Abelian Groups, Gauss Periods, and Normal Bases

AbstractA result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity

Monomial Dynamical Systems in # P-complete

In this paper, we study boolean monomial dynamical systems. Colón-Reyes, Jarrah, Laubenbacher, and Sturmfels(2006) studied fixed point structure of boolean monomial dynamical systems of f by associating the dynamical systems of f with its dependency graph χf and Jarrah, Laubenbacher, and Veliz-Cuba(2010) extended it and presented lower and upper bound for the number of cycles of a given length for general boolean monomial dynamics. But, it is even difficult to determine the exact number of fixed points of boolean monomial dynamics. We show that the problem of counting fixed points of a boolean monomial dynamical systems is #P-complete, for which no efficient algorithm is known. This is proved by a 1-1 correspondence between fixed points of f sand antichains of the poset of strongly connected components of χf.

Johnson-Lindenstrauss projection of high dimensional data

Johnson and Lindenstrauss (1984) proved that any finite set of data in a high dimensional space can be projected into a low dimensional space with the Euclidean metric information of the set being preserved within any desired accuracy. Such dimension reduction plays a critical role in many applications with massive data. There have been extensive effort in the literature on how to find explicit constructions of Johnson-Lindenstrauss projections. In this poster, we show how algebraic codes over finite fields can be used for fast Johnson-Lindenstrauss projections of data in high dimensional Euclidean spaces. This is joint work with Shuhong Gao and Yue Mao

Efficient Fully Homomorphic Encryption Scheme

Since Gentry discovered in 2009 the first fully homomorphic encryption scheme, the last few years have witnessed dramatic progress on designing more efficient homomorphic encryption schemes, and some of them have been implemented for applications. The main bottlenecks are in bootstrapping and large cipher expansion (the ratio of the size of ciphertexts to that of messages). Ducas and Micciancio (2015) show that homomorphic computation of one bit operation on LWE ciphers can be done in less than a second, which is then reduced by Chillotti et al. (2016, 2017) to 13ms. This paper presents a compact fully homomorphic encryption scheme that has the following features: (a) its cipher expansion is 6 with private-key encryption and 20 with public-key encryption; (b) all ciphertexts after any number (unbounded) of homomorphic bit operations have the same size and are always valid with the same error size; (c) its security is based on the LWE and RLWE problems (with binary secret keys) and the cost of breaking the scheme by the current approaches is at least $2^{160}$ bit operations. The scheme protects function privacy and provides a simple solution for secure two-party computation and zero knowledge proof of any language in NP

From Hall's Matching Theorem to Optimal Routing on Hypercubes

AbstractWe introduce a concept of so-called disjoint ordering for any collection of finite sets. It can be viewed as a generalization of a system of distinctive representatives for the sets. It is shown that disjoint ordering is useful for network routing. More precisely, we show that Hall's “marriage” condition for a collection of finite sets guarantees the existence of a disjoint ordering for the sets. We next use this result to solve a problem in optimal routing on hypercubes. We give a necessary and sufficient condition under which there are internally node-disjoint paths each shortest from a source node to any others(s⩽n) target nodes on ann-dimensional hypercube. When this condition is not necessarily met, we show that there are always internally node-disjoint paths each being either shortest or near shortest, and the total length is minimum. An efficient algorithm is also given for constructing disjoint orderings and thus disjoint short paths. As a consequence, Rabin's information disposal algorithm may be improved

The Complexity of Subdivision for Diameter-Distance Tests

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight