentication scheme and prove its security in our security model

Characteristics of Early-Onset vs Late-Onset Colorectal Cancer: A Review.

The incidence of early-onset colorectal cancer (younger than 50 years) is rising globally, the reasons for which are unclear. It appears to represent a unique disease process with different clinical, pathological, and molecular characteristics compared with late-onset colorectal cancer. Data on oncological outcomes are limited, and sensitivity to conventional neoadjuvant and adjuvant therapy regimens appear to be unknown. The purpose of this review is to summarize the available literature on early-onset colorectal cancer. Within the next decade, it is estimated that 1 in 10 colon cancers and 1 in 4 rectal cancers will be diagnosed in adults younger than 50 years. Potential risk factors include a Westernized diet, obesity, antibiotic usage, and alterations in the gut microbiome. Although genetic predisposition plays a role, most cases are sporadic. The full spectrum of germline and somatic sequence variations implicated remains unknown. Younger patients typically present with descending colonic or rectal cancer, advanced disease stage, and unfavorable histopathological features. Despite being more likely to receive neoadjuvant and adjuvant therapy, patients with early-onset disease demonstrate comparable oncological outcomes with their older counterparts. The clinicopathological features, underlying molecular profiles, and drivers of early-onset colorectal cancer differ from those of late-onset disease. Standardized, age-specific preventive, screening, diagnostic, and therapeutic strategies are required to optimize outcomes

Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background

The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests of general relativity, including direct study of the polarization of gravitational waves. While general relativity allows for only two tensor gravitational-wave polarizations, general metric theories can additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition of cosmological and individually unresolved astrophysical sources. Using data recorded by Advanced LIGO during its first observing run, we search for a stochastic background of generically polarized gravitational waves. We find no evidence for a background of any polarization, and place the first direct bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-uniform priors for the energy in each polarization, we limit the energy densities of tensor, vector, and scalar modes at 95% credibility to Ω0T<5.58×10-8, Ω0V<6.35×10-8, and Ω0S<1.08×10-7 at a reference frequency f0=25 Hz. © 2018 American Physical Society