Cybersecurity and Cyberforensics Conference最新文献

{"title":"Limits of Minimum Circuit Size Problem as Oracle","authors":"Shuichi Hirahara, O. Watanabe","doi":"10.4230/LIPIcs.CCC.2016.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.18","url":null,"abstract":"The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-Turing reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP ≠ EXP, which is a major open problem in computational complexity. \u0000 \u0000In this paper, we provide strong evidence that current techniques cannot establish NP-hardness of MCSP, even under polynomial-time Turing reductions or randomized reductions: Specifically, we introduce the notion of oracle-independent reduction to MCSP, which captures all the currently known reductions. We say that a reduction to MCSP is oracle-independent if the reduction can be generalized to a reduction to MCSPA for any oracle A, where MCSPA denotes an oracle version of MCSP. We prove that no language outside P is reducible to MCSP via an oracle-independent polynomial-time Turing reduction. We also show that the class of languages reducible to MCSP via an oracle-independent randomized reduction that makes at most one query is contained in AM ∩ coAM. Thus, NP-hardness of MCSP cannot be established via such oracle-independent reductions unless the polynomial hierarchy collapses. \u0000 \u0000We also extend the previous results to the case of more general reductions: We prove that establishing NP-hardness of MCSP via a polynomial-time nonadaptive reduction implies ZPP ≠ EXP, and that establishing NP-hardness of approximating circuit complexity via a polynomial-time Turing reduction also implies ZPP ≠ EXP. Along the way, we prove that approximating Levin's Kolmogorov complexity is provably not EXP-hard under polynomial-time Turing reductions, which is of independent interest.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134026203","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":"New Non-Uniform Lower Bounds for Uniform Classes","authors":"L. Fortnow, R. Santhanam","doi":"10.4230/LIPIcs.CCC.2016.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.19","url":null,"abstract":"We strengthen the nondeterministic hierarchy theorem for non-deterministic polynomial time to show that the lower bound holds against sub-linear advice. More formally, we show that for any constants d and d' such that 1 ≤ d < d', and for any time-constructible bound t = o(nd), there is a language in NTIME(nd) which is not in NTIME(t)/n1/d'. The best known earlier separation of Fortnow, Santhanam and Trevisan could only handle o(log(n)) bits of advice in the lower bound, and was not tight with respect to the time bounds. \u0000 \u0000We generalize our hierarchy theorem to work for other syntactic complexity measures between polynomial time and polynomial space, including alternating polynomial time with any fixed number of alternations. We also use our technique to derive an almost-everywhere hierarchy theorem for non-deterministic classes which use a sub-linear amount of non-determinism, i.e., the lower bound holds on all but finitely many input lengths rather than just on infinitely many. \u0000 \u0000As one application of our main result, we derive a new lower bound for NP against NP-uniform non-deterministic circuits of size O(nk) for any fixed k. This result is a significant strengthening of a result of Kannan, which states that not all of NP can be solved with P-uniform circuits of size O(nk) for any fixed k. As another application, we show strong non-uniform lower bounds for the complexity class RE of languages decidable in randomized linear exponential time with one sided error.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116710340","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":"Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits","authors":"Ruiwen Chen, R. Santhanam, S. Srinivasan","doi":"10.4086/toc.2018.v014a009","DOIUrl":"https://doi.org/10.4086/toc.2018.v014a009","url":null,"abstract":"We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is ed > 0 such that Parity has correlation at most 1/nΩ(1) with depth-d threshold circuits which have at most n1+ed wires, and the Generalized Andreev Function has correlation at most 1/2nΩ(1) with depth-d threshold circuits which have at most n1+ed wires. Previously, only worst-case lower bounds in this setting were known [22]. \u0000 \u0000We use our ideas to make progress on several related questions. We give satisfiability algorithms beating brute force search for depth-d threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2. We also show that Parity cannot be computed by polynomial-size AC0 circuits with no(1) general threshold gates. Previously no lower bound for Parity in this setting could handle more than log(n) gates. This result also implies subexponential-time learning algorithms for AC0 with no(1) threshold gates under the uniform distribution. In addition, we give almost optimal bounds for the number of gates in a depth-d threshold circuit computing Parity on average, and show average-case lower bounds for threshold formulas of any depth. \u0000 \u0000Our techniques include adaptive random restrictions, anti-concentration and the structural theory of linear threshold functions, and bounded-read Chernoff bounds.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116203679","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":"Learning Algorithms from Natural Proofs","authors":"M. Carmosino, R. Impagliazzo, Valentine Kabanets, A. Kolokolova","doi":"10.4230/LIPIcs.CCC.2016.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.10","url":null,"abstract":"Based on Hastad's (1986) circuit lower bounds, Linial, Mansour, and Nisan (1993) gave a quasipolytime learning algorithm for AC0 (constant-depth circuits with AND, OR, and NOT gates), in the PAC model over the uniform distribution. It was an open question to get a learning algorithm (of any kind) for the class of AC0[p] circuits (constant-depth, with AND, OR, NOT, and MODp gates for a prime p). Our main result is a quasipolytime learning algorithm for AC0[p] in the PAC model over the uniform distribution with membership queries. This algorithm is an application of a general connection we show to hold between natural proofs (in the sense of Razborov and Rudich (1997)) and learning algorithms. We argue that a natural proof of a circuit lower bound against any (sufficiently powerful) circuit class yields a learning algorithm for the same circuit class. As the lower bounds against AC0[p] by Razborov (1987) and Smolensky (1987) are natural, we obtain our learning algorithm for AC0[p].","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125727654","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":"New Characterizations in Turnstile Streams with Applications","authors":"Yuqing Ai, Wei Hu, Yi Li, David P. Woodruff","doi":"10.4230/LIPIcs.CCC.2016.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.20","url":null,"abstract":"Recently, [Li, Nguyen, Woodruff, STOC'2014] showed any 1-pass constant probability streaming algorithm for computing a relation f on a vector x ∈ {−m, − (m − 1), ..., m}n presented in the turnstile data stream model can be implemented by maintaining a linear sketch A · × mod q, where A is an r × n integer matrix and q = (q1, ..., qr) is a vector of positive integers. The space complexity of maintaining A · × mod q, not including the random bits used for sampling A and q, matches the space of the optimal algorithm. \u0000 \u0000We give multiple strengthenings of this reduction, together with new applications. In particular, we show how to remove the following shortcomings of their reduction: \u0000 \u00001. The Box Constraint. Their reduction applies only to algorithms that must be correct even if ∥;x∥;∞ = maxi∈[n] |xi| is allowed to be much larger than m at intermediate points in the stream, provided that x ∈ {−m, −(m − 1), ..., m}n at the end of the stream. We give a condition under which the optimal algorithm is a linear sketch even if it works only when promised that x ∈ {−m, −(m − 1), ..., m}n at all points in the stream. Using this, we show the first super-constant Ω(log m) bits lower bound for the problem of maintaining a counter up to an additive em error in a turnstile stream, where e is any constant in (0, ½). Previous lower bounds are based on communication complexity and are only for relative error approximation; interestingly, we do not know how to prove our result using communication complexity. More generally, we show the first super-constant Ω(log m) lower bound for additive approximation of ep-norms; this bound is tight for 1 ≤ p ≤ 2. \u0000 \u00002. Negative Coordinates. Their reduction allows xi to be negative while processing the stream. We show an equivalence between 1-pass algorithms and linear sketches A · x mod q in dynamic graph streams, or more generally, the strict turnstile model, in which for all i ∈ [n], xi ≥ 0 at all points in the stream. Combined with [Assadi, Khanna, Li, Yaroslavtsev, SODA'2016], this resolves the 1-pass space complexity of approximating the maximum matching in a dynamic graph stream, answering a question in that work. \u0000 \u00003. 1-Pass Restriction. Their reduction only applies to 1-pass data stream algorithms in the turnstile model, while there exist algorithms for heavy hitters and for low rank approximation which provably do better with multiple passes. We extend the reduction to algorithms which make any number of passes, showing the optimal algorithm is to choose a new linear sketch at the beginning of each pass, based on the output of previous passes.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"126 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128064538","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":"Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation","authors":"Richard Ryan Williams","doi":"10.4230/LIPIcs.CCC.2016.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.2","url":null,"abstract":"We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,ldots,x_n)$ of size $s$ and degree $d$ over a field ${mathbb F}$, and any inputs $a_1,ldots,a_K in {mathbb F}^n$, \u0000$bullet$ the Prover sends the Verifier the values $C(a_1), ldots, C(a_K) in {mathbb F}$ and a proof of $tilde{O}(K cdot d)$ length, and \u0000$bullet$ the Verifier tosses $textrm{poly}(log(dK|{mathbb F}|/varepsilon))$ coins and can check the proof in about $tilde{O}(K cdot(n + d) + s)$ time, with probability of error less than $varepsilon$. \u0000For small degree $d$, this \"Merlin-Arthur\" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in $c^{n}$ time (for various $c < 2$) for the Permanent, $#$Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of $0$-$1$ linear programs. In general, the value of any polynomial in Valiant's class ${sf VP}$ can be certified faster than \"exhaustive summation\" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. \u0000We also give a three-round (AMA) proof system for quantified Boolean formulas running in $2^{2n/3+o(n)}$ time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in $n^{k/2+O(1)}$-time for counting $k$-cliques in graphs. \u0000We point to some potential future directions for refuting the Nondeterministic Strong ETH.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133663457","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":"On the sum-of-squares degree of symmetric quadratic functions","authors":"Troy Lee, A. Prakash, R. D. Wolf, H. Yuen","doi":"10.4230/LIPIcs.CCC.2016.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.17","url":null,"abstract":"We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $ell_{infty}$-norm as well as in $ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"38 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120923344","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":"Sculpting Quantum Speedups","authors":"S. Aaronson, S. Ben-David","doi":"10.4230/LIPIcs.CCC.2016.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.26","url":null,"abstract":"Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the \"sculpting problem.\" In this work, we give a full characterization of sculptable functions in the query complexity setting. We show that a total function f can be restricted to a promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only if f has a large number of inputs with large certificate complexity. The proof uses some interesting techniques: for one direction, we introduce new relationships between randomized and quantum query complexity in various settings, and for the other direction, we use a recent result from communication complexity due to Klartag and Regev. We also characterize sculpting for other query complexity measures, such as R(f) vs. R_0(f) and R_0(f) vs. D(f). \u0000Along the way, we prove some new relationships for quantum query complexity: for example, a nearly quadratic relationship between Q(f) and D(f) whenever the promise of f is small. This contrasts with the recent super-quadratic query complexity separations, showing that the maximum gap between classical and quantum query complexities is indeed quadratic in various settings - just not for total functions! \u0000Lastly, we investigate sculpting in the Turing machine model. We show that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in PromiseBPP. Under a weaker assumption, that some problem in BQP is hard on average for P/poly, we show that every paddable language outside BPP is sculptable in this way.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125187709","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":"Polynomial bounds for decoupling, with applications","authors":"R. O'Donnell, Yu Zhao","doi":"10.4230/LIPIcs.CCC.2016.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.24","url":null,"abstract":"Let f(x) = f(x_1, ..., x_n) = sum_{|S| <= k} a_S prod_{i in S} x_i be an n-variate real multilinear polynomial of degree at most k, where S subseteq [n] = {1, 2, ..., n}. For its \"one-block decoupled\" version, \u0000f~(y,z) = sum_{|S| <= k} a_S sum_{i in S} y_i prod_{j in Si} z_j, \u0000we show tail-bound comparisons of the form \u0000Pr[|f~(y,z)| > C_k t] t]. \u0000Our constants C_k, D_k are significantly better than those known for \"full decoupling\". For example, when x, y, z are independent Gaussians we obtain C_k = D_k = O(k); when x, y, z, Rademacher random variables we obtain C_k = O(k^2), D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is known in these settings. \u0000We describe consequences of these results for query complexity (related to conjectures of Aaronson and Ambainis) and for analysis of Boolean functions (including an optimal sharpening of the DFKO Inequality).","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129615963","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":"Reconstruction of Real Depth-3 Circuits with Top Fan-In 2","authors":"Gaurav Sinha","doi":"10.4230/LIPIcs.CCC.2016.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.31","url":null,"abstract":"We present a polynomial time randomized algorithm for reconstructing $SigmaPiSigma(2)$ circuits over $mathbb{R}$, i.e. depth 3 circuits with fan in 2 at the top addition gate and having real coefficients. The algorithm needs only a blackbox query access to the polynomial $fin mathbb{R}[x_1,ldots,x_n]$ of degree d in n variables, computable by a $SigmaPiSigma(2)$ circuit C. In addition, we assume that the simple rank of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time $poly(n,d)$ and returns an equivalent $SigmaPiSigma(2)$ circuit(with high probability). Our main techniques are based on the use of Quantitative Syslvester Gallai Theorems from the work of Barak et.al.([3]) to find a small collection of nice subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the nice subspaces can be glued. We also use Brills Equations([8]) to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [14].","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122480103","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}