In case this is way to old for some of you and don’t really know what is it about check this post: Visiting Stuttgart. Also, you can check the posts made for the first 2 days of the project with the HFTFirst day and Second day. And now it is the time for the 3rd day, and this is a Crypto day mostly, but it also has bits of finance; unfortunately I decided to leave it together in the finance  day, which will be the next one. This cryptography part was one of my favorite thing in all the week. A small thing about me : I will do cryptography as my project for my 4th year at university, so I am sure you get it.

We started with a little history, so in case you did not know, somewhere in 1940 Godfrey Hardy wrote an essay ‘Mathematician’s Apology’ where he motivated his interest in number theory with the fact that number theory was the only mathematical discipline which had absolutely no practical applications. Really strange for this now, because this is like one of the most important discipline in mathematics, but it was way back. Now modern cryptography is based on number theory. All this has some history in it, and it goes way back to secret messages and ciphers, but we mostly discussed about the more modern things, such as ‘Enigma’ machine. (The fun part is that I made a presentation on the Enigma machine in my 2nd year at university, if you would like to see it and read it leave a comment in the box below, it is really interesting, I assure you…). Going to our modern world we have online banking, internet shopping, secure mailing systems, telephone banking etc and it is essential to be able to secure messages and information from unauthorized outsiders.

So there are a couple of methods to use for encoding messages. Firstly we can talk about a secret words and simple shift of alphabet for the other letters we need, but unfortunately this is not a secure way because it is extremely easy to just try all the combinations and way you go, the answer is close by. And from this method there are things derived and in history there were a lot of derivations, but with a good eye and some frequencies of letters everything is solved quickly. A system which requires a secret key word known to both parties is called a symmetric encryption system. For more examples and a history approach check this: www.cryptool-online.org. But nowadays, we are not using this thing at all.

For the modern one we need modular arithmetic, or the clock arithmetic. I talked before about this in this post: Time. And this is like the key to everything, it is like the mother of modern cryptography, or like the alphabet for it… Extremely important. But there not everything is solved, we still have special things, such as: the discrete exponential problem (which is not feasible without a calculator, even if there are ways to make it bearable, there are still a lot of calculations) and the discrete logarithm problem. And from these things comes the well know and not that easy key exchange system by Diffie and Hellman. (Totally recommend reading this Wikipedia article about this: Diffie and Hellman key exchange; also this problem is explained in most Modular arithmetic lectures because students seems to love it and find it really interesting… I know that I was the same when I first saw it, fascinated).

If you want to play more with it, try this:  http://dkerr.home.mindspring.com/diffie_hellman_calc.html .

Though with all this modular arithmetic and with the nice concepts that we have today, it seems that we also need something better. So for this, Elliptic curves become more and more attractive for cryptography.  They require even more basics from pure mathematics (algebraic number theory and algebraic geometry). For a small description, elliptic curves are geometric objects with a group law. If the coordinates of the points are considered modulo p, the set of points forms a finite abelian group. Also take a close look to this lecture: An Introduction to the Theory of Elliptic Curves from the Summer School on Computational Number Theory and Applications to Cryptography from University of Wyoming.

Have a great week. Hope you liked this post, and if you would like to see more let me know. Any feedback is appreciated so use the like and share buttons and also comment below with your favorites, too.

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