In this thesis, we explore the pairing-based cryptography on elliptic curves from the theoretical and implementation point of view. In this respect, we first study so-called pairing-friendly elliptic curves used in pairing-based cryptography. We classify these curves according to their construction methods and study them in details. Inspired of the work of Koblitz and Menezes, we study the elliptic curves in the form $y^{2}=x^{3}-c$ over the prime field $F_{q}$ and compute explicitly the number of points $#E(mathbb{F}_{q})$. In particular, we show that the elliptic curve $y^{2}=x^{3}-1$ over $mathbb{F}_{q}$ for the primes $q$ of the form $27A^{2}+1$ has an embedding degree $k=1$ and belongs to Scott-Barreto families in our classification. Finally, we give examples of those primes $q$ for which the security level of the pairing-based cryptographic protocols on the curve $y^{2}=x^{3}-1$ over $mathbb{F}_{q}$ is equivalent to 128-, 192-, or 256-bit AES keys. From the implementation point of view, it is well-known that one of the most important part of the pairing computation is final exponentiation. In this respect, we show explicitly how the final exponentiation is related to the linear recurrence relations. In particular, this correspondence gives that finding an algoritm to compute final exponentiation is equivalent to finding an algorithm to compute the $m$-th term of the associated linear recurrence relation. Furthermore, we list all those work studied in the literature so far and point out how the associated linear recurrence computed efficiently.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 85-105).

Die vorliegende retrospektive Studie untersuchte die lokale Rezidiv- und Überlebensrate von Patienten mit Plattenepithelkarzinomen in Abhängigkeit vom Randstatus und Resektionsabstand der Tumorresektion sowie von der postoperativen Therapie. Dabei wurden Daten von 351 Patienten mit Plattenepithelkarzinomen des Oropharynx, Hypopharynx und Larynx ausgewertet, welche zwischen 2005 und 2009 reseziert wurden. Der gesundheitliche Zustand der Patienten wurde anschließend bis Dezember 2011 beobachtet. Im Ergebnis konnte kein Einfluss des Randstatus und des Resektionsabstandes auf die lokale Rezidivrate nachgewiesen werden. Beide Faktoren hatten jedoch einen signifikanten Einfluss auf die Letalitätsrate. Die postoperative Therapie zeigte einen positiven Einfluss auf die lokale Rezidivrate. Sie konnte aber die negativen Prognosefaktoren, die eine Indikation dieser Therapie darstellen, bezüglich der Letalitätsrate nicht ausgleichen. Weiterhin war ein positiver Einfluss der postoperativen Therapie auf das rezidivfreie Überleben (DFS-disease-free survival) sowohl uni- als auch multivariat nachweisbar. Die Gesamtüberlebensrate (OS-overall survival) wurde signifikant positiv durch einen geringeren Randstatus und einen zunehmenden Resektionsabstand (ASR Klassifikation) beeinflusst. Das krankheitsspezifische Überleben (DSS-disease-specific survival) sank bei Auftreten eines lokalen Rezidivs. Aus diesen Ergebnissen kann geschlussfolgert werden, dass eine postoperative Therapie zur Kontrolle lokaler Rezidive von hoher Bedeutung ist. Da lokale Rezidive die Überlebensrate signifikant senken, hat die postoperative Therapie indirekt einen positiven Effekt auf das Überleben. Des Weiteren zeigt sich bei Patienten mit freien Resektionsrändern die höchste Überlebensrate, wobei diese mit zunehmendem Abstand des Karzinoms vom Resektionsrand steigt.

The random Fibonacci sequence is defined by t_1 = t_2 = 1 and t_n = ± t_{n–1} + t_{n–2} , for n ? 3, where each ± sign is chosen at random with P(+) = P(–) = 1/2. We can think of all possible such sequences as forming a binary tree T. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate 1.13198824.... He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. We will discuss a new computation of Viswanath's constant which is based on a formula due to Kalmár-Nagy, and uses an interesting reduction R of the tree T developed by Rittaud. Also, we will focus on the growth rate of the expected value of a random Fibonacci sequence, which was studied by Rittaud. This differs from the almost sure growth rate in that we first find an expression for the average of the n^th term in a sequence, and then calculate its growth. We will derive this growth rate using a slightly different and more simplified method than Rittaud, using the tree R and a Pascal-like array of numbers, for which we can further give an explicit formula. We will also consider what happens to random Fibonacci sequences when we remove the randomness. Specifically, we will choose coefficients which belong to the set {1, –1} and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order, for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length. Lastly we will look at two ways to completely characterize the trace, given the coefficient cycle, by breaking the matrix product up into blocks.

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The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given. / Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms.

Disertacijoje nagrinėjamos tiesinės rekurenčiosios sekos. Ieškoma tokių pradinių narių, kurie generuotų rekurenčiąsias sekas sudarytas iš sudėtinių skaičių. Pilnai išnagrinėtos antros eilės tiesinės rekurenčiosos ir tribonačio tipo sekos, patiekiami pavyzdžiai. Gauti rezultatai ir k-bonačio tipo sekoms. / The main objects studied in this thesis are linear recurrence sequences of composite numbers. We have studied the second order (binary) linear recurrence, tribonacci – like and higher order sequences. Many examples have been given.

<p>Chemotherapy remains the only available treatment for triple-negative (TN) breast cancer. Although some TN breast cancers respond initially to neoadjuvant chemotherapy, the majority of patients die within three years of treatment due to recurrent tumor growth. Developing ex vivo models for TN breast cancer recurrence and defining responsible molecules will be crucial to developing effective combination therapies for TN breast cancer patients. We have developed an in vitro model of TN breast cancer dormancy/recurrence. Short-term exposure of tumor cells to chemotherapy at clinically relevant doses enriches for a dormant tumor cell population. Several days after removing chemotherapy, dormant tumor cells regain proliferative ability and establish colonies, resembling tumor recurrence. Tumor cells from "recurrent" colonies exhibit increased chemotherapy resistance, resembling therapy resistance of recurrent tumors in patients. Furthermore, we identify a novel signaling axis [nuclear bFGF/DNA-dependent protein kinase (DNA-PK)] supported by chemotherapy-enriched dormant TN breast cancer cells. This signaling axis drives accelerated DNA repair in chemo-residual TN breast cancer cells. Targeting this axis with either with a bFGF shRNA or DNA-PK small molecule inhibitor blocks recurrent colony formation. Using the Oncomine gene expression database, we found that bFGF expression in tumor samples from TN breast cancer patients predicts five year tumor recurrence following neoadjuvant chemotherapy treatment. Finally, we demonstrate that recurrent tumor cells exhibit increased invasiveness, reflecting the aggressive behavior of recurrent tumors in patients. Collectively, these studies identify a novel signaling axis in TN breast cancer that likely contributes to tumor recurrence and provide molecular targets for developing future therapeutics against TN breast cancer.</p> / Dissertation

In this work, we study the statistical properties of deterministic and stochastic dynamical systems. We are particularly interested in extreme values and recurrence. We prove the existence of Extreme Value Laws (EVLs) and Hitting Time Statistics (HTS)/ ReturnTime Statistics (RTS) for systems with decay of correlations against L1 observables. We also carry out the study of the convergence of Rare Event Point Processes (REPP). In the first part, we investigate the problem for deterministic dynamics and completely characterise the extremal behaviour of expanding systems by giving a dichotomy relying on the existence of an Extremal Index (EI). Namely, we show that the EI is strictly less than 1 for periodic centres and is equal to 1 for non-periodic ones. In a more general setting, we prove that the REPP converges to a standard Poisson if the centre is non-periodic, and to a compound Poisson with a geometric multiplicity distribution for the periodic case. Moreover, we perform an analysis of the convergence of the REPP at discontinuity points which gives the convergence to a compound Poisson with a multiplicity distribution different than the usual geometric one.In the second part, we consider stochastic dynamics by randomly perturbing a deterministic system with additive noise. We present two complementary methods which allow us to obtain EVLs and statistics of recurrence in the presence of noise. The first approach is more probabilistically oriented while the second one uses spectral theory. We conclude that, regardless of the centre chosen, the EI is always equal to 1 and the REPP converges to the standard Poisson.

Locally recurrent renal cell carcinoma and multiple pulmonary metastases were successfully resected in a patient 20 years after nephrectomy. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.