A Toy Keyless Encryption Protocol

Cryptography is a natural application of number theory and so I'd like to write down a few thoughts about it in this blog. (The fact that there are real world applications to number theory deserves some appreciation in itself, but it would throw us too much off track here.) One particularly nice feature of cryptography is the ability to explain its inner workings with real world analogies about security.

For instance, one way two parties (who we, by convention, call Alice and Bob) could hide their secrete communication is if Alice writes a letter, puts it in a box, and locks it with a padlock for which both she and Bob have a key, but no one else. Then Alice can send this box safely through any public means as anyone who intercepts it will not be able to open the box, rendering it useless to them. This is the principle behind symmetric cryptography. The obvious problem is that Alice and Bob will only be able to communicate if they managed to obtain identical keys beforehand.

Alternatively, Bob could distribute padlocks for which only he possesses the key to anyone who's interested. Now Alice can obtain such a padlock, use it to close the box with her letter and send it off to Bob. She won't be able to open the box, but neither will anybody else -- except Bob. It sounds tedious to ship padlocks for every single communication, but the advantage is that there's no need whatsoever for Bob to restrict distribution to trusted parties, making it possible for anyone to send him messages securely. This is the basis of public key cryptography (a field which is extremely number theory heavy -- a course about it will feature extensively theorems on primes, groups, elliptic curves, lattices, and many more really abstract concepts).

But there is yet another way parties could communicate securely. Let's imagine Alice uses a lock for which only she has a key. She ships this of to Bob who cannot unlock it himself -- instead, he'll add another padlock himself and sends everything back to Alice. She couldn't open it herself anymore, but she can remove her padlock and still send it off to Bob once more knowing no one except Bob will be able to open her secret. Finally, Bob opens his lock and can read Alice's message. No need to meet up and share keys, no shipping padlocks. Continue reading A Toy Keyless Encryption Protocol

Prime Generating Sequences

A couple of months ago (really, a long, long time ago) I found an interesting question on Mathematics Stack Exchange (another site to effectively waste away hours of your life). It reminded me of my Bachelor's thesis (which I wrote a really, really long time ago) about the sequence

where $a_1=7$ and

Here, $\mathrm{gcd}(a,b)=(a,b)$ stands for the greatest common divisor,1 i.e., the largest integer $d$ that divides both $a$ and $b$. This may not seem terribly interesting at first sight, but if you look at the first few values for $g_n$ you'll notice something curious:

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101...

There are for sure a lot of ones in there, but other than that, all the numbers are primes. This is not a bias in the first few example -- Eric Rowland proved that all values of $g_n$ are either $1$ or a prime in a beautiful little paper back in 2008. Continue reading Prime Generating Sequences

1. It's common to abbreviate gcd(a,b)=(a,b) in number theory, and I shall do so in the remainder of the article. Similarly, it's convention to write lcm(a,b)=[a,b].

Applying the Explicit Formula

It's quite some time since we arrived at Riemann's main result, the explicit formula

where $J(x)$ is the prime power counting function introduced even earlier. It's high time we applied this!

First, let's take a look at $J(x)$ when calculating it exactly:

In the Beginning, There Was... Euler's Formula!

I will start this blog the way Bernhard Riemann started his paper: with Euler's product formula that John Derbyshire called the golden key:

This holds for any complex number $s$ with $\Re s > 1$. If you look up a proof in any modern textbook, you will find a number technical rearrangements that end up in an examination of the absolute convergence on both sides. But actually, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes. Let's start by writing out the sum on the left hand side: Continue reading In the Beginning, There Was... Euler's Formula!