Stopping time signatures for some algorithms in cryptography

We consider the normalized distribution of the overall running times of some cryptographic algorithms, and what information they reveal about the algorithms. Recent work of Deift, Menon, Olver, Pfrang, and Trogdon has shown that certain numerical algorithms applied to large random matrices exhibit a characteristic distribution of running times, which depends only on the algorithm but are independent of the choice of probability distributions for the matrices. Different algorithms often exhibit different running time distributions, and so the histograms for these running time distributions provide a time-signature for the algorithms, making it possible, in many cases, to distinguish one algorithm from another. In this paper we extend this analysis to cryptographic algorithms, and present examples of such algorithms with time-signatures that are indistinguishable, and others with time-signatures that are clearly distinct.Comment: 20 page

On Global Solutions of a Zakharov type System

We consider a class of wave-Schroedinger systems with a Zakharov-Schulman type coupling. This class of systems is indexed by a parameter gamma which measures the strength of the null form in the nonlinearity of the wave equation. The case gamma = 1 corresponds to the well-known Zakharov system, while the case gamma = -1 corresponds to the Yukawa system. Here we show that sufficiently smooth and localized Cauchy data lead to pointwise decaying global solutions which scatter, for any gamma in (0,1].Comment: 19 pages. Minor changes, including titl

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I

We study the determinant $\det(I-\gamma K_s), 0<\gamma <1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $\beta=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-\gamma)$ supported on an interval of length $\frac{2s}{\pi}$. We evaluate, in particular, the double scaling limit of $\det(I-\gamma K_s)$ as $s\rightarrow\infty$ and $\gamma\uparrow 1$, in the region $0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta$, for any fixed $0<\delta<1$. This problem was first considered by Dyson in \cite{Dy1}.Comment: 49 pages, 15 figures. Version 2 contains an extended introduction and corrects typo