Böcker av Bhupendra Nath Tiwari

In this research, we study primality testing of arbitrary integers via number theory, randomized algorithms and optimization theories. Concerning fundamentals of modern cryptography, we focus on the cryptanalysis, cryptosystems and RSA keys, which are widely used in secure communications, e.g., banking systems and other platforms with an online security. For a given n-bit integer N, our consideration is realized as a decision problem, viz. an optimized algorithm rendering the output YES, if N is a prime, and NO otherwise. In order to design such an algorithm, we begin by examining essential ingredients from the number theory, namely, divisibility, modular arithmetic, integer representations, distributions of primes, primality testing algorithms, greatest common divisor, least common multiplication, pseudoprimes, Np-hard discrete logarithm problem, residues and others. Hereby, we give special attention to the congruence relations, Chinese remainder theorem and Fermat's little theorem towards the optimal primality testing of an integer. Finally, we anticipate optimized characterizations of Cunningham numbers in the light of randomization theory and their applications to cryptography.

From the perspective of D-brane physics, we consider the role of the real intrinsic Riemannian geometry and describe the statistical nature of gauge and exotic instanton vacuum fluctuations. For the Veneziano-Yankielowiz/ Affleck-Dine-Seiberg and non-perturbative instanton superpotentials, the issue of the wall (in)stabilities is analysed for marginal and threshold like vacua, and their arbitrary linear combinations. Physically, for both the stationary and non-stationary statistical configurations with and without the statistical fluctuations of the gauge and exotic instanton curves, the Gaussian fluctuations over equilibrium (non)-stationary vacua accomplish a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds for statistically admissible domains of (i) one loop renormalized mass and vacuum expectation value of the chiral field for the stationary vacua and (ii) the corresponding contributions of the instanton curves for the non-stationary vacua. As a function of the vacuum expectation value of the chiral field, the global ensemble stability and phase transition criteria algebraically reduce to the invariance of the quadratic and quartic polynomials.

The present research offers thermodynamic geometric properties of black holes in string theory and M-theory. It systematically investigates the state-space and conformally related chemical geometries for extremal and non-extremal black holes, black strings, black rings and supertubes in four and higher spacetime dimensions. From the perspective of the intrinsic differential geometry, the questions of stability, regularity, existence of critical phenomena and phase transitions have been analyzed for charged anticharged black holes in a given basin of attractor. For the two parameter half BPS giants and superstars, an ensemble of arbitrary liquid droplets or irregular shaped fuzzballs shows that the chemical correlations involve ordinary summations, while the state-space correlations are depicted by standard polygamma functions. The state-space geometry provides definite stability character to black brane configurations under both the Planck length corrections and higher derivative stringy corrections. Perspective research includes D-branes, attractor mechanism, topological trees, Calabi-Yau compactifications, moduli space geometry and algebraic geometry.