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Nonlinear dynamics of wave packets in parity-time-symmetric optical lattices near the phase-transition point is analytically studied. A nonlinear Klein–Gordon equation is derived for the envelope of these wave packets. A variety of phenomena known to exist in this envelope equation are shown to also exist in the full equation, including wave blowup, periodic bound states, and solitary wave solutions.

The SFLASH signature scheme stood for a decade as the most successful cryptosystem based on multivariate polynomials, before an efficient attack was finally found in 2007. In this paper, we review its recent cryptanalysis and we notice that its weaknesses can all be linked to the fact that the cryptosystem is built on the structure of a large field. As the attack demonstrates, this richer structure can be accessed by an attacker by using the specific symmetry of the core function being used. Then, we investigate the effect of restricting this large field to a purely linear subset and we find that the symmetries exploited by the attack are no longer present. At a purely defensive level, this defines a countermeasure which can be used at a moderate overhead. On the theoretical side, this informs us of limitations of the recent attack and raises interesting remarks about the design itself of multivariate schemes.

In this paper, we present a new algorithm, such that, for the small integer solution (SIS) problem, if the solution is bounded ( by an integer \beta in l_\infty norm, which we call a bounded SIS (BSIS) problem, {\it and if the difference between the row dimension n and the column dimension m of the corresponding matrix is relatively small with respect the row dimension m}, we can solve it easily with a complexity of polynomial in m.

Given a loop on a surface, its homotopy class can be specified as a word consisting of letters representing the homotopy group generators. One of the interesting problems is how to compute the shortest word for a given loop. This is an NP-hard problem in general. However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently. In this paper, we propose an efficient algorithm to compute the shortest words for loops given on triangulated surface meshes. The design of this algorithm is inspired and guided by the work of Dehn and BirmanSeries. In support of the shortest word algorithm, we also propose efficient algorithms to compute shortest paths and shortest loops under hyperbolic metrics using a novel