### About Crypto Giants

Crypto Giants is a group of young, enthusiastic, diverse and mathematically-minded individuals who are passionate about the complex properties of numbers as well as Mathematics as a whole and its application in daily life. They are embarking on a problem-solving journey by tackling various mathematical problems and research projects, which will help them to enhance their mathematical skills and analytical thinking processes, thus discovering different approaches to problems at hand. They strive to learn more from the experts whom are willing to impart their knowledge to them concerning their different (research) fields. The field of mathematics covered in these projects are the following: Elementary number theory, basic analysis, creative mathematics and logic and set theory. The applications of the above fields are found in Cryptography(RSA) and Mathematical optics. Students are challenged to understand the basic theory of mathematics and applying it in physics, biology, finance and computer science for example. The students challenged to solve Cryptography problems using Mathematica and Matlab, and to write their report using latex. They are also challenged to solve logic puzzles and discussions about logical fallacies forms part of the program. Fuzzy-logical problem-solving through lateral thinking is a structured session as well. The program is structured to prepare students for honours level mathematics, and they are expected to present their literature review to the group and come up with their own examples where applicable.

## Nowhere Differentiable Functions

By: Jan Wuzyk

In this article I am going to show that nowhere differentiable functions do in fact exists and give a few examples, some of which are relatively modern. But first I’m going to try to answer a question that is, in my opinion, too rarely discussed in mathematics classes, ”Why do we care?”.

# Why we care

To answer this question we have to look into the history of mathematics. In 1821 Augustin-Louis Cauchy published his seminal book, Cours d’Analyse, this is generally recognized as the first serious attempt to put calculus on a rigours footing[Com][1] , mainly through introducing rigorous definitions of limits,continuity and differentiability among others, and the definitions that go with them2. This was also time the integral was defined as an area instead of simply as the antiderivative.

It should be noted that Cauchy by no means closed the issue of rigour in analysis but he provided a starting point.…

## Verification of the RSA Algorithm

During the course of this year the Crypto Giants have been looking at the RSA Algorithm. They were required to make use of MATHEMATICA software in order to verify the mechanics of the RSA public key crypto system, that decryption of a message produces the original message. The project was supervised by Dr Gaza Maluleka. Below are 2 reports from their members, Ntombifuthi Khoza and Pilane Koma, in which they describe how they carried out the task.

CRYPTO GIANTS PROJECT

Verification of RSA Algorithm.

N Khoza

30 October 2015

Project Supervisor:Dr Gaza Maluleka

Contents

1 Historic Background on the

RSA Algorithm ………………………………………………………………………….. 3

2 Introduction …………………………………………………………………………….. 4

3 Task procedure …………………………………………………………………… 4

3.1 Prime numbers……………………………………………………………………………………………… 4

3.2 Computing n…………………………………………………………………………………………………… 4

3.3 Phi φ………………………………………………………………………………………………………………… 5

3.4 Public Key(e)………………………………………………………………………………………………….. 5

3.5 Private Key(d)……………………………………………………………………………………………….. 5

3.6 Encryption……………………………………………………………………………………………………… 5

3.7 Decryption……………………………………………………………………………………………………… 6

4 Conclusion …………………………………………………………………………………… 7

5 Reference ……………………………………………………………………………………… 8

1 Historic Background on the RSA

## Algorithm

The RSA algorithm was developed by professor Ron River, Adi Shamir and Leonard Adlemen in 1977 at MIT.…

## Just some Maths of Finance fun

Friday the 31st of July 2015 was truly a day to be remembered as Crypto Giants was visited by none other than Prof. C.Labuschagne from the University of Johannesburg. Having been someone who has done research in the pure maths field for the better part of 20 years, Prof. C.Labuschagne now does research primarily in the mathematics of finance field which he believes has a lot more that is unknown when compared with pure maths. His presentation was truly remarkable as he informed us on some interesting facts on the mathematics of finance. These facts include that there exists an interaction between the maths of finance, probability theory (including stochastic processes) and functional analysis. We were also informed on subtle differences between functional analysts and mathematicians: functional analysts basically study the notion of distance (and what properties describe distance) whereas mathematicians study concepts such as order (e.g. partial order or ordered spaces).…

## Paying it Forward

Crypto Giant members together with the founder of Under Twenty Mathematicians programme, Professor Jean Lubuma participated in Mathematics awareness initiative that target high school Learners.  The aim of the project is to identify schools where there is a lack of information about tertiary education opportunities and careers in Mathematics, and to identify the learners which they can groom into pursuing a career in Mathematical Sciences.

In 2014, the group saw a need to pay it forward since they’d gained much from others. Thus they started tutoring classes for calculus and linear algebra to first year students every Saturday morning.  This is where they identified the gap that existed between high school mathematics and first year mathematics. Some students struggled with problem solving approaches, others weren’t even sure why they were there and most concerning was the deteriorating love of Mathematics. Some members of the group also identified with the struggles that the students were facing.…