A Telescoping method for Double Summations

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term $F(n,i,j)$, we aim to find a difference operator $L=a_0(n) N^0 + a_1(n) N^1 +...+a_r(n) N^r$ and rational functions $R_1(n,i,j),R_2(n,i,j)$ such that $L F = \Delta_i (R_1 F) + \Delta_j (R_2 F)$. Based on simple divisibility considerations, we show that the denominators of $R_1$ and $R_2$ must possess certain factors which can be computed from $F(n, i,j)$. Using these factors as estimates, we may find the numerators of $R_1$ and $R_2$ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, Carlitz's identities, the Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic

Applicability of the qq-Analogue of Zeilberger's Algorithm

The applicability or terminating condition for the ordinary case of Zeilberger's algorithm was recently obtained by Abramov. For the $q$-analogue, the question of whether a bivariate $q$-hypergeometric term has a $qZ$-pair remains open. Le has found a solution to this problem when the given bivariate $q$-hypergeometric term is a rational function in certain powers of $q$. We solve the problem for the general case by giving a characterization of bivariate $q$-hypergeometric terms for which the $q$-analogue of Zeilberger's algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate $q$-hypergeometric term has a $qZ$-pair.Comment: 15 page

Universal linear-temperature resistivity: possible quantum diffusion transport in strongly correlated superconductors

The strongly correlated electron fluids in high temperature cuprate superconductors demonstrate an anomalous linear temperature ($T$) dependent resistivity behavior, which persists to a wide temperature range without exhibiting saturation. As cooling down, those electron fluids lose the resistivity and condense into the superfluid. However, the origin of the linear-$T$ resistivity behavior and its relationship to the strongly correlated superconductivity remain a mystery. Here we report a universal relation $d\rho/dT=(\mu_0k_B/\hbar)\lambda^2_L$, which bridges the slope of the linear-$T$-dependent resistivity ($d\rho/dT$) to the London penetration depth $\lambda_L$ at zero temperature among cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and heavy fermion superconductors CeCoIn$_5$, where $\mu_0$ is vacuum permeability, $k_B$ is the Boltzmann constant and $\hbar$ is the reduced Planck constant. We extend this scaling relation to different systems and found that it holds for other cuprate, pnictide and heavy fermion superconductors as well, regardless of the significant differences in the strength of electronic correlations, transport directions, and doping levels. Our analysis suggests that the scaling relation in strongly correlated superconductors could be described as a hydrodynamic diffusive transport, with the diffusion coefficient ($D$) approaching the quantum limit $D\sim\hbar/m^*$, where $m^*$ is the quasi-particle effective mass.Comment: 8 pages, 2 figures, 1 tabl

Memory-Based High-Level Synthesis Optimizations Security Exploration on the Power Side-Channel

High-level synthesis (HLS) allows hardware designers to think algorithmically and not worry about low-level, cycle-by-cycle details. This provides the ability to quickly explore the architectural design space and tradeoffs between resource utilization and performance. Unfortunately, security evaluation is not a standard part of the HLS design flow. In this article, we aim to understand the effects of memory-based HLS optimizations on power side-channel leakage. We use Xilinx Vivado HLS to develop different cryptographic cores, implement them on a Spartan-6 FPGA, and collect power traces. We evaluate the designs with respect to resource utilization, performance, and information leakage through power consumption. We have two important observations and contributions. First, the choice of resource optimization directive results in different levels of side-channel vulnerabilities. Second, the partitioning optimization directive can greatly compromise the hardware cryptographic system through power side-channel leakage due to the deployment of memory control logic. We describe an evaluation procedure for power side-channel leakage and use it to make best-effort recommendations about how to design more secure architectures in the cryptographic domain