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Person# Mahdi Cheraghchi Bashi Astaneh

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Measurement

Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or

Number

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words.

Construction

Construction is a general term meaning the art and science to form objects, systems, or organizations, and comes from Latin constructio (from com- "together" and struere "to pile up") and Old French

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Mahdi Cheraghchi Bashi Astaneh, Mohammad Amin Shokrollahi

A wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message, while knowledge of the entire encoding reveals the message using the decoder. In this paper, the notion of efficiently invertible extractors is studied and it is shown that a wiretap protocol can be constructed from such an extractor. Then, invertible extractors for symbol-fixing, affine, and general sources are constructed and used to create wiretap protocols with asymptotically optimal trade-offs between their rate (ratio of the length of the message versus its encoding) and resilience (ratio of the observed positions of the encoding and the length of the encoding). The results are further applied to create wiretap protocols for challenging communication problems, such as active intruders who change portions of the encoding, network coding, and intruders observing arbitrary Boolean functions of the encoding.

2012Mahdi Cheraghchi Bashi Astaneh, Amin Karbasi, Soheil Mohajerzefreh

Non-adaptive group testing involves grouping arbitrary subsets of $n$ items into different pools. Each pool is then tested and defective items are identified. A fundamental question involves minimizing the number of pools required to identify at most $d$ defective items. Motivated by applications in network tomography, sensor networks and infection propagation, a variation of group testing problems on graphs is formulated. Unlike conventional group testing problems, each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper, a test is associated with a random walk. In this context, conventional group testing corresponds to the special case of a complete graph on $n$ vertices. For interesting classes of graphs a rather surprising result is obtained, namely, that the number of tests required to identify $d$ defective items is substantially similar to what required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if $T(n)$ corresponds to the mixing time of the graph $G$, it is shown that with $m=O(d^2T^2(n)\log(n/d))$ non-adaptive tests, one can identify the defective items. Consequently, for the Erd\H{o}s-R'enyi random graph $G(n,p)$, as well as expander graphs with constant spectral gap, it follows that $m=O(d^2\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Next, a specific scenario is considered that arises in network tomography, for which it is shown that $m=O(d^3\log^3n)$ non-adaptive tests are sufficient to identify $d$ defective items. Noisy counterparts of the graph constrained group testing problem are considered, for which parallel results are developed.

2012