Project Title: Divergent Sums and Analytic Continuation
BASIS Advisor: Mr. Rappelfeld
My Posts
Week 12: Presentation
Jun 20, 2022
Hello everyone and welcome back to my blog! I spent this week preparing my slide show presentation and modifying it to introduce the topic of cybersecurity and analytic continuation in a more engaging way. My original presentation was much more like a lecture on the different primality tests, encryption algorithms and summation methods that I […]
Week 11: Understanding a very strange behavior
Jun 20, 2022
Hello everyone and welcome back to my blog! This week I am pleased to have made significant progress on partially explaining a phenomenon that was part of the motivation to work on this topic in the first place. At the start of the project I had seen (and coded) an example of the Euler-Maclaurin series […]
Week 10: Analytic Continuation
Jun 20, 2022
Hello everyone and welcome back to my blog! This week I showed my off site mentor an example of a “proof” that relies on assigning a divergent series a finite value and explained how I planned to tweak it to be correct. The problem I was coming up against was the final step of this […]
Week 9: First few proofs successfully tweaked!
Jun 02, 2022
Hello everyone and welcome back to my blog! This week I managed to find a way to tweak a large amount of proofs by generating functions of equations involving n!, such as the equation 0*0!+1*1!+2*2!+…+n*n!=(n+1)!-1. While this particular equation and others can be proven much more directly using induction or in this case writing the […]
Week 8: Borel Summation
Jun 02, 2022
Hello everyone and welcome back to my blog! This week I learned about Borel Summation and made substantial progress on tweaking a number of proofs that I had wanted to fix. Like Cesaro summation, Borel Summation assigns values to divergent sums by having the property that if a series is convergent, the Borel sum of […]
Week 7: Coding ElGamal Encryption and Running Tests
May 31, 2022
Hello everyone and welcome back to my blog! This week I built off of some of the coding I had done for RSA encryption and coded ElGamal encryption and ran some tests which seemed to reveal more about the similarities and differences between ElGamal and RSA encryption. When these algorithms are used in the real […]
Week 6: Cesaro Summation and Tweaking Proofs
May 20, 2022
Hello everyone and welcome back to my blog! This week I learned about Cesaro summation. Most of them time, when we talk about the sum of a series, we are referring to its partial sums or infinite sum (which is the limit of its partial sums). The Cesaro sum of a series is a different […]
Week 5: ElGamal Encryption
May 20, 2022
Hello everyone and welcome back to my blog! This week I learned about ElGamal encryption, an alternative to RSA encryption. The math behind ElGamal encryption is much more accessible than RSA encryption which requires knowledge of Fermat’s Little Theorem or Euler’s Totient Theorem, so I may want to present the math that is used in […]
Week 4: Coding
Apr 27, 2022
Hello everyone and welcome back to my blog! This week I implemented a much more efficient way to calculate large exponents into my code, which is crucial to decrypting messages encrypted with RSA and making the algorithm practical. My computer can now computer absurdly large exponents extremely quickly, and now the main problem my code […]
Week 3: More Primality Tests and a Lead on How to Exponentiate Large Numbers
Apr 20, 2022
Hello everyone, and welcome back to my blog! This week I learned about some more primality tests that are used in the generation of prime numbers for use in RSA encryption including: the Miller-Rabin test, Baillie-PSW test and Solovay-Strassen test. These tests tend to follow this pattern: start with some proven mathematical theorem about primes […]
Week 2: RSA in Practice
Apr 08, 2022
Hello everyone and welcome back to my blog! As I discussed last time, my research this week has been devoted to seeing how issues in the use of RSA have been addressed making the encryption method useful in practice. This week I learned about several “primality tests”, methods used to test whether a number is […]
Week 1: Starting My Research: Computational Hardness Assumptions
Apr 07, 2022
Hello everyone and welcome back to my blog! After the website deleted my first draft, I have finally mustered up the will to redo this. I began my senior project by researching the math behind cybersecurity and encryption as well as the assumptions and principles used. I began with computational hardness assumptions-assumptions about how hard […]
Week 0: Introduction
Mar 30, 2022
Hello everyone, and welcome to my senior project blog. My name is Soren, and I enjoy doing math, playing piano, hiking and learning about economics. I am a student at BASIS Independent Brooklyn and participate in a math program every Saturday called Math-M-Addicts. For my senior project, I will be exploring if and how the […]
Week 0: Introduction
Mar 19, 2022
Hello everyone, and welcome to my senior project blog. My name is Soren, and I enjoy doing math, playing piano, hiking and learning about economics. I am a student at BASIS Independent Brooklyn and participate in a math program every Saturday called Math-M-Addicts. For my senior project, I will be exploring if and how the […]