Happy $\pi$ day. We’ll begin with an obligatory video.

One of the reasons I enjoyed Summer Wars so much is because the main character’s superpower is math. Well, okay, you say, he’s really good at math, but so what? A lot of people complain about the implausibility of OZ, but those of us with a basic understanding of cryptography and number theory will have been drawn to Kenji’s quick problem solving work with an eyebrow raised. So let’s talk about why Kenji is a wizard.

We’ll start with modular arithmetic, which Kenji mentions to Natsuki on the train ride to Ueda. When we divide numbers, we often end up with remainders. Suppose we divide some integer $k$ by $N$ and we get a remainder of $r$. Then we say that $k$ and $r$ are equivalent $\bmod{N}$ and we denote that by $k = r \bmod{N}$. Because it’s how division works, for any integer $k, r$ will be some number from $0$ to $N-1$. It turns out a lot of arithmetic operations work the same way in modular arithmetic: adding, subtracting, and multiplying numbers and then taking the modulus of the result will give you the same number as adding, subtracting, and multiplying the moduli of the numbers you started out with.

However, division doesn’t work as we would expect it to. So we have to think about division (or the equivalent operation) differently. Instead of thinking of division as splitting a group of stuff into smaller groups, we’ll think of it as multiplying by an inverse. What’s an inverse? Well, we can try thinking of it in terms of addition. It’s pretty intuitive that subtraction is the opposite of addition. If we have some integer $k$, then the additive inverse of $k$ is $-k$. When we add $k$ and $-k$, we get $0$, the additive identity. The identity is just the special number that we can add to anything and get that same thing back unchanged ($n+0 = n$). In the same way, if we multiply $k$ by its inverse, $k^{-1}$, then we’ll get $1$, since $k \times 1$ is just $k$ again. What this means is that the inverse of $k \bmod{N}$ is just some other number $j$ from $0$ to $N-1$ such that $j\cdot k = 1 \bmod{N}$ and it’s just multiplication again.

Now, the problem with this is that it’s not guaranteed that there’s always an inverse hanging around in $\bmod{N}$. In particular, if $k$ and $N$ share any divisors, then $k$ won’t have an inverse $\bmod{N}$. This is interesting because it also tells us that if we consider integers mod a prime number $P$, then every integer $\bmod{P}$ has an inverse, since $P$ doesn’t share any divisors with any integers from $0$ to $P-1$. We call these things that have inverses units. So if we have a unit $k$, then $k^m$ is also a unit, for any integer $m$. We even have a funny function $\phi$ defined such that $\phi(n)$ is the number of units in $\bmod{n}$.

So taking everything we’ve learned, we can set up a cryptosystem! The one we’ll be looking at is called RSA, after the guys who invented it. We have Bob who wants to securely send a message $M$ to Alice. Alice chooses two prime numbers $p$ and $q$ and figures out $m = pq$. She also goes and figures out $\phi(m)$, which happens to be $(p-1)(q-1)$. Finally, she picks some integer $k$, a unit in $\bmod{\phi(m)}$. She lets everyone know $m$ and $k$, but she keeps $p$, $q$, and $\phi(m)$ secret.

So Bob wants to send $M$, which is just his message conveniently in number form. He makes $M$ into a number between $0$ and $m$, and if $M$ is too big, he can just break it up into chunks. Bob figures out the smallest $b$ such that $b = a^k \bmod{m}$ and sends $b$ over to Alice. Now, since Alice has $k$ and $\phi(m)$, she can also find $k^{-1}$ pretty easily. Once she has that, she can get the original message by figuring out $b^{k^{-1}} = (M^k)^{k{-1}} = M \bmod{m}$, since $kk^{-1} = 1 \bmod \phi(m)$.

The interesting thing here is that all of the information is out there for someone to encrypt a message to send to Alice, but no one is able to decrypt it. Well, they’re able to decrypt it if they know what $p$ and $q$ are, since once they’ve got that, they can get $\phi(m)$. But it turns out getting $p$ and $q$ from $m$ (which Alice just throws up on the interwebs) is really hard. And it really works for reals, because RSA is pretty widely deployed for things like keeping your credit card information safe while you send it through the tubes to Amazon.

Let’s go back and think about units some more. Of course, there are only $N$ numbers in the integers $\bmod{N}$, so there’s a point at which $k^m$ is just $1$ again and starts over. If $k^m = 1 \bmod{N}$, we say that $m$ is the order of $k$. But why do we care about finding the order of $k$?

It turns out finding the order of elements is very, very similar to factoring an integer into primes and other related problems, like discrete logarithms. If we can find orders of elements, it won’t be too hard to figure out how to factor a number. In this case, the eavesdropper wants to figure out what $p$ and $q$ are, so they’ll want to factor $m$. And it turns out a lot of other public-key cryptosystems (like elliptic curves) are based on the difficulty of factoring.

How hard could it be? Well, we could just check every possibility, which doesn’t seem that bad for a number like 48, but once we get into numbers that are hundreds of digits long, that might start to suck. It turns out the fastest known algorithms for order finding take approximately $e^{O(\log N \log \log N)^{\frac{1}{2}}}$ steps. Current key lengths for RSA are at least 1024 bits, which would give us about 4.4 x 10^{29} operations. Assuming three trillion operations per second (3 GHz), it’d take a PC about 4.7 billion years. Sure, you could just throw more powerful computers at it, but they’d just double the key size and suddenly, you’d need to do 10^{44} operations.

Well, that’s not entirely true. One of the breakthroughs in quantum computing was coming up with a fast algorithm for factoring. It turns out quantum order finding takes $O((\log N)^2 \log \log N \log \log \log N)$ steps, which, for a 1024-bit key is just over 60 operations. Doubling the key-size to 2048 bits only increases the number of operations by just over 20. Unfortunately (or fortunately, because we’d be pretty screwed if someone could easily break RSA right now), we haven’t built any quantum computers that large yet, nor are we capable of doing so anytime soon.

tl;dr – Kenji is a quantum computer.