Curves, codes, and cryptography

This thesis deals with two topics: elliptic-curve cryptography and code-based cryptography. In 2007 elliptic-curve cryptography received a boost from the introduction of a new way of representing elliptic curves. Edwards, generalizing an example from Euler and Gauss, presented an addition law for the curves x2 + y2 = c2(1 + x2y2) over non-binary fields. Edwards showed that every elliptic curve can be expressed in this form as long as the underlying field is algebraically closed. Bernstein and Lange found fast explicit formulas for addition and doubling in coordinates (X : Y : Z) representing (x, y) = (X/Z, Y/Z) on these curves, and showed that these explicit formulas save time in elliptic-curve cryptography. It is easy to see that all of these curves are isomorphic to curves x2 + y2 = 1 + dx2y2 which now are called "Edwards curves" and whose shape covers considerably more elliptic curves over a finite field than x2 + y2 = c2(1 + x2y2). In this thesis the Edwards addition law is generalized to cover all curves ax2 +y2 = 1+dx2y2 which now are called "twisted Edwards curves." The fast explicit formulas for addition and doubling presented here are almost as fast in the general case as they are for the special case a = 1. This generalization brings the speed of the Edwards addition law to every Montgomery curve. Tripling formulas for Edwards curves can be used for double-base scalar multiplication where a multiple of a point is computed using a series of additions, doublings, and triplings. The use of double-base chains for elliptic-curve scalar multiplication for elliptic curves in various shapes is investigated in this thesis. It turns out that not only are Edwards curves among the fastest curve shapes, but also that the speed of doublings on Edwards curves renders double bases obsolete for this curve shape. Elliptic curves in Edwards form and twisted Edwards form can be used to speed up the Elliptic-Curve Method for integer factorization (ECM). We show how to construct elliptic curves in Edwards form and twisted Edwards form with large torsion groups which are used by the EECM-MPFQ implementation of ECM. Code-based cryptography was invented by McEliece in 1978. The McEliece public-key cryptosystem uses as public key a hidden Goppa code over a finite field. Encryption in McEliece’s system is remarkably fast (a matrix-vector multiplication). This system is rarely used in implementations. The main complaint is that the public key is too large. The McEliece cryptosystem recently regained attention with the advent of post-quantum cryptography, a new field in cryptography which deals with public-key systems without (known) vulnerabilities to attacks by quantum computers. The McEliece cryptosystem is one of them. In this thesis we underline the strength of the McEliece cryptosystem by improving attacks against it and by coming up with smaller-key variants. McEliece proposed to use binary Goppa codes. For these codes the most effective attacks rely on information-set decoding. In this thesis we present an attack developed together with Daniel J. Bernstein and Tanja Lange which uses and improves Stern’s idea of collision decoding. This attack is faster by a factor of more than 150 than previous attacks, bringing it within reach of a moderate computer cluster. We were able to extract a plaintext from a ciphertext by decoding 50 errors in a [1024, 524] binary code. The attack should not be interpreted as destroying the McEliece cryptosystem. However, the attack demonstrates that the original parameters were chosen too small. Building on this work the collision-decoding algorithm is generalized in two directions. First, we generalize the improved collision-decoding algorithm for codes over arbitrary fields and give a precise analysis of the running time. We use the analysis to propose parameters for the McEliece cryptosystem with Goppa codes over fields such as F31. Second, collision decoding is generalized to ball-collision decoding in the case of binary linear codes. Ball-collision decoding is asymptotically faster than any previous attack against the McEliece cryptosystem. Another way to strengthen the system is to use codes with a larger error-correction capability. This thesis presents "wild Goppa codes" which contain the classical binary Goppa codes as a special case. We explain how to encrypt and decrypt messages in the McEliece cryptosystem when using wild Goppa codes. The size of the public key can be reduced by using wild Goppa codes over moderate fields which is explained by evaluating the security of the "Wild McEliece" cryptosystem against our generalized collision attack for codes over finite fields. Code-based cryptography not only deals with public-key cryptography: a code-based hash function "FSB"was submitted to NIST’s SHA-3 competition, a competition to establish a new standard for cryptographic hashing. Wagner’s generalized birthday attack is a generic attack which can be used to find collisions in the compression function of FSB. However, applying Wagner’s algorithm is a challenge in storage-restricted environments. The FSBday project showed how to successfully mount the generalized birthday attack on 8 nodes of the Coding and Cryptography Computer Cluster (CCCC) at Technische Universiteit Eindhoven to find collisions in the toy version FSB48 which is contained in the submission to NIST

Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation

We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling regime, focusing on the long time properties. By a saddle point analysis of the mode-coupling equations, we derive exact results for the correlation function in the long time limit - a limit which is hard to study using simulations. The correlation function at wavevector k in dimension d is found to behave asymptotically at time t as C(k,t)\simeq 1/k^{d+4-2z} (Btk^z)^{\gamma/z} e^{-(Btk^z)^{1/z}}, with \gamma=(d-1)/2, A a determined constant and B a scale factor.Comment: RevTex, 4 pages, 1 figur

Relativistic D-brane Scattering is Extremely Inelastic

We study the effects of quantum production of open strings on the relativistic scattering of D-branes. We find strong corrections to the brane trajectory from copious production of highly-excited open strings, whose typical oscillator level is proportional to the square of the rapidity. In the corrected trajectory, the branes rapidly coincide and remain trapped in a configuration with enhanced symmetry. This is a purely stringy effect which makes relativistic brane collisions exceptionally inelastic. We trace this effect to velocity-dependent corrections to the open-string mass, which render open strings between relativistic D-branes surprisingly light. We observe that pair-creation of open strings could play an important role in cosmological scenarios in which branes approach each other at very high speeds.Comment: 30 pages; added references and a comment about velocity-dependent masse

Allosteric inhibition of carnosinase (CN1) by inducing a conformational shift

In humans, low serum carnosinase (CN1) activity protects patients with type 2 diabetes from diabetic nephropathy. We now characterized the interaction of thiol-containing compounds with CN1 cysteine residue at position 102, which is important for CN1 activity. Reduced glutathione (GSH), N-acetylcysteine and cysteine (3.2 \uc2\ub1 0.4, 2.0 \uc2\ub1 0.3, 1.6 \uc2\ub1 0.2 \uc2\ub5mol/mg/h/mM; p <.05) lowered dose-dependently recombinant CN1 (rCN1) efficiency (5.2 \uc2\ub1 0.2 \uc2\ub5mol/mg/h/mM) and normalized increased CN1 activity renal tissue samples of diabetic mice. Inhibition was allosteric. Substitution of rCN1 cysteine residues at position 102 (Mut1C102S) and 229 (Mut2C229S) revealed that only cysteine-102 is influenced by cysteinylation. Molecular dynamic simulation confirmed a conformational rearrangement of negatively charged residues surrounding the zinc ions causing a partial shift of the carnosine ammonium head and resulting in a less effective pose of the substrate within the catalytic cavity and decreased activity. Cysteine-compounds influence the dynamic behaviour of CN1 and therefore present a promising option for the treatment of diabetes

Conductance of Mesoscopic Systems with Magnetic Impurities

We investigate the combined effects of magnetic impurities and applied magnetic field on the interference contribution to the conductance of disordered metals. We show that in a metal with weak spin-orbit interaction, the polarization of impurity spins reduces the rate of electron phase relaxation, thus enhancing the weak localization correction to conductivity. Magnetic field also suppresses thermal fluctuations of magnetic impurities, leading to a recovery of the conductance fluctuations. This recovery occurs regardless the strength of the spin-orbit interaction. We calculate the magnitudes of the weak localization correction and of the mesoscopic conductance fluctuations at an arbitrary level of the spin polarization induced by a magnetic field. Our analytical results for the ``$h/e$'' Aharonov-Bohm conductance oscillations in metal rings can be used to extract spin and gyromagnetic factor of magnetic impurities from existing experimental data.Comment: 18 pages, 8 figure

Dynamic landscapes and human dispersal patterns : tectonics, coastlines, and the reconstruction of human habitats

Studies of the impact of physical environment on human evolution usually focus on climate as the main external forcing agent of evolutionary and cultural change. In this paper we focus on changes in the physical character of the landscape driven by geophysical processes as an equally potent factor. Most of the landscapes where finds of early human fossils and artefacts are concentrated are ones that have been subjected to high levels of geological instability, either because of especially active tectonic processes associated with faulting and volcanic activity or because of proximity to coastlines subject to dramatic changes of geographical position and physical character by changes of relative sea level. These processes can have both beneficial effects, creating ecologically attractive conditions for human settlement, and deleterious or disruptive ones, creating barriers to movement, disruption of ecological conditions, or hazards to survival. Both positive and negative factors can have powerful selective effects on human behaviour and patterns of settlement and dispersal. We consider both these aspects of the interaction, develop a framework for the reconstruction and comparison of landscapes and landscape change at a variety of scales, and illustrate this with selected examples drawn from Africa and Arabia

Treatment Guidance for Patients With Lung Cancer During the Coronavirus 2019 Pandemic.

The global coronavirus disease 2019 pandemic continues to escalate at a rapid pace inundating medical facilities and creating substantial challenges globally. The risk of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection in patients with cancer seems to be higher, especially as they are more likely to present with an immunocompromised condition, either from cancer itself or from the treatments they receive. A major consideration in the delivery of cancer care during the pandemic is to balance the risk of patient exposure and infection with the need to provide effective cancer treatment. Many aspects of the SARS-CoV-2 infection currently remain poorly characterized and even less is known about the course of infection in the context of a patient with cancer. As SARS-CoV-2 is highly contagious, the risk of infection directly affects the cancer patient being treated, other cancer patients in close proximity, and health care providers. Infection at any level for patients or providers can cause considerable disruption to even the most effective treatment plans. Lung cancer patients, especially those with reduced lung function and cardiopulmonary comorbidities are more likely to have increased risk and mortality from coronavirus disease 2019 as one of its common manifestations is as an acute respiratory illness. The purpose of this manuscript is to present a practical multidisciplinary and international overview to assist in treatment for lung cancer patients during this pandemic, with the caveat that evidence is lacking in many areas. It is expected that firmer recommendations can be developed as more evidence becomes available

Statistical Theory of Spin Relaxation and Diffusion in Solids

A comprehensive theoretical description is given for the spin relaxation and diffusion in solids. The formulation is made in a general statistical-mechanical way. The method of the nonequilibrium statistical operator (NSO) developed by D. N. Zubarev is employed to analyze a relaxation dynamics of a spin subsystem. Perturbation of this subsystem in solids may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between the particles or with a thermal bath (lattice). The generalized kinetic equations were derived previously for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. In this paper, these results are used to describe the relaxation and diffusion of nuclear spins in solids. The aim is to formulate a successive and coherent microscopic description of the nuclear magnetic relaxation and diffusion in solids. The nuclear spin-lattice relaxation is considered and the Gorter relation is derived. As an example, a theory of spin diffusion of the nuclear magnetic moment in dilute alloys (like Cu-Mn) is developed. It is shown that due to the dipolar interaction between host nuclear spins and impurity spins, a nonuniform distribution in the host nuclear spin system will occur and consequently the macroscopic relaxation time will be strongly determined by the spin diffusion. The explicit expressions for the relaxation time in certain physically relevant cases are given.Comment: 41 pages, 119 Refs. Corrected typos, added reference