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Introduction to cryptography - RSA

It has been a while since I've written a blogpost, but I find it interesting that everyone hangs out at RSA conference but so few know how RSA really works.
In this blogpost I'll try to explain why RSA is so powerful, as well as explain the math behind its basic form.

Motivation

Until this point, we have described several cryptosystems that use the same key for both encryption and decryption. Those are known as Symmetric encryption schemas.
The great disadvantage is that two parties (traditionally, Alice and Bob) have to exchange a secret somehow (the key), which is a non-trivial task assuming distances.
For instance, when you browse, you most commonly use the HTTPS protocol, which is just http over TLS. However, you never "met" with the web server and exchanged a secret.
This emphasizes the necessity of exchanging secrets securely without physically meeting, and while key-exchange is a slightly different problem, it's completely solvable assuming an Asymmetric encryption schema.
In short, an Asymmetric encryption is a cryptosystem in which decryption and encryption use different keys. Traditionally we call one a private key and that is known only to one party (who owns the key) and the other is known as a public key, which is, well, public.
Assuming such a system, it's easy to exchange a secret:

1. Alice publishes her public key, anyone is able to use it, but keeps her private key to herself.
2. Bob creates a random secret and uses Alice's public key to encrypt it.
3. Bob sends encrypted secret to Alice, and Alice decrypts it with her private key.
4. Now Alice and Bob both know a joint secret, and can start using Symmetric encryption.

The main takeaway from these steps is that Bob sends the encrypted secret, and because only Alice has the private key, only Alice can decrypt the message. There are several challenges with that (e.g. how does Bob know that the public key indeed belongs to Alice) but those are solvable (in some sense, I will discuss that in a later blogpost).
The important part is that Asymmetric encryption could be used for key-exchanges.
Of course, the main challenge is - how to create such a cryptosystem? Well, using math.

The algorithm

RSA works within the multipicative group mod n, which contains all the coprimes to n, with a multipication operation mod n.
If you don't know what those are then I refer you to my previous blogposts, and specifically modular arithmetics which we will be using a lot today.
Well, how does the algorithm work? Let us first describe the algorithm before explaining how it works:

1. Alice creates two random large primes, p and q (how to create a "random" prime? I will discuss that in a later blogpost), and defines n = p*q. Note that "large" is a relative term, today 2048 bit values are considered "safe" (but not Quantum resilient).
2. Alice uses a number that we'll call e (that is not the e, remember, we are working with integers mod n!). In reality, e is really constant and "baked into" the cryptosystem (commonly 65537, I will explain why later).
3. Alice calculates a number d such that d is the multipicative inverse of e mod phi(n), where phi is the Euler Totient Function. Alice could calculate phi(n) directly: phi(n) = (p-1)(q-1).
4. Alice publishes her public key = (e, n) and keeps her private key = (d, n), and gets rid of p and q.
5. At this point, for every x in the multipicative group mod n, the following is true: (x^d)^e = (x^e)^d = x (mod n), where ^ is exponentiation.
6. When Bob wants to use Alice's public key to encrypt such a value x, all he does is send x^e (mod n).
7. Alice can then decrypt Bob's message by raising it to the d-th power (mod n), thus getting x back.

This sounds rather complicated but is actually quite simple. Let us explain how values are picked.

The math

So, why does RSA work? This question could be answered if we understand how the values are picked.

1. As I said, p and q are chosen randomly. There are good well-known algorithms for creating a random prime numbers (they are not trivial), but let us assume that is known. Therefore, n = pq simply ensures that n is a composite number but is computationally hard to factorize.
2. The value of phi(n) is simply the number of elements in the multipicative group Z*n, and is easy for Alice to calculate but hard for anyone else. Since the Euler Totient function is multipicative, phi(n) = phi(p) * phi(q), and since for every prime phi(p) = p-1 we get phi(n) = (p-1)(q-1). Note that an attacker that can calculate phi(n) can crack RSA; it is believed today that the problem of calculating phi(n) is equivalent to the problem of the factorization of n, and that is believed to be a computationally difficult problem (for a large value of n).
3. As I mentioned, e is really constant, but it has to be coprime to phi(n). In most cases it'd be 65537, since it's a prime number it's very likely for it to be a coprime of phi(n), as well as the fact that it's a power of two plus one (65537 = 2^16+1), so it's very efficient to use it in exponentiation. I have seen cases where e=3, but it's not commonly used and might actually pose security issues, for example, for small values of x it's trivial to get x out of x^e (mod n).
4. The value of d can be efficiently determined using the Extended Euclidean algorithm, since Alice knows phi(n). Again keep in mind that an attacker that can determine phi(n) can conclude d from it and therefore break the entire cipher.

Why does (x^d)^e = (x^e)^d = x (mod n)? Well, according to Lagrange, every the size of a sub-Group divides the size of the original Group. For example, the only possible sub-Groups of a Group of size 15 must be of sizes 3 or 5 (without including the trivial cases of 1 and 15).
Now, let's think of some x that belongs to a finite Group G, and let's start looking at its powers: x^0, x^1, x^2, x^3, .... Since this can go on forever but the size of G is finite, it means that at a certain point we'll get x^a = x^b for some integers a and b. This means that x^(a-b) = 1 (within Group G and its operation), and that means that there's some power of x c such that x^c = 1.
The lowest integer (excluding 0) that maps x^c to 1 is called the order of G and marked as c = ord(G).
Well, the order of a Group always divides the Group size, which means x^n = x^(cm) = (x^c)^m = 1 for some integer m.
To conclude, for every Group G with k elements, if a = b (mod k) then x^a = x^b. In our case:

• k is really phi(n).
• ed = 1 (mod k (because that's how we chose d).
• Therefore, (x^e)^d = x^(ed) = x^1 = x.

Why not using just RSA then?

Why should Alice and Bob not just continue conversing over RSA? Because there are several disadvantages:

1. RSA is computationally expansive. Raising large numbers to the power of large numbers mod large numbers takes non-trivial processing power.
2. You can't use any x as input to RSA - you can only use x that is coprime to n - that is a very large set but not arbitrarily large.
3. Key sizes for securing RSA are huge compared to symmetric ciphers - common AES (symmetric) key is around 256 bits, while modern RSA requires around 2048 or even 4096 bits.
4. Unlike symmetric ciphers, that are relatively okay, RSA is not Quantum-resistent. This is not a Quantum computation blogpost (and I am not an expert on the matter), but using Shor's algorithm assuming a Quantum computer (which is a big assumption nowadays) is very promising for breaking RSA by means of factorization; on the other hand, symmetric ciphers are quite safe - the best known Quantum attack against them is Grover's algorithm which can be solved by just doubling the size of the (already small) symmetric key size.

Summary

In this blogpost I've described how basic RSA works; in truth, there are more sophisticated versions of RSA but they still use the same fundemental principals.
There are many uses for RSA even today, and I wouldn't exaggarate if I said it's one of those fundemental algorithms that make the internet as we know it today, possible. It's also used extensively for digital signatures, which I plan on covering in the future.
I still owe a lot of future blogposts - from attacks on RSA to interesting uses, as well as random prime number generation, there's a lot to cover!

Stay tuned!

Jonathan Bar Or