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July 2015
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280 Reads
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4 Citations
Journal of Pure and Applied Algebra
It is shown that the Jacobian Conjecture (in all dimensions) is equivalent to the following statement: for almost all prime numbers p and each Keller map (i.e. ), the induced map is not the zero map.
September 2013
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16 Reads
Efficient secure password protocols are constructed that remain secure against offline dictionary attacks even when a large, but bounded, part of the storage of a server responsible for password verification is retrieved by an adversary through a remote or local connection. A registration algorithm and a verification algorithm accomplish the goal of defeating a dictionary attack. A password protocol where a server, on input of a login and a password, carefully selects several locations from the password files, properly combines their content according to some special function, and stores the result of this function as a tag that can be associated with this password and used in a verification phase to verify access by users. Two main instantiations of our method are given; in one, a combination of mathematical tools, called dispersers and pairwise-independent hash functions is used to achieve security against adaptive intrusions (dispersers make sure that the password of each user depends on randomly chosen locations in a large password file, and pairwise-independent hash functions help in making this dependency sufficiently random); in a second one, a combination of mathematical tools, called k-wise independent hash functions and locally-computable and strong extractors (k-wise independent hash functions make sure that the locations chosen in the large password file from each password are sufficiently random, and locally-computable and strong extractors are used to combine the contents of these locations to generate a single long random value, which makes verification harder for the adversary to foil).
February 2013
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21 Reads
A method and software product for limit privacy loss due to data shared in a social network, where the basic underlying assumptions are that users are interested in sharing data and cannot be assumed to constantly follow appropriate privacy policies. Social networks deploy an additional layer of server-assisted access control which, even under no action from a user, automatically evolves over time, by restricting access to the user's data. The evolving access control mechanism provides non-trivial quantifiable guarantees for formally specified requirements of utility (i.e., users share as much data as possible to all other users) and privacy (i.e., users expose combinations of sensitive data only with low probability and over a long time).
January 2013
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19 Reads
The claim in August 2010 by Dr. Vinay Deolalikar of HP Labs to have proved P not equal to NP, which would solve one of the million-dollar Clay Millennium Prize Problems, commanded the attention of much of the world mathematical community. The popular press also picked up the larger story of people coming together to review Deolalikar’s paper during twelve intense days, communicating via comments in several blogs, and primarily in “Gödel’s Lost Letter and P=NP.” Ken Regan tells the story of his and Dick Lipton’s role in hosting, summarizing, and rolling with these Internet events, and what lessons they hold for future mass scientific collaboration.
January 2013
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11 Reads
The main complexity lower and upper bound frontiers seem to be moved only once a decade lately, and Ryan Williams kicked off this decade by proving a new lower bound against Boolean circuits with modular counting. This chapter surveys his proof strategy and why he was able to break through a twenty-year barrier.
January 2013
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5 Reads
This chapter reflects on three big results from 2010, on unique games, Boolean circuits with modular counting, and lower bounds against the simplex method of linear programming. The last connects to foundational work by Klee with George Minty.
January 2013
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7 Reads
This chapter discusses the sensitive question of whether researchers should share their ideas, especially ideas that are not “complete.” The example of how Hamilton’s work on the Poincaré Conjecture was used by Grigori Perlman is raised before revisiting the Unique Games Conjecture.
January 2013
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6 Reads
A pseudorandom generator famously designed by Noam Nisan underlies many results and problems about low space-bounded complexity classes. Matei David and two co-authors recently proved a promising extension of the generator, enabling one to analyze multi-pass computations in these classes.
January 2013
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10 Reads
This chapter discusses aspects of Chattopadhyay’s results with Avi Wigderson on representing Boolean functions by polynomials in the integers modulo a composite number. When the modulus is prime many answers are known, but the composite case brings issues at the frontier of complexity lower bounds.
January 2013
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67 Reads
This chapter outlines Björklund’s paper at the FOCS 2010 conference, which solved a decades-old problem about Hamiltonian cycles in graphs. Sums of determinants of randomly extended matrices are key to breaking a power-of-2 barrier in algorithmic running time for detecting these cycles.
... Within a soccer team (HV Ribeiro et al 2010), baseball, American football league, and others, the members know that the opponents, taking advantage of their skills will do everything they can to get the triumph in a competition, hence it is transparent to the participants that everyone play against each other, in other words at the end of the race is recognized as winner to the team that generates the highest score taking the circumstances of each sport, with exceptions such as the contest of diving. The sports games have a number of complications in practice (Bregie et al 2011), starting with the selection of the best players (Lipton 2013) to participate in a tournament taking into consideration the target to win a contest. ...
January 2013
... Indeed, Strassen was trying to prove, by process of (intelligently exhaustive) elimination, that such an algorithm could not exist (e.g., [Lan08, Remark 1.1.1] or [LR13]). In his paper it is presented as follows, which "one easily sees" [Str69,p. ...
Reference:
Designing Strassen's algorithm
January 2013
... The min-gap pair needed to implement a min-gap test can be identified in optimal expected O(N ) time and space using Rabin's randomized closestpair algorithm [22,23]. Unlike the Gonzalez algorithm for max-gap, Rabin's algorithm generalizes efficiently to higher dimensions. ...
August 2010
... Hence the comparison ofX andŶ can be done in time O(N M ). It is known that if there existed faster methods for obtaining the automata intersection, then significant improvements would be implied to many long standing open problems [20]. Hence an immediate reduction to the problem of NFA intersection does not particularly help. ...
Reference:
Comparing Degenerate Strings
August 2010
... 3 As detailed in [31], the Positivity Problem (and assorted variants) has applications in a wide array of scientific areas, including theoretical biology, economics, software verification, probabilistic model checking, quantum computing, discrete linear dynamical systems, combinatorics, formal languages, statistical physics, generating functions, etc. Positivity also bears an important relationship to the well-known Skolem Problem: does a given LRS have a zero? The decidability of the Skolem Problem is generally considered to have been open since the 1930s (notwithstanding the fact that algorithmic decision issues had not at the time acquired the importance that they have today-see [20] for a discussion on this subject; see also [38, p. 258] and [23], in which this state of affairs-the enduring openness of decidability for the Skolem Problem-is described as "faintly outrageous" by Tao and a "mathematical embarrassment" by Lipton). A breakthrough occurred in the mid-1980s, when Mignotte et al. [27] and Vereshchagin [41] independently showed decidability for LRS of order 4 or less. ...
August 2010
... O bien, desde otra perspectiva, puede pensarse en las relaciones entre P y NP en términos de perteque los problemas P pertenecen a, o están incluidos en, NP?Es decir, ¿puede decirse que los problemas NP pertenecen a los problemas P? ¿O bien, que están incluidos los primeros en los segundos?Más radicalmente, ¿puede sostenerse razonablemente que los problemas fáciles -P-y los problemas difíciles -NP-son iguales, o quizás, en caso contrario, distintos(Lipton, 2010)?Revista LOGOS CIENCIA & TECNOLOGÍAISSN 2145-549X, Vol. 4. No. 2, Enero -Junio, 2013 ...
August 2010
... Hence, the Tate-Jacobian conjecture fails for important rings such as the p-adic rings. However, by refining the assumptions of the Tate-Jacobian conjecture and applying a result from A. van den Essen and R. J. Lipton [5], we can establish an equivalence for the Jacobian conjecture within the framework of Tate algebras over p-adic rings: ...
July 2015
Journal of Pure and Applied Algebra
... Alas, all FMM algorithms rely on computing the underlying SNF recursively using divide-and-conquer, which makes them impractical (asymptotic, IO-intensive [HYG18] and numerically unstable [BL79]), hence n ω -type algorithms are unlikely to be resolved on any imaginable hardware [LR13]. Our proposed operator avoids this shortcoming by applying the (SNF) corresponding to f using a single level of subdivision into tiles (more on this below). ...
January 2013
... Theorem 5.8 (Corollary of Theorem 3.1 from [LW12]). There exists a (polylog-time uniform) universal circuit family {C n } of size O (n · poly log(n)), such that for any bounded fan-in circuit ...
Reference:
Outcome Indistinguishability
June 2012
Proceedings of the Annual IEEE Conference on Computational Complexity
... In our main simulation (Theorem 1.1) we will utilize a construction to simplify the analysis of multitape Turing machines. The construction was originally used by Hopcroft,Paul,and Valiant [HPV75] in their O(t/ log t)-space simulation of t time, and it has been used in many subsequent works to "decompose" time-bounded computations (such as [PPST83,KLV03,KLRS12,LW13,MW17]). A time-t(n) block-respecting multitape Turing machine with blocks of length b(n) has the property that on all inputs of length n, the computation is partitioned into O(t(n)/b(n)) time blocks of length b(n), while each tape is partitioned into O(t(n)/b(n)) contiguous tape blocks of length b(n). ...
Reference:
Simulating Time With Square-Root Space
June 2012
computational complexity
