### Forward security for journalists (03 Dec 2013)

*A number of journalists have been writing stories about forward security recently. Some of them ended up talking to me and I completely failed to come up with a good metaphor that didn't cause more problems than it solved. Eventually I wrote the following as an introduction to forward security for journalists. It's *deeply* simplified but I think it captures the broad strokes.*

*It remains unclear whether anyone actually came away with a better understanding after reading it, however.*

A great many metaphors have been mangled, and analogies tortured, trying to explain what forward security is. But, fundamentally, it doesn't involve anything more complex than high-school math and anybody should be able to understand it. So this document attempts to describe what forward security *is*.

In the 1970s, cryptography was purely the concern of the military. The NSA existed (unofficially) and believed that it was the sole, legitimate center of cryptographic knowledge in the US. In this environment, Whitfield Diffie set out from MIT across the country to try and gather up what scraps of cryptography has escaped into the public realm. If we omit huge amounts of detail (see Steven Levy's book, “Crypto” for the details) we can say that he ended up at Stanford with a revolutionary idea in his head: public key cryptography.

Up to that point, cryptography had involved secret keys that both the sender and receiver were in possession of. Diffie wondered about a system where one could publish part of a key (the *public* part) and keep another part of the key private. However, he didn't have any concrete implementation of this.

At Stanford he found Martin Hellman and, together they published *New Directions in Cryptography* (1976), which contained the Diffie-Hellman key-agreement algorithm. This is possibly the most significant work in cryptographic history and is still a core algorithm today. (Later we learnt that a researcher at GCHQ had discovered the idea a few years earlier. But GCHQ didn't do anything with it and never published it.)

We'll need modular arithmetic, which is like working with times. What's 8pm plus five hours? One o'clock. Because 8+5=13 and 13 divided by 12 has a remainder of one. Likewise, 8pm plus 17 hours is still one o'clock, because adding multiples of 12 doesn't make any difference to a clock. We'll be using the same idea, but with numbers (called the *modulus*) other than 12.

To explain Diffie-Hellman we'll work modulo 11. So 5+10=4 now, because 15 divided by 11 has a remainder of 4.

Next, exponentiation: 2^{3} means 2, multiplied by itself three times. So 2^{3}=2×2×2=8. Now we have everything that we need to implement Diffie-Hellman.

In classic, cryptographic tradition we'll imagine two parties and we'll call them Alice and Bob. Alice generates a random number between 1 and 9 and calls it *a*. This is her *private* key; let's say that it's five. Alice calculates her public key as A=2^{a}=2^{5}=32=10 mod 11.

Bob does the same but his private key is called *b* and his public key is B. Let's say that his private key is six which means that B=2^{b}=2^{6}=64=9 mod 11.

Alice and Bob can freely publish A and B because, given them, no other party can work backwards and calculate their private keys. Well, in this case they can because we're using a toy example with a tiny modulus. A real modulus has over 500 digits. *Then* it's not possible with any current computer.

Once Alice has Bob's public key, or vice-versa, she can perform *key-agreement*. She combines her private key with Bob's public key like this: B^{a}=9^{5}=59049=1 mod 11. Bob can combine his private key with Alice's public key in the same way: A^{b}=10^{6}=1000000=1 mod 11. Both Alice and Bob arrived at the same, shared value (one) which nobody else can calculate. And without exchanging any secret information!

This was a huge breakthrough but it still left something to be desired: signatures. It seemed that it would be very useful for Alice to be able to calculate some function of a message that could be broadcast and would serve as a *signature*. So it had to be something that only Alice (or, rather, the entity holding the private part of Alice's public key) could have generated. Diffie-Hellman doesn't provide this.

In 1977, three researchers at MIT, Rivest, Shamir and Adleman, produced such a scheme. Now it's called by their initials: RSA. Again, the basics are high-school math.

We'll be working with modular arithmetic again but this time it'll be modulo the product of two primes. Like the example above, the numbers involved are usually huge, but we're going to use tiny numbers for the purposes of the examples.

So our two primes will be p=11 and q=17. This will serve as Alice's RSA private key. In order to calculate an RSA public key, one calculates n=11×17=187.

The core of RSA is that, when working modulo 187, it's very easy to calculate the cube of a number. But, unless you know that 187=11×17, it's very difficult to calculate the cube *root* of a number. Let's say that our message is 15. The cube is easy: 15^{3}=3375=9 mod 187. But to calculate the cube root you need to calculate the modular inverse of 3 and (p-1)(q-1)=10×16=160. We're getting a little past high-school math here so trust me that it's 107. By calculating 9^{107} mod 187 we find that we get our original number again: 15.

Think about what this means. If Bob knows that Alice's public key is 187, he can cube a message and send it to Alice and only Alice can calculate the cube root and recover the message. Also, if *Alice* starts with a message, she can calculate the cube root of it and publish that. Everyone else can be convinced that only Alice could have calculated the cube root of the message and so that functions as a digital signature of the message!

Now we have everything that we need to understand forward security so hang on while we wrap it all up.

RSA and Diffie-Hellman are both too slow to use for large amounts of data. Bulk data transfer uses *secret-key* cryptography: the old style cryptography where both sides need to know the secret key in order to process the ciphertext. But, in order to get the secret key to the other party, RSA or Diffie-Hellman is used.

Traditional HTTPS has the browser use RSA to transmit a secret key to the server. The browser picks a random, secret key, cubes it modulo the server's RSA modulus and sends it to the server. Only the server knows the two numbers that, multiplied together, equal its modulus so only the server can calculate the cube root and get the secret key for the connection.

However, if anyone were to record the HTTPS connection and later learn those two numbers (called *factors*) then they can retrospectively decrypt the recorded connections. They can also watch new connections to the server and decrypt those because they can calculate cube roots just as well as the real server can.

With forward security, the method of establishing the secret key for the connection is changed. When a client connects, the server generates a brand new Diffie-Hellman private key just for that connection. (It's called the *ephemeral* private key.) Then the server calculates its Diffie-Hellman public key (by 2^{a} mod m*, *remember?) *signs* the public key with its RSA key (by calculating the cube root of it, modulo its RSA modulus) and sends all that to the browser. The browser verifies the signature (by cubing it and checking that the result equals the public key) and then generates its own Diffie-Hellman private and public keys. Now, the client has the server's Diffie-Hellman public key, so it can combine it with its Diffie-Hellman private key and calculated the shared-secret. It sends its Diffie-Hellman public key to the server and the server ends up with the same, shared secret. That secret is used as the secret key for the connection.

The advantage here is that, should the server's, long-term, RSA private key be compromised in the future it isn't as bad. The confidentiality of the connections is protected by the Diffie-Hellman secrets which were generated fresh for the connection and destroyed immediately afterwards. Also, the attacker, even with the RSA private key cannot passively decrypt connections in real-time. The attacker has to impersonate the server, which is much more work.

That's why forward-secrecy is often written as DHE: it stands for Diffie-Hellman ephemeral. There's also ECDHE (Elliptic Curve, Diffie-Hellman ephemeral) which is the same idea in a more efficient mathematical structure.