Cybersecurity and Cyberforensics Conference最新文献

{"title":"Hardness vs Randomness for Bounded Depth Arithmetic Circuits","authors":"Chi-Ning Chou, Mrinal Kumar, Noam Solomon","doi":"10.4230/LIPIcs.CCC.2018.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.13","url":null,"abstract":"In this paper, we study the question of hardness-randomness tradeoffs for bounded depth arithmetic circuits. We show that if there is a family of explicit polynomials {fn}, where fn is of degree O(log2 n/log2 logn) in n variables such that fn cannot be computed by a depth Δ arithmetic circuits of size poly(n), then there is a deterministic sub-exponential time algorithm for polynomial identity testing of arithmetic circuits of depth Δ − 5. This is incomparable to a beautiful result of Dvir et al.[SICOMP, 2009], where they showed that super-polynomial lower bounds for depth Δ circuits for any explicit family of polynomials (of potentially high degree) implies sub-exponential time deterministic PIT for depth Δ − 5 circuits of bounded individual degree. Thus, we remove the \"bounded individual degree\" condition in the work of Dvir et al. at the cost of strengthening the hardness assumption to hold for polynomials of low degree. The key technical ingredient of our proof is the following property of roots of polynomials computable by a bounded depth arithmetic circuit : if f(x1, x2, ..., xn) and P(x1, x2, ..., xn, y) are polynomials of degree d and r respectively, such that P can be computed by a circuit of size s and depth Δ and P(x1, x2, ... , xn, f) ≡ 0, then, f can be computed by a circuit of size poly [EQUATION] and depth Δ + 3. In comparison, Dvir et al. showed that f can be computed by a circuit of depth Δ + 3 and size poly(n, s, r, dt), where t is the degree of P in y. Thus, the size upper bound in the work of Dvir et al. is non-trivial when t is small but d could be large, whereas our size upper bound is non-trivial when d is small, but t could be large.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130667198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A New Approach for Constructing Low-Error, Two-Source Extractors","authors":"Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, A. Ta-Shma","doi":"10.4230/LIPIcs.CCC.2018.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.3","url":null,"abstract":"Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework (STOC 2016), and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient there and in follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck. The parameters we require from t-non-malleable extractors for our reduction to work hold in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123005491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linear Sketching over F_2","authors":"Sampath Kannan, Elchanan Mossel, Swagato Sanyal, G. Yaroslavtsev","doi":"10.4230/LIPIcs.CCC.2018.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.8","url":null,"abstract":"We initiate a systematic study of linear sketching over F2. For a given Boolean function treated as f: Fn2 → F2 a randomized F2-sketch is a distribution M over d × n matrices with elements over F2 such that Mx suffices for computing f(x) with high probability. Such sketches for d L n can be used to design small-space distributed and streaming algorithms. Motivated by these applications we study a connection between F2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F2-sketching is optimal (up to constant factors) for uniformly distributed inputs. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F2 can be constructed as F2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133680507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Tight Lower Bound for Entropy Flattening","authors":"Yi-Hsiu Chen, Mika Göös, S. Vadhan, Jiapeng Zhang","doi":"10.4230/LIPIcs.CCC.2018.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.23","url":null,"abstract":"We study entropy flattening: Given a circuit CX implicitly describing an n-bit source X (namely, X is the output of CX on a uniform random input), construct another circuit CY describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have CY evaluate CX altogether Θ(n2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit CY for entropy flattening that repeatedly queries CX as an oracle requires ω(n2) queries. Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [12, 22, 13, 6, 11, 10, 7, 24]. It is also used in reductions between problems complete for statistical zero-knowledge [19, 23, 4, 25]. The Θ(n2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114128003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Worst-Case to Average Case Reductions for the Distance to a Code","authors":"Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf","doi":"10.4230/LIPIcs.CCC.2018.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.24","url":null,"abstract":"Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions [EQUATION], given as oracles, from a linear error correcting code V. The soundness of such systems relies on methods that act \"locally\" on u and map it to a single function u* that is, roughly, as far from V as are u1, ..., uk. Motivated by these applications to efficient proof systems, we study a natural worst-case to average-case reduction of distance for linear spaces, and show several general cases in which the following statement holds: If some member of a linear space U = span(u1, ..., uk) is Δ-far from (all elements) of V in relative Hamming distance, then nearly all elements of U are (1 − ϵ)Δ-far from V; the value of ϵ depends only on the distance of the code V and approaches 0 as that distance approaches 1. Our results improve on the previous state-of-the-art which showed that nearly all elements of U are 1/2Δ-far from V [Rothblum, Vadhan and Wigderson, STOC 2013]. When V is a Reed-Solomon (RS) code, as is often the case for algebraic proof systems, we show how to boost distance via a new \"local\" transformation that may be useful elsewhere. Relying on the affine-invariance of V, we map a vector u to a random linear combination of affine transformations of u, and show this process amplifies distance from V. Assuming V is an RS code with sufficiently large distance, this amplification process converts a function u that is somewhat far from V to one that is (1 − ϵ)-far from V; as above, ϵ depends only on the distance of V and approaches 0 as the distance of V approaches 1. We give two concrete application of these techniques. First, we revisit the axis-parallel low-degree test for bivariate polynomials of [Polischuk-Spielman, STOC 1994] and prove a \"list-decoding\" type result for it, when the degree of one axis is extremely small. This result is similar to the recent list-decoding-regime result of [Chiesa, Manohar and Shinkar, RANDOM 2017] but is proved using different techniques, and allows the degree in one axis to be arbitrarily large. Second, we improve the soundness analysis of the recent RS proximity testing protocol of [Ben-Sasson et al., ICALP 2018] and extend it to the \"list-decoding\" regime, bringing it closer to the Johnson bound.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"316 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133077839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hardness Amplification for Non-Commutative Arithmetic Circuits","authors":"M. Carmosino, R. Impagliazzo, Shachar Lovett, Ivan Mihajlin","doi":"10.4230/LIPIcs.CCC.2018.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.12","url":null,"abstract":"We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire. This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127191888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A PRG for Boolean PTF of Degree 2 with Seed Length Subpolynomial in epsilon and Logarithmic in n","authors":"D. Kane, Sankeerth Rao","doi":"10.4230/LIPIcs.CCC.2018.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.2","url":null,"abstract":"We construct and analyze a pseudorandom generator for degree 2 boolean polynomial threshold functions. Random constructions achieve the optimal seed length of [EQUATION], however the best known explicit construction of [8] uses a seed length of O(log n · ϵ−8). In this work we give an explicit construction that uses a seed length of [EQUATION]. Note that this improves the seed length substantially and that the dependence on the error ϵ is additive and only grows subpolynomially as opposed to the previously known multiplicative polynomial dependence. Our generator uses dimensionality reduction on a Nisan-Wigderson based pseudorandom generator given by Lu, Kabanets [18].","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"110 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134026478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lower Bounds on Non-Adaptive Data Structures Maintaining Sets of Numbers, from Sunflowers","authors":"Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao","doi":"10.4230/LIPIcs.CCC.2018.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.27","url":null,"abstract":"We prove new cell-probe lower bounds for dynamic data structures that maintain a subset of {1, 2, ..., n}, and compute various statistics of the set. The data structure is said to handle insertions non-adaptively if the locations of memory accessed depend only on the element being inserted, and not on the contents of the memory. For any such data structure that can compute the median of the set, we prove that: [EQUATION] where tins is the number of memory locations accessed during insertions, tmed is the number of memory locations accessed to compute the median, and w is the number of bits stored in each memory location. When the data structure is able to perform deletions non-adaptively and compute the minimum non-adaptively, we prove [EQUATION] where tmin is the number of locations accessed to compute the minimum, and tdel is the number of locations accessed to perform deletions. For the predecessor search problem, where the data structure is required to compute the predecessor of any element in the set, we prove that if computing the predecessors can be done non-adaptively, then [EQUATION] where tpred is the number of locations accessed to compute predecessors. These bounds are nearly matched by Binary Search Trees in some range of parameters. Our results follow from using the Sunflower Lemma of Erdős and Rado [11] together with several kinds of encoding arguments.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122351582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth","authors":"A. Lingas","doi":"10.4230/LIPIcs.CCC.2018.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.26","url":null,"abstract":"We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most ϵlogn conjunction-depth computing the n-dimensional Boolean vector convolution has ω(n2−4ϵ) and-gates. Analogously, any normalized Boolean circuit of at most ϵlogn conjunction-depth computing the n × n Boolean matrix product has ω(n3−4ϵ) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130338994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials","authors":"Richard Ryan Williams","doi":"10.4230/LIPIcs.CCC.2018.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2018.6","url":null,"abstract":"We consider the problem of representing Boolean functions exactly by \"sparse\" linear combinations (over $mathbb{R}$) of functions from some \"simple\" class ${cal C}$. In particular, given ${cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${cal C}$ is \"overcomplete\" and the set of functions is not linearly independent. We focus on the cases where ${cal C}$ is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. \u0000We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for \"semi-explicit\" Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: \u0000$bullet$ Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. \u0000$bullet$ Depth-two neural networks with ReLU activation function. \u0000$bullet$ $mathbb{R}$-linear combinations of $O(1)$-degree $mathbb{F}_p$-polynomials, for every prime $p$ (related to problems regarding Higher-Order \"Uncertainty Principles\"). We also obtain a function in $E^{NP}$ requiring $2^{Omega(n)}$ linear combinations. \u0000$bullet$ $mathbb{R}$-linear combinations of $ACC circ THR$ circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). \u0000(The above is a shortened abstract. For the full abstract, see the paper.)","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121745172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}