Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{\Omega(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(\Delta(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $\Delta(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi(H)$ does not exceed $\operatorname{tw}(H)$ and $\Delta(H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(\chi(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

Correlated-Source Extractors and Cryptography with Correlated-Random Tapes

In this paper, we consider the setting where a party uses correlated random tapes across multiple executions of a cryptographic algorithm. We ask if the security properties could still be preserved in such a setting. As examples, we introduce the notion of correlated-tape zero knowledge, and, correlated-tape multi-party computation, where, the zero-knowledge property, and, the ideal/real model security must still be preserved even if a party uses correlated random tapes in multiple executions. Our constructions are based on a new type of randomness extractor which we call correlated-source extractors. Correlated-source extractors can be seen as a dual of non-malleable extractors, and, allow an adversary to choose several tampering functions which are applied to the randomness source. Correlated-source extractors guarantee that even given the output of the extractor on the tampered sources, the output on the original source is still uniformly random. Given (seeded) correlated-source extractors, and, resettably-secure computation protocols, we show how to directly get a positive result for both correlated-tape zero-knowledge and correlated-tape multi-party computation in the CRS model. This is tight considering the known impossibility results on cryptography with imperfect randomness. Our main technical contribution is an explicit construction of a correlated-source extractor where the length of the seed is independent of the number of tamperings. Additionally, we also provide a (non-explicit) existential result for correlated source extractors with almost optimal parameters

Oksidacijski stres u lakirera izloženih niskim razinama olova

Lead toxicity is a public health problem particularly to the children and to occupationally exposed adults. Evidence is mounting successively regarding the adverse health effects of lead at low levels. This study was undertaken to assess the antioxidant status of lead-exposed residential and commercial painters of Lucknow city in Uttar Pradesh, India. Thirty-five painters aged 20 to 50 years who had blood lead levels ≤400 µg L-1 were selected for the study from a population of 56 male painters initially screened for blood lead. The control group included an equal number of subjects of the same age group without any occupational exposure to lead. We studied the association between low lead level exposure and antioxidant status and found that blood lead levels in painters were approximately seven times as high as in controls [(219.2 ± 61.9) µg L-1 vs. (30.6±10.1) µg L-1, respectively]. Among the biomarkers of lead toxicity a significant decrease in the level of delta-aminolevulinic acid dehydratase [(9.13±4.62) UL-1 vs. (39.38±5.05) UL-1] and an increase in the level of zinc protoporphyrin [(187.9±49.8) µg L-1 vs. (26.4±5.5) µg L-1] were observed in painters compared to controls. Among antioxidant enzymes, painters showed a significant decrease in catalase [(56.77±11.11) UL-1 vs. (230.30±42.55) UL-1] and superoxide dismutase [(0.64±0.19) UL-1 vs. (2.68±0.62) UL-1] compared to controls. Lipid peroxidation was monitored by measuring thiobarbituric acid reactive substances (TBARS) that were expressed in terms of malondialdehyde (MDA) equivalents. Concentration of MDA in plasma was higher in painters than in controls [(7.48±1.31) nmol mL-1 vs. (3.08±0.56) nmol mL-1]. Significant changes were also observed in reduced and oxidised glutathione levels. The strong association between blood lead levels and oxidative stress markers in this population suggests that oxidative stress should be considered in the pathogenesis of lead-related diseases among people with low level environmental exposure to lead.Toksičnost olova javnozdravstveni je problem, napose u djece i odraslih osoba koje su im izložene profesionalno. Sve je više dokaza o štetnom djelovanju olova pri niskim razinama. Svrha je ovog ispitivanja bila procijeniti antioksidacijski status u lakirera iz grada Lucknowa u indijskoj pokrajini Uttar Pradesh. Iz skupine od 56 muškaraca lakirera u dobi od 20 do 50 godina s pozitivnim početnim nalazima olova u krvi, za ispitivanje su izabrana 35-orica čije su razine iznosile ≤400 µg L-1. Izabran je i jednaki broj kontrolnih ispitanika iz iste dobne skupine, koji nisu bili profesionalno izloženi olovu. Ispitana je povezanost izme|u izloženosti niskim razinama olova i antioksidacijskoga stanja te je utvrđeno da su razine olova u krvi lakirera [(219,2±61,9) µg L-1] bile oko sedam puta više negoli u kontrolnih ispitanika [(30,6±10,1) µg L-1]. Od biopokazatelja toksičnosti olova u lakirera je zamijećen značajan pad razina delta- ALAD [(9,13±4,62) UL-1 prema (39,38±5,05) UL-1] te rast razina cinkova protoporfirina [(187,9±49,8) µg L-1 prema (26,4±5,5) µg L-1] u odnosu na kontrolne ispitanike. Od antioksidacijskih enzima u lakirera je značajno pala aktivnost katalaze [(56,77±11,11) UL-1 prema (230,30±42,55) UL-1] i superoksid dismutaze [(0,64±0.19) UL-1 prema (2,68±0,62) UL-1] u odnosu na kontrolu, dok je produkt lipidne peroksidacije u plazmi (izv. thiobarbituric acid reactive substances, TBARS) izražen kao koncentracija malondialdehida (MDA) porastao [(7,48±1,31) nmol mL-1 prema (3,08±0,56) nmol mL-1]. Značajne su promjene također zamijećene u smanjenim razinama glutationa i njihovoj oksidaciji. Snažna povezanost razina olova u krvi s pokazateljima oksidacijskoga stresa upućuje na to da u osoba s niskom razinom izloženosti olovu iz okoliša kod razmatranja patogeneze bolesti povezane s olovom u obzir valja uzeti oksidacijski stres

On Quadratic Threshold CSPs

Abstract. ApredicateP: {−1, 1} k →{0, 1} can be associated with a constraint satisfaction problem Max CSP(P). P is called “approximation resistant ” if Max CSP(P) cannot be approximated better than the approximation obtained by choosing a random assignment, and “approximable” otherwise. This classification of predicates has proved to be an important and challenging open problem. Motivated by a recent result of Austrin and Mossel (Computational Complexity, 2009), we consider a natural subclass of predicates defined by signs of quadratic polynomials, including the special case of predicates defined by signs of linear forms, and supply algorithms to approximate them as follows. In the quadratic case we prove that every symmetric predicate is approximable. We introduce a new rounding algorithm for the standard semidefinite programming relaxation of Max CSP(P) for any predicate P: {−1, 1} k →{0, 1} and analyze its approximation ratio. Our rounding scheme operates by first manipulating the optimal SDP solution so that all the vectors are nearly perpendicular and then applying a form of hyperplane rounding to obtain an integral solution. The advantage of this method is that we are able to analyze the behaviour of a set of k rounded variables together as opposed to just a pair of rounded variables in most previous methods. In the linear case we prove that a predicate called “Monarchy ” is approximable. This predicate is not amenable to our algorithm for the quadratic case, nor to other LP/SDP-based approaches we are aware of.

Conference — TCC 2014. This is the full version. Continuous Non-malleable Codes

Non-malleable codes are a natural relaxation of error correcting/detecting codes that have useful applications in the context of tamper resilient cryptography. Informally, a code is non-malleable if an adversary trying to tamper with an encoding of a given message can only leave it unchanged or modify it to the encoding of a completely unrelated value. This paper introduces an extension of the standard non-malleability security notion – so-called continuous non-malleability – where we allow the adversary to tamper continuously with an encoding. This is in contrast to the standard notion of non-malleable codes where the adversary only is allowed to tamper a single time with an encoding. We show how to construct continuous non-malleable codes in the common split-state model where an encoding consist of two parts and the tampering can be arbitrary but has to be independent with both parts. Our main contributions are outlined below: 1. We propose a new uniqueness requirement of split-state codes which states that it is computationally hard to find two codewords C = (X0, X1) and C ′ = (X0, X ′ 1) such that both codwords are valid, but X0 is the same in both C and C ′. A simple attack shows that uniqueness is necessary to achieve continuous non-malleability in the split-state model. Moreover