93. Joshua Brandon Holden, Math Professor and Author of The Mathematics of Secrets

Joshua Brandon Holden, the author of The Mathematics of Secrets, Cryptography from Caesar Ciphers to Digital Encryption, graduated with a degree in pure math and went on to teach at the University of Massachusetts and Duke. He discovered that he was spending most of his time on teaching, so he sought jobs where they would reward teaching. He then worked at the Rose Hulman Institute of Technology, where he did both teaching and research. 

Common Misconceptions about Cryptography

Joshua discusses common misconceptions about cryptography and its connection to the internet. He explains that people often knew about cryptography in ancient times but don't know about the throughline. Older theories of cryptography were implicitly mathematical but not explicitly, while new theories are very explicitly mathematical. Joshua aims to open up the connection between older forms of cryptography and the new ones, stating that everyone has some ability to do all of it in varying amounts. He talks about the current state of cryptography online, including public key cryptography, which originated in the 70s and gained popularity in the 90s with internet commerce. Public key cryptography allows users to send secret messages through a one-way key, which is only decrypted by the sender who has a different key. This is important for sending credit card information to companies like Amazon or Walmart. However, end-to-end encryption means middlemen are no longer able to decrypt messages, so it's crucial to look carefully at providers' policies to determine if they stay in the loop. Joshua talks about the networks and relationships within the cryptography field, including the opportunities for professionals to work in private camps, government agencies, and academia. He notes that while there is money and space in the field, there is also a lot of space for professionals to stay updated on the latest theories and developments.

Quantum Computers in Cryptography

The conversation turns to the potential of quantum computers in cryptography and the potential for breaking encryption systems. He believes that quantum computers are expected to be better at breaking the problems used in creating mathematical problems used in special public key systems, such as encryption used by browsers to protect credit card information and communications. He also discusses the development of quantum resistant cryptography, which is a more complex system but the basic principles of quantum resistance systems are still relatively graspable for anyone with high school algebra and a willingness to dig deep. By applying enough computing power to end-to-end encryption systems, it is possible to break them. The only way to achieve perfect secrecy is to have a secret key, which is as long as the conversation. This method was supposedly used for the famous red phone between the White House and the Kremlin during the Cold War.

Keeping Your Data Safe

In terms of security, Joshua advises people to know their threat model and consider the potential threats they face. Some people may worry about powerful governments trying to break their communications, while others may be concerned about corporate spies, children, or random people passing by. For those worried about corporate espionage, it is recommended to look for end-to-end encryption systems. While quantum computers may not be easy to break, they do not guarantee that someone can’t break the system with enough computing power.

Class Field Towers Explained

Joshua talks about his research in the field of mathematics, specifically in the area of class field towers. He explains why imaginary numbers are not square roots but rather arbitrary choices. He also discusses the concept of Galois groups, which track the number of ways complex numbers can be shuffled around without making a difference. He explains that class field towers consist of rational numbers, real numbers with irrational decimals, and complex numbers on top of them. These towers record the complexity of each jump made in the tower. 

Joshua talks about the role of computers in mathematical research, stating that there is more computer usage in this area due to improved software tools and more applications in cryptography. He identifies two traits that are most useful for being successful in mathematical research: perseverance and curiosity. Perseverance is the reason most people persist. In graduate school or postgraduate school, those who stick with their passion and interest in math may be more likely to succeed in mathematical research. He encourages students to not give up on problems that require a different kind of math, even if it's not necessary for their career. He believes that having a sense of curiosity about everything comes from the fact that in mathematics, all one needs is to  just think hard about things and talk to others. This gives one a sense of confidence that they can figure things out without the need for special abilities or tools.

Influential Harvard Professors and Courses

Joshua mentions Math 25, an honors calculus course. He also enjoyed Professor McConnell, who he still maintains a friendship with. He also shares his experience with changing his name, which was the first of his non-professional wanderings. 

Timestamps:

04:33 Cryptography and its applications in online security

11:57 Cryptography, public key systems, and quantum computing

21:07 Encryption, mathematics, and data security

27:49 Mathematical research and talent

33:41 Math education, career choices, and personal growth

Links:

Website: https://wordpress.rose-hulman.edu/holden/the-mathematics-of-secrets/