# Elliptic Curve Cryptography

by Alana Drumgold*, Lauren Flowers, Emily Huang, Tamarr Moore*, and Ashleigh Sico.

*Alana and Tamarr helped the group work through the problem set but were unfortunately unable to attend camp during the blog writing.

For years, people have been trying to find a way to send secret messages. This may have been easy to do in the ancient times of the Roman Empire, where you could write a message, and then hand-deliver it to your recipient.  This way, you could be certain that nobody else could intercept it. However, this becomes a lot more difficult in today’s online tech-driven world. People no longer hand-deliver letters; rather, we email or text our friends.  So how do we make sure that nobody else can intercept your text message as it travels the internet before finally landing on your friend’s cell-phone? The answer is found in cryptography, a technology that is becoming more and more important in today’s world.  Today, we are going to focus on one particular form of cryptography: elliptic curve cryptography.

# Mathematical Epidemiology

By Camilla Fratta, Ananya Jain, Sydney Mason, Gabby Matejowsky, and Nevaeh Pinkney*.

*Nevaeh helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Mathematical Epidemiology explores the realm of mathematics applied to public health. It relies on modeling to use known information about certain scenarios regarding the spread of diseases and then uses it to predict future outcomes. By the end of the problem set, our group learned about the challenging process that comes with trying to predict population sizes in order to control the spreading of diseases. The equations that are faced in this branch of mathematics are at the heart of mathematical modeling.

Mathematical Models and Modeling

A mathematical model is an equation used to predict or model the most likely results to occur in a real-world situation.  We used these types of equations to model the spread of a disease in a population, tracking the flow of populations from susceptible to infected to recovered.  In real life scenarios, there are too many variables to fully account for, so we only were able to place a few in our equations. This made the models less accurate, but at the same time very useful to us in our problem set.  They gave us a good idea of how things worked in an actual epidemic and helped us to understand what mathematical modeling really is.

# Quantum Mechanics

By Izzy Cox, Divya Iyer, Wgoud Mansour, Ashleigh Sico, and Elizabeth Whetzel.

Quantum Mechanics is the physics of molecular and microscopic particles. However, it has applications in everyday life as well. If someone asked you if a human was a particle or a wave, what would you think? What about a ball? What about light? Not so easy now, is it? It turns out that all of those things, and in fact, everything around us, can be expressed in physics as both a particle and a wave. This might seem a little unbelievable, but for now, let’s start with the basics.

Classical Physics

Although Classical Physics sounds like a complicated idea, it’s the most simple branch of physics. It’s what you think of when someone says “physics”. Classical Physics lays the basic foundation to Quantum Physics with a few basic laws.

# Knot Theory

By Jillian Byrnes, Monique Dacanay, Kaycee DeArmey,  Alana Drumgold, Ariyana Smith*, and Wisdom Talley*.

*Ariyana and Wisdom helped the group work through the problem set but were unfortunately unable to attend camp during the blog writing.

A mathematical knot is a loop in three-dimensional space that doesn’t intersect itself, and knot theory is the topological study of these knots. Two knots are considered to be equivalent if they can be stretched or bent into each other without cutting or passing  through themselves. The simplest of these knots is known as the unknot, which is just a circle or its equivalence. Similar to a knot is a link, which is multiple knots intersecting each other. Both knots and links are often described in the form of knot diagrams, which are two-dimensional representations of the three-dimensional shape. There are an infinite number of both knots and links, but here are a few examples in diagram form:

# Classification of Surfaces

By Ayanna Blake, Lisa Oommen*, Myla Marve, Tamarr Moore, Caylah Vickers, and Lily Zeng.

*Lisa helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

The Girls Talk Math camp is about female high school students from different places who discuss mathematics, mathematicians, and theories. We were split up into groups and were assigned different math topics to learn. Our topic was classification of surfaces, which is listed under the umbrella topic of abstract geometry.

We thought the surfaces project was very interesting and cool to learn about, because it introduced us to college level math and allowed us to understand different parts of geometry. Along with gaining knowledge of surfaces, we also got to learn about other groups topics. Campers presented their topics on the last day and helped us to perceive the significance of the different subjects.

# Elliptic Curve Cryptography

By Mukta Dharmapurikar, Anagha Jandhyala, Savanna Jones, and Ciara Renaud.

Have you ever wondered how your credit card number stays secure after shopping online? Every day millions of people’s personal information is entered online or stored in databases, where it seems like anyone could access it. However, a process called cryptography keeps theft from occurring.

Cryptography is the ancient art of keeping secret messages secure. Elliptic curve cryptography is one type of encryption that we spent the last two weeks learning about. It has some advantages over the more common cryptography method, known as RSA.

RSA relies on the difficulty of factoring very large prime numbers. Despite the current security, it’s feasible that one day a method could be invented that makes factoring large prime numbers realistic. In this blog post, we will be explaining the essential math behind how elliptic curves work and how they are used to encrypt messages.

# RSA Encryption Cryptography

By Camille Clark, Layke Jones, Isabella Lane, Aza McFadden*, and Lizbeth Otero.

*Aza helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Cryptography is a field of coding and decoding information. It relies on the framework of number theory. Therefore, it can be used to connect theories as well as teaching others the fundamental properties of integers. Relevant number theory topics are modular arithmetic, prime factorization, greatest common divisor, and theorems such as the Chinese Remainder Theorem and Euler’s Theorem. This blog post will focus on the first three topics.

# Network Science

By Myla James, Shania Johnson, Maya Mukerjee, and Savitha Saminathan.

Graph Theory

Nodes – vertex/point.
Edges – lines connecting vertices.
Adjacent – two nodes (vertices) are adjacent if they share an edge (line).
Degree – number of edges adjacent to a particular node.

We started this problem set with learning about the difference between connected and disconnected graphs.

Connected Graph – able to travel from one node to any other through its edges.
Disconnected graph – more complex; it has components.
Components – parts of the graphs that are connected.

# Number Systems

By Miranda Copenhaver, Nancy Hindman*, Efiotu Jagun, and Gloria Su.

*Nancy helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Number systems are how we represent numbers like 1, 32, and 75. We use the base ten (decimal) system for our numbers most of the time. It’s called base ten because it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. But what if I told you that 1001101 and 4D both mean seventy-seven? Crazy, right? There are countless number systems, but today we will be focusing on two: hexadecimal (base sixteen) and binary (base two)!

As we’ve said before, the binary system is base two; it only uses 0 and 1. Since only 1 or 0 can be used, the placement of each digit is important. Computers use binary to store and transfer information. It is used in communication (Morse code, braille) and everything electronic like computers, lights, calculators, MP3s, MIDI, JPEG, etc.

# Elliptic Curve Cryptography

By Noa Bearman, Kimberly Cruz Lopez, Tina Lin, Xintong Xiang, and Maria Neri Otero*

*Maria helped the group work through the problem set but was unfortunately unable to attend camp during the blog writing.

Introduction

Have you ever tried to send a secret message to a friend? Did it work? Was it secure? Well, one way to do so in a more secure way is by using Elliptic Curve Cryptography (ECC). Most people have never heard of ECC before, and two weeks ago, neither did we. However, in the past two weeks, we have been learning how to use this exciting application of the techniques of algebraic geometry and abstract algebra applied to the ancient art of keeping messages secure. ECC was first introduced by Victor Miller and Neal Koblitz in 1985. It was proposed as an alternative to other forms of cryptography with public-key systems such as DSA and RSA. Public-key systems involve the use of two different kinds of keys: a public key that is available to the public and a private key in which only the owner knows. The applications of ECC has been growing and has recently gained a lot of attention in industry and academia. The following information below will go more in-depth on what ECC is, how it works, its advantages, its disadvantages, and our progression throughout this course.