Pawel Lorek

Strong stationary duality for Möbius monotone Markov chains

For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of Möbius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.

A Robust Algorithm for Segmenting and Tracking Clustered Cells in Time-Lapse Fluorescent Microscopy

We present herein a robust algorithm for cell tracking in a sequence of time-lapse 2-D fluorescent microscopy images. Tracking is performed automatically via a multiphase active contours algorithm adapted to the segmentation of clustered nuclei with obscure boundaries. An ellipse fitting method is applied to avoid problems typically associated with clustered, overlapping, or dying cells, and to obtain more accurate segmentation and tracking results. We provide quantitative validation of results obtained with this new algorithm by comparing them to the results obtained from the established CellProfiler, MTrack2 (plugin for Fiji), and LSetCellTracker software.

Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler’s ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.

Strong stationary times and its use in cryptography

This paper presents applicability of Strong Stationary Times (SST) techniques in the area of cryptography. The applicability is in three areas: (1) Propositions of a new class of cryptographic algorithms (pseudo-random permutation generators) which do not run for the predefined number of steps. Instead, these algorithms stop according to a stopping rule defined as SST, for which one can obtain provable properties: a) results are perfect samples from uniform distribution, b) immunity to timing attacks (no information about the resulting permutation leaks through the information about the number of steps SST algorithm performed). (2) We show how one can leverage properties of SST-based algorithms to construct an implementation (of a symmetric encryption scheme) which is immune to the timing-attack by reusing implementations which are not secure against timing-attacks. In symmetric key cryptography researchers mainly focus on constant time (re)implementations. Our approach goes in a different direction and explores ideas of input masking. (3) Analysis of idealized (mathematical) models of existing cryptographic schemes—i.e., we improve a result by Mironov [21] .

Randomized Stopping Times and Provably Secure Pseudorandom Permutation Generators

Conventionally, key-scheduling algorithm (KSA) of a cryptographic scheme runs for predefined number of steps. We suggest a different approach by utilization of randomized stopping rules to generate permutations which are indistinguishable from uniform ones. We explain that if the stopping time of such a shuffle is a Strong Stationary Time and bits of the secret key are not reused then these algorithms are immune against timing attacks.

RiffleScrambler – A Memory-Hard Password Storing Function

We introduce RiffleScrambler: a new family of directed acyclic graphs and a corresponding data-independent memory hard function with password independent memory access. We prove its memory hardness in the random oracle model.

Statistical Testing of PRNG: Generalized Gambler's Ruin Problem.

We present a new statistical test (GGRTest) which is based on the generalized gambler’s ruin problem (with arbitrary winning/losing probabilities). The test is able to detect non-uniformity of the outputs generated by the pseudo-random bit generators (PRNGs).

Leakage-Resilient Riffle Shuffle.

Analysis of various card-shuffles – finding its mixing-time is an old mathematical problem. The results show that e.g., it takes \(\mathcal {O}(\log n)\) riffle-shuffles (Aldous and Diaconis, American Mathematical Monthly, 1986) to shuffle a deck of n cards while one needs to perform \(\varTheta (n \log n)\) steps via cyclic to random shuffle (Mossel et al., FOCS, 2004).