OpenRaiser/Intern-Atlas huggingface dataset:
Refuting the Pseudo Attack on the REESSE1+ Cryptosystem
Venue: arXiv cs.CR 2010
Year: 2010

We illustrate through example 1 and 2 that the condition at theorem 1 in [8] dissatisfies necessity, and the converse proposition of fact 1.1 in [8] does not hold, namely the condition Z/M - L/Ak < 1/(2 Ak^2) is not sufficient for f(i) + f(j) = f(k). Illuminate through an analysis and ex.3 that there is a logic error during deduction of fact 1.2, which causes each of fact 1.2, 1.3, 4 to be invalid. Demonstrate through ex.4 and 5 that each or the combination of qu+1 > qu * D at fact 4 and table 1 at fact 2.2 is not sufficient for f(i) + f(j) = f(k), property 1, 2, 3, 4, 5 each are invalid, and alg.1 based on fact 4 and alg.2 based on table 1 are disordered and wrong logically. Further, manifest through a repeated experiment and ex.5 that the data at table 2 is falsified, and the example in [8] is woven elaborately. We explain why Cx = Ax * W^f(x) (% M) is changed to Cx = (Ax * W^f(x))^d (% M) in REESSE1+ v2.1. To the signature fraud, we point out that [8] misunderstands the existence of T^-1 and Q^-1 % (M-1), and forging of Q can be easily avoided through moving H. Therefore, the conclusion of [8] that REESSE1+ is not secure at all (which connotes that [8] can extract a related private key from any public key in REESSE1+) is fully incorrect, and as long as the parameter Omega is fitly selected, REESSE1+ with Cx = Ax * W^f(x) (% M) is secure.
Tropical Implicitization and Mixed Fiber Polytopes
Venue: arXiv cs.SC 2010
Year: 2010

The software TrIm offers implementations of tropical implicitization and tropical elimination, as developed by Tevelev and the authors. Given a polynomial map with generic coefficients, TrIm computes the tropical variety of the image. When the image is a hypersurface, the output is the Newton polytope of the defining polynomial. TrIm can thus be used to compute mixed fiber polytopes, including secondary polytopes.
Faster Least Squares Approximation
Venue: arXiv cs.DS 2010
Year: 2010

Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets $n$ be the number of constraints and $d$ be the number of variables, with $n \gg d$. Then, existing exact methods find a solution vector in $O(nd^2)$ time. We present two randomized algorithms that provide very accurate relative-error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms. Both of our algorithms preprocess the data with the Randomized Hadamard Transform. One then uniformly randomly samples constraints and solves the smaller problem on those constraints, and the other performs a sparse random projection and solves the smaller problem on those projected coordinates. In both cases, solving the smaller problem provides relative-error approximations, and, if $n$ is sufficiently larger than $d$, the approximate solution can be computed in $O(nd \log d)$ time.
Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs
Venue: arXiv cs.CG 2010
Year: 2010

We consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that $\lfloor\frac{n+1}{3}\rfloor$ diagonal guards or $\lfloor\frac{2n+1}{5}\rfloor$ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: a diagonal 2-dominating set of size $\lfloor\frac{n+1}{3}\rfloor$ in linear time, an edge 2-dominating set of size $\lfloor\frac{2n+1}{5}\rfloor$ in $O(n^2)$ time, and an edge 2-dominating set of size $\lfloor\frac{3n}{7}\rfloor$ in O(n) time. Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: a mobile guard set of size $\lfloor\frac{n+1}{3}\rfloor$ in $O(n\log{}n)$ time, an edge guard set of size $\lfloor\frac{2n+1}{5}\rfloor$ in $O(n^2)$ time, and an edge guard set of size $\lfloor\frac{3n}{7}\rfloor$ in $O(n\log{}n)$ time. Finally, we show that $\lfloor\frac{n}{3}\rfloor$ mobile or $\lceil\frac{n}{3}\rceil$ edge guards are sometimes necessary. When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: $\lceil\frac{n+1}{4}\rceil$ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most $\lceil\frac{n+1}{4}\rceil$, can be computed in O(n) time.
Fixed Point and Aperiodic Tilings
Venue: arXiv cs.CC 2010
Year: 2010

An aperiodic tile set was first constructed by R.Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals) We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. The flexibility of this construction allows us to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. This property was not known for any of the existing aperiodic tile sets.
Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability
Venue: arXiv cs.CG 2010
Year: 2010

A \emph{binary tanglegram} is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an $O(n^3)$-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.
Data linkage dynamics with shedding
Venue: arXiv cs.LO 2010
Year: 2010

We study shedding in the setting of data linkage dynamics, a simple model of computation that bears on the use of dynamic data structures in programming. Shedding is complementary to garbage collection. With shedding, each time a link to a data object is updated by a program, it is determined whether or not the link will possibly be used once again by the program, and if not the link is automatically removed. Thus, everything is made garbage as soon as it can be viewed as garbage. By that, the effectiveness of garbage collection becomes maximal.
Algorithms for Dynamic Spectrum Access with Learning for Cognitive Radio
Venue: arXiv cs.NI 2010
Year: 2010

We study the problem of dynamic spectrum sensing and access in cognitive radio systems as a partially observed Markov decision process (POMDP). A group of cognitive users cooperatively tries to exploit vacancies in primary (licensed) channels whose occupancies follow a Markovian evolution. We first consider the scenario where the cognitive users have perfect knowledge of the distribution of the signals they receive from the primary users. For this problem, we obtain a greedy channel selection and access policy that maximizes the instantaneous reward, while satisfying a constraint on the probability of interfering with licensed transmissions. We also derive an analytical universal upper bound on the performance of the optimal policy. Through simulation, we show that our scheme achieves good performance relative to the upper bound and improved performance relative to an existing scheme. We then consider the more practical scenario where the exact distribution of the signal from the primary is unknown. We assume a parametric model for the distribution and develop an algorithm that can learn the true distribution, still guaranteeing the constraint on the interference probability. We show that this algorithm outperforms the naive design that assumes a worst case value for the parameter. We also provide a proof for the convergence of the learning algorithm.
Proposition Algebra with Projective Limits
Venue: arXiv cs.LO 2010
Year: 2010

Sequential propositional logic deviates from ordinary propositional logic by taking into account that during the sequential evaluation of a propositional statement,atomic propositions may yield different Boolean values at repeated occurrences. We introduce `free valuations' to capture this dynamics of a propositional statement's environment. The resulting logic is phrased as an equationally specified algebra rather than in the form of proof rules, and is named `proposition algebra'. It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The four axioms for free valuation congruence are then combined with other axioms in order define a few more valuation congruences that gradually identify more propositional statements, up to static valuation congruence (which is the setting of conventional propositional logic). Proposition algebra is developed in a fashion similar to the process algebra ACP and the program algebra PGA, via an algebraic specification which has a meaningful initial algebra for which a range of coarser congruences are considered important as well. In addition infinite objects (that is propositional statements, processes and programs respectively) are dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return.
The Balanced Unicast and Multicast Capacity Regions of Large Wireless Networks
Venue: arXiv cs.IT 2010
Year: 2010

We consider the question of determining the scaling of the $n^2$-dimensional balanced unicast and the $n 2^n$-dimensional balanced multicast capacity regions of a wireless network with $n$ nodes placed uniformly at random in a square region of area $n$ and communicating over Gaussian fading channels. We identify this scaling of both the balanced unicast and multicast capacity regions in terms of $Θ(n)$, out of $2^n$ total possible, cuts. These cuts only depend on the geometry of the locations of the source nodes and their destination nodes and the traffic demands between them, and thus can be readily evaluated. Our results are constructive and provide optimal (in the scaling sense) communication schemes.
Universal Secure Network Coding via Rank-Metric Codes
Venue: arXiv cs.IT 2010
Year: 2010

The problem of securing a network coding communication system against an eavesdropper adversary is considered. The network implements linear network coding to deliver n packets from source to each receiver, and the adversary can eavesdrop on μarbitrarily chosen links. The objective is to provide reliable communication to all receivers, while guaranteeing that the source information remains information-theoretically secure from the adversary. A coding scheme is proposed that can achieve the maximum possible rate of n-μpackets. The scheme, which is based on rank-metric codes, has the distinctive property of being universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying network code. The only requirement of the scheme is that the packet length be at least n, which is shown to be strictly necessary for universal communication at the maximum rate. A further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network, and the network may suffer from a rank deficiency of at most ρ. In this case, the proposed scheme can be extended to achieve the rate of n-ρ-2t-μpackets. This rate is shown to be optimal under the assumption of zero-error communication.
Throughput Scaling of Wireless Networks With Random Connections
Venue: arXiv cs.IT 2010
Year: 2010

This work studies the throughput scaling laws of ad hoc wireless networks in the limit of a large number of nodes. A random connections model is assumed in which the channel connections between the nodes are drawn independently from a common distribution. Transmitting nodes are subject to an on-off strategy, and receiving nodes employ conventional single-user decoding. The following results are proven: 1) For a class of connection models with finite mean and variance, the throughput scaling is upper-bounded by $O(n^{1/3})$ for single-hop schemes, and $O(n^{1/2})$ for two-hop (and multihop) schemes. 2) The $Θ(n^{1/2})$ throughput scaling is achievable for a specific connection model by a two-hop opportunistic relaying scheme, which employs full, but only local channel state information (CSI) at the receivers, and partial CSI at the transmitters. 3) By relaxing the constraints of finite mean and variance of the connection model, linear throughput scaling $Θ(n)$ is achievable with Pareto-type fading models.
Thresholded Basis Pursuit: An LP Algorithm for Achieving Optimal Support Recovery for Sparse and Approximately Sparse Signals from Noisy Random Measurements
Venue: arXiv cs.IT 2010
Year: 2010

In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless $\ell_1$ problem, namely, $\min \|β\|_1 ~ s.t. ~ y=G β$ and quantizing the resulting solution. We show that the quantized solution perfectly reconstructs the sign pattern of a sufficiently sparse signal. Specifically, we show that the sign pattern of an arbitrary k-sparse, n-dimensional signal $x$ can be recovered with $SNR=Ω(\log n)$ and measurements scaling as $m= Ω(k \log{n/k})$ for all sparsity levels $k$ satisfying $0< k \leq αn$, where $α$ is a sufficiently small positive constant. Surprisingly, this bound matches the optimal \emph{Max-Likelihood} performance bounds in terms of $SNR$, required number of measurements, and admissible sparsity level in an order-wise sense. In contrast to our results, previous results based on LASSO and Max-Correlation techniques either assume significantly larger $SNR$, sublinear sparsity levels or restrictive assumptions on signal sets. Our proof technique is based on noisy perturbation of the noiseless $\ell_1$ problem, in that, we estimate the maximum admissible noise level before sign pattern recovery fails.
On the Complexity of Core, Kernel, and Bargaining Set
Venue: arXiv cs.GT 2010
Year: 2010

Coalitional games are mathematical models suited to analyze scenarios where players can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. A fundamental problem for coalitional games is to single out the most desirable outcomes in terms of appropriate notions of worth distributions, which are usually called solution concepts. Motivated by the fact that decisions taken by realistic players cannot involve unbounded resources, recent computer science literature reconsidered the definition of such concepts by advocating the relevance of assessing the amount of resources needed for their computation in terms of their computational complexity. By following this avenue of research, the paper provides a complete picture of the complexity issues arising with three prominent solution concepts for coalitional games with transferable utility, namely, the core, the kernel, and the bargaining set, whenever the game worth-function is represented in some reasonable compact form (otherwise, if the worths of all coalitions are explicitly listed, the input sizes are so large that complexity problems are---artificially---trivial). The starting investigation point is the setting of graph games, about which various open questions were stated in the literature. The paper gives an answer to these questions, and in addition provides new insights on the setting, by characterizing the computational complexity of the three concepts in some relevant generalizations and specializations.
Interpolation of Shifted-Lacunary Polynomials
Venue: arXiv cs.SC 2010
Year: 2010

Given a "black box" function to evaluate an unknown rational polynomial f in Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 < e2 < ... < et, and the coefficients c1,...,ct in Q\{0} such that f(x) = c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, and in particular is logarithmic in deg(f). Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.
Adaptive Uncertainty Resolution in Bayesian Combinatorial Optimization Problems
Venue: arXiv cs.DS 2010
Year: 2010

In several applications such as databases, planning, and sensor networks, parameters such as selectivity, load, or sensed values are known only with some associated uncertainty. The performance of such a system (as captured by some objective function over the parameters) is significantly improved if some of these parameters can be probed or observed. In a resource constrained situation, deciding which parameters to observe in order to optimize system performance itself becomes an interesting and important optimization problem. This general problem is the focus of this paper. One of the most important considerations in this framework is whether adaptivity is required for the observations. Adaptive observations introduce blocking or sequential operations in the system whereas non-adaptive observations can be performed in parallel. One of the important questions in this regard is to characterize the benefit of adaptivity for probes and observation. We present general techniques for designing constant factor approximations to the optimal observation schemes for several widely used scheduling and metric objective functions. We show a unifying technique that relates this optimization problem to the outlier version of the corresponding deterministic optimization. By making this connection, our technique shows constant factor upper bounds for the benefit of adaptivity of the observation schemes. We show that while probing yields significant improvement in the objective function, being adaptive about the probing is not beneficial beyond constant factors.
Decidability of the Equivalence of Multi-Letter Quantum Finite Automata
Venue: arXiv cs.FL 2010
Year: 2010

Multi-letter {\it quantum finite automata} (QFAs) were a quantum variant of classical {\it one-way multi-head finite automata} (J. Hromkovič, Acta Informatica 19 (1983) 377-384), and it has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages $(a+b)^{*}b$ that are unacceptable by the previous one-way QFAs. In this paper, we study the decidability of the equivalence of multi-letter QFAs, and the main technical contributions are as follows: (1) We show that any two automata, a $k_{1}$-letter QFA ${\cal A}_1$ and a $k_{2}$-letter QFA ${\cal A}_2$, over the same input alphabet $Σ$ are equivalent if and only if they are $(n^2m^{k-1}-m^{k-1}+k)$-equivalent, where $m=|Σ|$ is the cardinality of $Σ$, $k=\max(k_{1},k_{2})$, and $n=n_{1}+n_{2}$, with $n_{1}$ and $n_{2}$ being the numbers of states of ${\cal A}_{1}$ and ${\cal A}_{2}$, respectively. When $k=1$, we obtain the decidability of equivalence of measure-once QFAs in the literature. It is worth mentioning that our technical method is essentially different from that for the decidability of the case of single input alphabet (i.e., $m=1$). (2) However, if we determine the equivalence of multi-letter QFAs by checking all strings of length not more than $ n^2m^{k-1}-m^{k-1}+k$, then the worst time complexity is exponential, i.e., $O(n^6m^{n^2m^{k-1}-m^{k-1}+2k-1})$. Therefore, we design a polynomial-time $O(m^{2k-1}n^{8}+km^kn^{6})$ algorithm for determining the equivalence of any two multi-letter QFAs. Here, the time complexity is concerning the number of states in the multi-letter QFAs, and $k$ is thought of as a constant.
The Violation Heap: A Relaxed Fibonacci-Like Heap
Venue: arXiv cs.DS 2010
Year: 2010

We give a priority queue that achieves the same amortized bounds as Fibonacci heaps. Namely, find-min requires O(1) worst-case time, insert, meld and decrease-key require O(1) amortized time, and delete-min requires $O(\log n)$ amortized time. Our structure is simple and promises an efficient practical behavior when compared to other known Fibonacci-like heaps. The main idea behind our construction is to propagate rank updates instead of performing cascaded cuts following a decrease-key operation, allowing for a relaxed structure.
Linearly Parameterized Bandits
Venue: arXiv cs.LG 2010
Year: 2010

We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an $r$-dimensional random vector $\mathbf{Z} \in \mathbb{R}^r$, where $r \geq 2$. The objective is to minimize the cumulative regret and Bayes risk. When the set of arms corresponds to the unit sphere, we prove that the regret and Bayes risk is of order $Θ(r \sqrt{T})$, by establishing a lower bound for an arbitrary policy, and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases. The phase-based policy is also shown to be effective if the set of arms satisfies a strong convexity condition. For the case of a general set of arms, we describe a near-optimal policy whose regret and Bayes risk admit upper bounds of the form $O(r \sqrt{T} \log^{3/2} T)$.
Multicasting correlated multi-source to multi-sink over a network
Venue: arXiv cs.IT 2010
Year: 2010

The problem of network coding with multicast of a single source to multisink has first been studied by Ahlswede, Cai, Li and Yeung in 2000, in which they have established the celebrated max-flow mini-cut theorem on non-physical information flow over a network of independent channels. On the other hand, in 1980, Han has studied the case with correlated multisource and a single sink from the viewpoint of polymatroidal functions in which a necessary and sufficient condition has been demonstrated for reliable transmission over the network. This paper presents an attempt to unify both cases, which leads to establish a necessary and sufficient condition for reliable transmission over a network multicasting correlated multisource to multisink. Here, the problem of separation of source coding and channel coding is also discussed.