arXiv subjects

Saad Quader

Publications and source records attributed to Saad Quader.

6 recordsLinked to original sources

arXiv2020

Consistency of Proof-of-Stake Blockchains with Concurrent Honest Slot Leaders

We improve the fundamental security threshold of eventual consensus Proof-of-Stake (PoS) blockchain protocols under the longest-chain rule by showing, for the first time, the positive effect of rounds with concurrent honest leaders. Current security analyses reduce consistency to the dynamics of an abstract, round-based block creation process that is determined by three events associated with a round: (i) event $A$: at least one adversarial leader, (ii) event $S$: a single honest leader, and (iii) event $M$: multiple, but honest, leaders. We present an asymptotically optimal consistency analysis assuming that an honest round is more likely than an adversarial round (i.e., $\Pr[S] + \Pr[M] > \Pr[A]$); this threshold is optimal. This is a first in the literature and can be applied to both the simple synchronous communication as well as communication with bounded delays. In all existing consistency analyses, event $M$ is either penalized or treated neutrally. Specifically, the consistency analyses in Ouroboros Praos (Eurocrypt 2018) and Genesis (CCS 2018) assume that $\Pr[S] - \Pr[M] > \Pr[A]$; the analyses in Sleepy Consensus (Asiacrypt 2017) and Snow White (Fin. Crypto 2019) assume that $\Pr[S] > \Pr[A]$. Moreover, all existing analyses completely break down when $\Pr[S] < \Pr[A]$. These thresholds determine the critical trade-off between the honest majority, network delays, and consistency error. Our new results can be directly applied to improve the security guarantees of the existing protocols. We also provide an efficient algorithm to explicitly calculate these error probabilities in the synchronous setting. Furthermore, we complement these results by analyzing the setting where $S$ is rare, even allowing $\Pr[S] = 0$, under the added assumption that honest players adopt a consistent chain selection rule.

cs.DC↗

arXiv2019

Linear Consistency for Proof-of-Stake Blockchains

The blockchain data structure maintained via the longest-chain rule---popularized by Bitcoin---is a powerful algorithmic tool for consensus algorithms. Such algorithms achieve consistency for blocks in the chain as a function of their depth from the end of the chain. While the analysis of Bitcoin guarantees consistency with error $2^{-k}$ for blocks of depth $O(k)$, the state-of-the-art of proof-of-stake (PoS) blockchains suffers from a quadratic dependence on $k$: these protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that depth $Θ(k^2)$ is sufficient. Whether this quadratic gap is an intrinsic limitation of PoS---due to issues such as the nothing-at-stake problem---has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctness. We give an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longest-chain rule and achieve, in broad generality, $Θ(k)$ dependence on depth in order to achieve consistency error $2^{-k}$. In particular, for the first time, we show that PoS protocols can match proof-of-work protocols for linear consistency. We analyze the associated stochastic process, give a recursive relation for the critical functionals of this process, and derive tail bounds in both i.i.d. and martingale settings via associated generating functions.

cs.CR↗

arXiv2019

Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite Fields

We establish an uncertainty principle for functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ with constant support (where $p \mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: \mathbb{Z}/p \rightarrow \mathbb{F}_q$ for which $|\text{supp}\; {f}| = S$ must satisfy $|\text{supp}\; \hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredi's theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.

math.CO↗

arXiv2018

How to Realize a Graph on Random Points

We are given an integer $d$, a graph $G=(V,E)$, and a uniformly random embedding $f : V \rightarrow \{0,1\}^d$ of the vertices. We are interested in the probability that $G$ can be "realized" by a scaled Euclidean norm on $\mathbb{R}^d$, in the sense that there exists a non-negative scaling $w \in \mathbb{R}^d$ and a real threshold $θ> 0$ so that \[ (u,v) \in E \qquad \text{if and only if} \qquad \Vert f(u) - f(v) \Vert_w^2 < θ\,, \] where $\| x \|_w^2 = \sum_i w_i x_i^2$. These constraints are similar to those found in the Euclidean minimum spanning tree (EMST) realization problem. A crucial difference is that the realization map is (partially) determined by the random variable $f$. In this paper, we consider embeddings $f : V \rightarrow \{ x, y\}^d$ for arbitrary $x, y \in \mathbb{R}$. We prove that arbitrary trees can be realized with high probability when $d = Ω(n \log n)$. We prove an analogous result for graphs parametrized by the arboricity: specifically, we show that an arbitrary graph $G$ with arboricity $a$ can be realized with high probability when $d = Ω(n a^2 \log n)$. Additionally, if $r$ is the minimum effective resistance of the edges, $G$ can be realized with high probability when $d=Ω\left((n/r^2)\log n\right)$. Next, we show that it is necessary to have $d \geq \binom{n}{2}/6$ to realize random graphs, or $d \geq n/2$ to realize random spanning trees of the complete graph. This is true even if we permit an arbitrary embedding $f : V \rightarrow \{ x, y\}^d$ for any $x, y \in \mathbb{R}$ or negative weights. Along the way, we prove a probabilistic analog of Radon's theorem for convex sets in $\{0,1\}^d$. Our tree-realization result can complement existing results on statistical inference for gene expression data which involves realizing a tree, such as [GJP15].

cs.DM↗

arXiv2014

A (Somewhat Dated) Comparative Study of Betweenness Centrality Algorithms on GPU

The problem of computing the Betweenness Centrality (BC) is important in analyzing graphs in many practical applications like social networks, biological networks, transportation networks, electrical circuits, etc. Since this problem is computation intensive, researchers have been developing algorithms using high performance computing resources like supercomputers, clusters, and Graphics Processing Units (GPUs). Current GPU algorithms for computing BC employ Brandes' sequential algorithm with different trade-offs for thread scheduling, data structures, and atomic operations. In this paper, we study three GPU algorithms for computing BC of unweighted, directed, scale-free networks. We discuss and measure the trade-offs of their design choices about balanced thread scheduling, atomic operations, synchronizations and latency hiding. Our program is written in NVIDIA CUDA C and was tested on an NVIDIA Tesla M2050 GPU.

cs.SI↗

arXiv2012

A Beginner's Guide to Counting Spanning Trees in a Graph

(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix. This is a 165 year old result in graph theory and the proof is conceptually simple. However, the elegance of this result is it connects many apparently unrelated concepts in linear algebra and graph theory. Our motivation behind this work was to make the proof accessible to anyone with beginner\slash intermediate grasp of linear algebra. Therefore in this paper we present proof of every single argument leading to the final result. For example, we prove the elementary properties of determinants, relationship between the roots of characteristic polynomial (that is, eigenvalues) and the minors, the Cauchy-Binet formula, the Laplace expansion of determinant, etc.

cs.DM↗