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Ever wonder what happens when you send your card details over the internet? How exactly does public-key encryption work? I will write a brief, non-technical introduction to these concepts and the mathematical background in them, containing some sample code and plenty of examples.
Before we describe the RSA algorithm, there is one important mathematical concept, which is prime numbers and their factorization. Recall that a number is prime if no other number divides other than itself and . For technical reasons we exclude the number from being prime. So lets see some examples. Is prime? Well, yes, no other number other than and divide it. Is prime? No, because .
There is a fundamental theorem in number theory which says that every number can be uniquely written as a product of prime numbers, i.e. where are prime. So again, a few examples cannot hurt. Take the number . We know that , but now is not prime so . Hence . That’s what a prime factorisation is, and what the theorem says is pretty basic, if a number , then .
At this point now I can state what is the fundamental idea behind RSA:
Factoring a number into prime factors is much harder than checking if a number is prime!
But why is this true? There are technical reasons for this but I prefer to think along the following lines. Computers are much like humans, so imagine if a human is given the task of factorising numbers and checking if numbers are prime.
The trolley dilemma is summed in two parts as follows. Suppose that a trolley is running down a hill at a fast speed, heading towards five people at the bottom of the street. When it reaches them it will surely kill all of them. You notice that there is a switch next to you that could direct the trolley to a side path where there is one man standing and once you do, it will be the one man that dies. Would you do it?
Most people would answer this question with an affirmative. Let us call this the switch scenario. The second scenario is that a trolley is again running down a hill at fast speed, aimed at five people at the bottom which it will surely kill. However this time you are standing on a bridge with a fat man next to you. If you push the fat man off the bridge the trolley will stop but kill that fat man. Would you do it?
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This post is just to link Tim Gowers’s recent blog post on Plünnecke-type sumset estimates in groups because he discusses the recent work of his soon-to-be-former student Giorgis Petridis.
I had the pleasure of being taught (outstandingly) by Giorgis on more than one occasion during my undergraduate years and it was great to be in the audience for his recent talk on his work at the Isaac Newton Institute. His new papers are on the ArXiv here. Congratulations!
I thought I’d do a quick note just saying what’s next for Blame It On The Analyst.
Soon I will finish the current series on Infinity and The Continuum Hypothesis and term will begin! For me, this means I’ll be taking graduate courses, learning as much relevant mathematics as I can, teaching undergraduates and thinking about certain mini-projects. I expect that the content of the blog will shift slightly. It will have to in order to accurately represent what I’ll be spending my time thinking about. Unfortunately, this means that the ‘Mathematics For Everyone’ content is likely to slow down and there’ll be some of something along the lines of ‘Mathematics for Mathematics Students’.
I have made progress with the rest of my musings about Inception and they are nearly ready to go up! The series has become a three-post series because the second post was becoming enourmous, but I’ll be posting the next two posts in fairly rapid succession. The first of them will be coming up very soon.
I have returned from my absence and have started writing again.
I apologise for the lengthy break.
We were in the middle of talking about Inception and in the next post I was going to discuss the work of Kurt Gödel. Unfortunately, Inception is nowhere near as hot a topic as it was when I wrote the first post about it a few weeks ago and doubly unfortunately, my half-baked ideas about how the two link together in some way are (as is often the case with half-baked things) flimsier than I thought, not least because the idea of explaining Gödels theorems is in itself an ambitious one, let alone linking them to Inception! This said I do feel I ought to follow through with the second post, but I am not going to spend too much time on it as I am keen to move on to things more relevant to the mathematics I do, given that I’ve lost a lot of time, rather than over-indulge myself in pseudo-academic critique.
For now, I’d better get writing.
I’m afraid that something important has arisen and I will be unable to post for a few weeks. I look forward to resuming my discussion of Gödel’s theorems in September!
I recently saw Christopher Nolan‘s latest blockbuster Inception. I thought it was exceptional and it may well have jumped straight into my favourite films’ shortlist. But, imagine the compounded joy when I began to realise it had some links with one of my favourite books of all time, namely Douglas Hofstadter‘s incredible Pulitzer Prize-winning Gödel, Escher, Bach: An Eternal Golden Braid, hereafter GEB . Hofstadter’s book weaves together the works and ideas of these three men in a beautiful way, touching on many many themes throughout, the overall complexity of which is some sort of metaphor for or representation of “how cognition and thinking emerge from well-hidden neurological mechanisms”.
The analogies in this post are tenuous at best and the braid I weave is thin and frayed in comparison but it’s all fun, so let’s weave away.
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