Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences

The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) x(3) equivalent to y(2)z (mod p), where p is a given prime number. A nontrivial solution must also satisfy x(3) not equal y(2)z and 1 <= x, y, z < p(alpha), where alpha is a given real number such that 1/3 < alpha <= 1/2. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as x equivalent to v(2)z (mod p) and y equivalent to v(3)z (mod p). In this paper, we give a deterministic polynomial-time (O(ln(3) p) bit-operations) algorithm to determine, for a given v, a nontrivial solution to CSC, if one exists. Previously it took (O) over tilde (p(alpha)) time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers N satisfying the fractional part inequality {theta N} < phi (theta and phi are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter z in the former problem corresponds to determining N in the latter problem. We also show in the alpha = 1/2 case of CSC that for a certain class of primes the CSC problem can be solved deterministically in <(O)over tilde>(p(1/3)) time compared to the previous best of (O) over tilde (p(1/2)). It is empirically observed that about one out of three primes is covered by the above class. (C) 2013 Elsevier B.V. All rights reserved.

Competing views on cancer

Despite intense research efforts that have provided enormous insight, cancer continues to be a poorly understood disease. There has been much debate over whether the cancerous state can be said to originate in a single cell or whether it is a reflection of aberrant behaviour on the part of a `society of cells'. This article presents, in the form of a debate conducted among the authors, three views of how the problem might be addressed. We do not claim that the views exhaust all possibilities. These views are (a) the tissue organization field theory (TUFT) that is based on a breakdown of tissue organization involving many cells from different embryological layers, (b) the cancer stem cell (CSC) hypothesis that focuses on genetic and epigenetic changes that take place within single cells, and (c) the proposition that rewiring of the cell's protein interaction networks mediated by intrinsically disordered proteins (IDPs) drives the tumorigenic process. The views are based on different philosophical approaches. In detail, they differ on some points and agree on others. It is left to the reader to decide whether one approach to understanding cancer appears more promising than the other.

Simvastatin Induced Neurite Outgrowth Unveils Role of Cell Surface Cholesterol and Acetyl CoA Carboxylase in SH-SY5Y Cells

Statins are known to modulate cell surface cholesterol (CSC) and AMP-activated protein kinase (AMPK) in nonneural cells; however no study demonstrates whether CSC and AMPK may regulate simvastatin induced neuritogenesis (SIN). We found that simvastatin (SIM) maintains CSC as shown by Fillipin III staining, Flotillin-2 protein expression / localization and phosphorylation of various receptor tyrosine kinases (RTKs) in the plasma membrane. Modulation of CSC revealed that SIN is critically dependent on this CSC. Simultaneously, phospho array for mitogen activated protein kinases (MAPKs) revealed PI3K / Akt as intracellular pathway which modulates lipid pathway by inhibiting AMPK activation. Though, SIM led to a transient increase in AMPK phosphorylation followed by a sudden decline; the effect was independent of PI3K. Strikingly, AMPK phosphorylation was regulated by protein phosphatase 2A (PP2A) activity which was enhanced upon SIM treatment as evidenced by increase in threonine phosphorylation. Moreover, it was observed that addition of AMP analogue and PP2A inhibitor inhibited SIN. Biocomposition of neurites shows that lipids form a major part of neurites and AMPK is known to regulate lipid metabolism majorly through acetyl CoA carboxylase (ACC). AMPK activity is negative regulator of ACC activity and we found that phosphorylation of ACC started to decrease after 6 hrs which becomes more pronounced at 12 hrs. Addition of ACC inhibitor showed that SIN is dependent on ACC activity. Simultaneously, addition of Fatty acid synthase (FAS) inhibitor confirmed that endogenous lipid pathway is important for SIN. We further investigated SREBP-1 pathway activation which controls ACC and FAS at transcriptional level. However, SIM did not affect SREBP-1 processing and transcription of its target genes likes ACC1 and FAS. In conclusion, this study highlights a distinct role of CSC and ACC in SIN which might have implication in process of neuronal differentiation induced by other agents.