Shirshendu Ganguly

A complete characterization of the evolution of RC4 pseudo random generation algorithm

Abstract In this paper, we provide a complete characterization of the RC4 Pseudo Random Generation Algorithm (PRGA) for one step: i = i + 1; j = j + S[i]; swap(S[i], S[j]); z = S[S[i] + S[j]]. This is the first time such an involved description is presented to get a concise view of how RC4 PRGA evolves. Considering all the permutations (we also keep in mind the Finney states), we find that the distribution of z is not uniform given i, j. A corollary of this result shows that information about j is always leaked from z. Next, studying two consecutive steps of RC4 PRGA, we prove that the index j is not produced uniformly at random given the value of j two steps ago. We also provide additional evidence of z leaking information on j. Further, we present a novel distinguisher for RC4 which shows that under certain conditions the equality of two consecutive bytes is more probable than by random association. Our analysis holds regardless of the amount of initial keystream bytes thrown away during the RC4 PRGA.

Upper tails and independence polynomials in random graphs

Abstract The upper tail problem in the Erdős–Renyi random graph G ∼ G n , p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1 + δ . Chatterjee and Dembo showed that in the sparse regime of p → 0 as n → ∞ with p ≥ n − α for an explicit α = α H > 0 , this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function c H ( δ ) such that, for p as above and any fixed δ > 0 , the upper tail probability is exp ⁡ [ − ( c H ( δ ) + o ( 1 ) ) n 2 p Δ log ⁡ ( 1 / p ) ] , where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, c H ( δ ) , is governed by the independence polynomial of H, defined as P H ( x ) = ∑ i H ( k ) x k where i H ( k ) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then c H ( δ ) is the minimum between 1 2 δ 2 / m and the unique positive solution of P H ( x ) = 1 + δ .

Escape rates for rotor walks in Z d

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z d for the initial conguration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order n in dimensions d 3, and of order n= logn in dimension 2.

SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

Competitive Erosion is Conformally Invariant

We study a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. We consider competitive erosion on discretizations of ‘smooth’, planar, simply connected, domains. The main result of this article shows that at stationarity, with high probability, the blue and the red regions are separated by the level curves of the Green function, with Neumann boundary conditions, which are orthogonal circular arcs on the disc and hyperbolic geodesics on a general, simply connected domain. This establishes conformal invariance of the model.

Convergence of Discrete Green Functions with Neumann Boundary Conditions

In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion.

Sequence assembly from corrupted shotgun reads

The prevalent technique for DNA sequencing consists of two main steps: shotgun sequencing, where many randomly located fragments, called reads, are extracted from the overall sequence, followed by an assembly algorithm that aims to reconstruct the original sequence. There are many different technologies that generate the reads: widely-used second-generation methods create short reads with low error rates, while emerging third-generation methods create long reads with high error rates. Both error rates and error profiles differ among methods, so reconstruction algorithms are often tailored to specific shotgun sequencing technologies. As these methods change over time, a fundamental question is whether there exist reconstruction algorithms which are robust, i.e., which perform well under a wide range of error distributions. Here we study this question of sequence assembly from corrupted reads. We make no assumption on the types of errors in the reads, but only assume a bound on their magnitude. More precisely, for each read we assume that instead of receiving the true read with no errors, we receive a corrupted read which has edit distance at most ε times the length of the read from the true read. We show that if the reads are long enough and there are sufficiently many of them, then approximate reconstruction is possible: we construct a simple algorithm such that for almost all original sequences the output of the algorithm is a sequence whose edit distance from the original one is at most O(ε) times the length of the original sequence.

Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

Let

Non-fixation for Conservative Stochastic Dynamics on the Line

X_k

Consistent nonparametric estimation for heavy-tailed sparse graphs

denote the number of