The Euclidean, Extended Euclidean, and RSA Algorithms

The Euclidean Algorithm is a methodical process for calculating the greatest common divisor of two integers. It does not require the factorization of the integers which is an advantage for large integers. The Extended Euclidean Algorithm is a methodical process for finding the integer coefficients in the linear representation of the gcd in terms of the original two integers. The RSA Algorithm is a public key cryptographic system in wide use in computer systems all over the world. The system's security rests on the mathematical fact that there is no known easy way to factor sufficiciently large integers. (If a way was discovered, there would be hell to pay in the world of secure data.) The system includes a so-called public key pair by which messages are encrypted and private key pair by which they are decrypted. When the public key pair is established and published, the privacy key of the private key pair is computed using the Extended Euclidean Algorithm. The paper The Euclidean and RSA Algorithms, gives a detailed description and drivation of the Euclidean Algorithms as well as a description and worked example of the RSA Algorithm.

Forces Introduced in Accelerated Frames

One of the basic axioms of the special theory of relativity is that the laws of physics are the same in all unaccelerated frames of reference. That axiom, along with the constancy of the speed of light, lead to the results of the previous two blog entries in this collection. In this entry, we will look at some accelerated frames, especially rotating frames, to see that Newton's second law must be modified to account for the acceleration of one frame with respect to another. This modification is the introduction of apparent (or so called "fictitious" forces) that are a direct result of the accelerating frame. In the paper, Forces Introduced in Accelerated Frames we look at the forces introduced in rotating frames.

A Relativistic Dynamics Primer

In early Physics classes the conservation of momentum and conservation of energy are fundmental laws used in all kinds of applications. Momentum is defined as the product of mass and velocity. But it turns out that momentum has to redefined when the velocities involved become relativistic. This new definition has consequences in the concepts that depend on it. This paper Relativistic Dynamics Primer summarizes the basic ideas of momentum and energy when relativistic speeds are in play. In particular, it explains why the formula for momentum used in classical mechanics must be altered to satisfy conservation under a Lorentz transformation and how this change in the definition of momentum leads to the concept of mass as energy in a new formulation of relativistic total energy. The concepts reduce to the familiar Newtonian ones when the velocity is not relativistic. It assumes the reader is familiar with the concepts in the previous blog entry - A Special Relativity Primer. A few examples are worked to apply the concepts discussed.

Special Relativity Primer

The consequences of Special Relativity follow from the assertion that the laws of physics are the same in all inertial frames. In particular, the speed of light is a limiting constant in all such frames. The consequences, deduced from this assertion, make up a body of results that have been repeatedly affirmed in practice since the early 20th century. I once tried to describe some of these consequences in a paper I wrote in a Philosophy of Science class. The professor gave me an A on the paper but asked, in the end, do these things really happen? This paper, Special Relativity Primer, explains the basic theory and provides exampes and worked exercises demonstrating the consequences.

The Polynomial of Best Fit

To test a sample data set for linearity, we can compute a number r in [-1,1] called the correlation coefficient whose absolute value is a measure of how much the data corresponds to a linear representation. We can also construct the so-called regression line that is the best linear fit to the data. If the correlation coefficient is small (closer to 0) the regression line will not correspond very well with the data. If the correlation coefficient is closer to 1 in absolute value, the regression line will correspond increasingly well with the data. This topic is discussed in detail, including the derivation of the regression line as the best fit with several examples, as well as the idea of "best fit" extended to higher degree polynomials, in the paper The Polynomial of Best Fit, where we can see that the linear case is just a special case of the general polynomial of best fit.

Calculating Square Roots By Hand

Before the days when students carried personal calculators that gave instantaneous square root calculations, that is, pre-1970's, most were taught a mysterious method that was accepted without understanding the logic behind it and therefore often forgotten. In the paper Algorithm to Compute the Square Root the technique is re-visited and examples and a justification are provided.

Phase Space Trajectories and Conservation of Energy

The motivation for the derivation of the Schroedinger equation arising from the Law of Conservation of Energy as seen in the blog entry of December 13, 2022, is made clearer when one looks at the phase space trajectories in familiar classical settings. In the paper, Phase Space Curves and Conservation of Energy, we go through a few examples of how Newton's 2nd Law of motion and the conservation of energy are closely related and how the phase space trajectory gives insight into the equation that defines motion in quantum settings.