A Transparent Square Root Algorithm to Beat Brute Force for Sufficiently Large Primes of the Form p = 4n + 1

Finding square roots in the modular integers is a well known problem that is the basis for many modern cryptosystems. For primes of the form p=4n+3, given C\in\mathbb Z_p^\times , finding solutions to x^2\equiv C \pmod{p} is deterministic. For primes of the form p=4n+1, no known deterministic computation exists for determining x given C. Tonelli (later improved by Shanks,) Cipolla, and Pocklington, among others, found sophisticated algorithms to perform this task. Brute force is a transparent approach, but offers no insights into the problem. In this thesis, we produce a transparent approach to this problem, visualized using a model built on Symplectic Geometry. One of the insights from viewing the problem in this way is a conjecture on the distribution of quadratic residues, which we exploit in our algorithm. Even though the conjecture is not essential to the workings of the algorithm, it gives it an edge over brute force for large enough primes. Finally, we follow this with examples of the algorithm\u27s execution

Parasitophorous vacuole poration precedes its rupture and rapid host erythrocyte cytoskeleton collapse in Plasmodium falciparum egress.

In the asexual blood stages of malarial infection, merozoites invade erythrocytes and replicate within a parasitophorous vacuole to form daughter cells that eventually exit (egress) by sequential rupture of the vacuole and erythrocyte membranes. The current model is that PKG, a malarial cGMP-dependent protein kinase, triggers egress, activating malarial proteases and other effectors. Using selective inhibitors of either PKG or cysteine proteases to separately inhibit the sequential steps in membrane perforation, combined with video microscopy, electron tomography, electron energy loss spectroscopy, and soft X-ray tomography of mature intracellular Plasmodium falciparum parasites, we resolve intermediate steps in egress. We show that the parasitophorous vacuole membrane (PVM) is permeabilized 10-30 min before its PKG-triggered breakdown into multilayered vesicles. Just before PVM breakdown, the host red cell undergoes an abrupt, dramatic shape change due to the sudden breakdown of the erythrocyte cytoskeleton, before permeabilization and eventual rupture of the erythrocyte membrane to release the parasites. In contrast to the previous view of PKG-triggered initiation of egress and a gradual dismantling of the host erythrocyte cytoskeleton over the course of schizont development, our findings identify an initial step in egress and show that host cell cytoskeleton breakdown is restricted to a narrow time window within the final stages of egress

A self-renewal assay for cancer stem cells

Cancers of epithelial origin are responsible for the majority of cancer-related deaths in the USA. Unfortunately, although chemotherapy and/or radiation therapy can sometimes shrink tumors, metastatic cancers of epithelial origin are essentially incurable. It is clear that new approaches are needed to treat these diseases. Although cancer cell lines provide invaluable information, their biological properties often differ in crucial ways from de novo cancer cells. Our laboratory has developed a novel mouse model that reliably permits individual cancer cells isolated directly from patients’ tumors to be assayed. This will allow the characterization of crucial signaling pathways involved in processes such as self-renewal that are critical for tumor formation by the cancer cells within de novo tumors. These tools should lead to new insights into the cellular and molecular mechanisms that drive human breast cancer growth and invasion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46932/1/280_2005_Article_97.pd

MIF as a glucocorticoid-induced modulator of cytokine production

Glucocorticoid hormones are important for vital functions and act to modulate inflammatory and immune responses. Yet, in contrast to other hormonal systems, no endogenous mediators have been identified that can directly counter-regulate their potent anti-inflammatory and immunosuppressive properties. Recent investigations of the protein macrophage migration inhibitory factor (MIF), which was discovered originally to be a T-lymphocyte-derived factor, have established it to be a pro-inflammatory pituitary and macrophage cytokine and a critical mediator of septic shock. Here we report the unexpected finding that low concentrations of glucocorticoids induce rather than inhibit MIF production from macrophages. MIF then acts to override glucocorticoid-mediated inhibition of cytokine secretion by lipopolysaccharide (LPS)-stimulated monocytes and to overcome glucocorticoid protection against lethal endotoxaemia. These observations identify a unique counter-regulatory system that functions to control inflammatory and immune responses