Elliptic Curve Cryptography

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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.

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Real World Cryptography

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By: Shai Caspin, Natalie Bowers, Bryana Dorsey, Nia Pierce, and Cana Perry

Cryptography is the act of encrypting and decrypting codes. It’s used to pass secret messages and keep outsiders from accessing information. Math is used to help encrypt codes using different methods. One common methods is to use RSA encryptions, which uses prime numbers and mod functions to make deciphering impossible. RSA encryptions are so successful since factoring large numbers into their prime factors is incredibly difficult, and there is yet a way to do so quickly and efficiently. 

We were all very interested in learning more about cryptography since it incorporates everyday math with real-world problems and situations.

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Elliptic Curve Cryptography

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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.

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Mathematical Epidemiology

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by Jillian Byrnes, La’Ziyah Henry, Delphine Liu, Sophie Ussery, and Elizabeth Whetzel.

What is Mathematical Epidemiology?

What is mathematical epidemiology? Well, mathematical epidemiology is when mathematicians use math to predict outcomes in various statistical problems. These problems include growth in infectious bacteria, change in population, and even the effects of climate change. Why is this used? It is used because it doesn’t need a complete set of data to figure out a solution, as long as you can create an equation and plug in the values.

Who uses it? Mathematicians and scientists use it in fields such as biotechnology, medical science, civil engineering, and as public health professionals.

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Network Science

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by Kayla Aguilar, Maris James, and Aynsley S.

Data is all around us, but it has to be studied in some way, right? How else are we supposed to know what it’s about? That’s what graph theory and network science are for! To organize and connect data mathematicians use networks and graphs as well as scientific computing (like coding).

Network science is an application-based study of graphs. To understand network science, we first have to understand the graphs:

Graph Theory

Graphs represent data through nodes, which are the separate points of a graph, and edges, which connect the nodes. There are two types of graphs: directed and undirected graphs. Directed graphs rely on the order of the vertices to be the same, while undirected graphs don’t rely on the order of the nodes.

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Elliptic Curve Cryptography

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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.

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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.

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The Art of Cryptography

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By: Nia Beverly, Makayla McDaniel, Yuanyuan Matherly, and Tyler Deegan

Introduction

Cryptography is defined as the art of writing and solving codes. Upon first thought, many people picture codes as an antiquated war time communication technique. However, the field of cryptography is alive and well,  and it has become pervasive in our everyday lives. The world is becoming more and more connected through technology, and with this, there is a greater need to protect information. Encryption is probably the most widely used application of cryptography, and it is used to protect information by making it so only one person with a key can understand what is transmitted. In the following paragraphs we will walk through the steps to mathematically understanding one widely used type of encryption.

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Network Science

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By Cameron Farrar, Elizabeth Gross, Shiropa Noor, and Rebecca Rozansky

Girls Talk Math was an eyeopening experience to a brand new world of mathematics. Over the past two weeks, we have been introduced to multiple topics and related professions. We learned about: quantum mechanics, surface classification, knot theory, computing & dynamics, elliptic curve cryptography, RSA encryption, special relativity and the most interesting of them all- NETWORK SCIENCE!

During our time at Girls Talk Math, we learned about the wonders of network science and graph theory. The difficult part of this otherwise enjoyable journey? Mathematica. Mathematica is a software created to make you suffer, especially if you already know computer science (AHEM BECKY). Basically, we created graphs, did calculations and got confused on Mathematica. Typing out all the commands took ages. We’ll show you some examples as we go through the different concepts we explored. Don’t worry- once you spend some time on Mathematica, you’ll get used to it.

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Classification of Surfaces

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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.

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