One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.Comment: 33 page

Large well-relaxed models of vitreous silica, coordination numbers and entropy

A Monte Carlo method is presented for the simulation of vitreous silica. Well-relaxed networks of vitreous silica are generated containing up to 300,000 atoms. The resulting networks, quenched under the BKS potential, display smaller bond-angle variations and lower defect concentrations, as compared to networks generated with molecular dynamics. The total correlation functions T(r) of our networks are in excellent agreement with neutron scattering data, provided that thermal effects and the maximum inverse wavelength used in the experiment are included in the comparison. A procedure commonly used in experiments to obtain coordination numbers from scattering data is to fit peaks in rT(r) with a gaussian. We show that this procedure can easily produce incorrect results. Finally, we estimate the configurational entropy of vitreous silica.Comment: 7 pages, 4 figures (two column version to save paper

Error Graphs and the Reconstruction of Elements in Groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions. In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.Comment: Journal of Combinatorial Theory A 200

Computational Indistinguishability between Quantum States and Its Cryptographic Application

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCDff, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCDff has three properties of cryptographic interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail proofs and follow-up of recent wor

Towards device-size atomistic models of amorphous silicon

The atomic structure of amorphous materials is believed to be well described by the continuous random network model. We present an algorithm for the generation of large, high-quality continuous random networks. The algorithm is a variation of the "sillium" approach introduced by Wooten, Winer, and Weaire. By employing local relaxation techniques, local atomic rearrangements can be tried that scale almost independently of system size. This scaling property of the algorithm paves the way for the generation of realistic device-size atomic networks.Comment: 7 pages, 3 figure