Wednesday, September 21, 2016

RC4 stream cipher variants and visualization of table permutation state

Here, I present some work I have been doing on two RC4 stream cipher variants. The first variant, as seen below, I wrote to help me visualize and understand what the RC4 tables was doing, and help me understand its properties.

Identity Permutation

The class that contains it is called SimpleTable and is exactly that; The simplest R4C implementation possible. It is notable in the fact that it does not use key scheduling at all, and its starting state is that of the Identity Permutation. The identity permutation is where the value at index zero equals zero, the value at index one is one, and so on. An easy way to remember what the Identity Permutation is, just recall the notion of a Multiplicative Identity (which is 1), where by multiplying a number N by the Multiplicative Identity gives you back the value of N, also known as the identity. Similarly, the Identity Permutation of an array A just gives you A. This is the trivial permutation. That is, there is no permuting of the array at all!

Anyways, this is done to see the perfectly ordered state, and how each round effects that state. In this way, we can visually check for the avalanche effect. In order to visualize the table, i just assign each value 0 to 255 a different shade of grey (I also have a rainbow-colored option that might be easier to tell apart similar values). At each step I create a Bitmap by looping through the table. Below, you can find an animated GIF of the first 100 steps of this cipher being applied to the identity permutation:

Notice how it takes a while to get going, and the first several values don't move much at all. After 256 steps, or one round, the cursor arrives back at index zero. Because the location of the first several values have not moved much or at all, we can clearly see that a mere 256 steps is insufficient at permuting the state enough to avoid leaking the first part of your key. Therefore it it is vital to permute the table for several rounds (256 steps per round) before you start using the stream.

Wired Equivalent Protocol

As some of you may know, WEP used RC4 with a weak key schedule. The key is spread out over 256 bytes using the following approach:

j = (j + table[i] + key[i mod keylength]) % 256;
SwapValues(table[i], table[j]);

and then it began streaming bytes from the table. Typically a nonce is concatenated to the key. Every time the table is set up and/or the nonce changes, some information about the key is leaked. Obviously a more secure procedure would use a hash of the key and the nonce, instead of the plain-text key, and to toss away the first 1024 bytes or so.

Cycle Length

Because each step in an RC4 cipher is a permutation, there is a limit to the number of unique bytes that can be produced before it begins repeating. This is called the permutation cycle. The length of the permutation cycle depends on the exact starting state, but we can get an upper bound.

Since there are 256 elements in the array, and two indices into the array (i and j), there is a maximum of
256! * 256^2 = 5.62 * 10^512 = 2^1700
possible states. That's 4.6 * 10^488 yottabytes!

This is the maximum possible states, however, and other starting states could have less. If the RC4 algorithm performed as a random permutation (which it does not, it performs worse), the cycle length would be half of the theoretical maximum above. Luckily the number above is so vast, that even some faction of it is still so many bytes that all of humanity has never and likely will never have that much total storage.

One thing to watch out for, however is something called Finney States. If an RC4 is started in one of these Finney States, the length of the cycle is much, much reduced. The chance of randomly generating one of these starting states, however, is VERY, very low.

Strengthening RC4

As stated, and visualized, above, it is vital to permute the table for several rounds (at 256 steps per round) after the key schedule, discarding the bytes, before you start using the stream. Also, it would be foolish to use the actual bytes of the key for permuting the starting state. It would be instead better to use a hash of the key + nonce or a key derivation function from the key instead the actual value of the key itself.

Another idea is, after shuffling the table enough rounds to hide the key, scramble the table an additional number of rounds, that value being some function of the key. This increases the possible starting states by whatever your range is.

In the classic RC4, each step would return one byte. The number of steps taken before returning each byte is configurable in my implementation.

Memory hardening

Check out the experimental branch for a memory hardened version. It stores the key class in memory, with the key XORed with a one-time pad, and then is protected in memory from access with the System.Security.Cryptography.ProtectedMemory class.

Other uses

The pseudo-random byte stream from the RC4 table is deterministic. Therefore if two remote computers with a shared secret, both computers can independently set up an RC4 table with exactly the same starting state and will get the same sequence of bytes which would be difficult to guess, given just the stream of bytes. If the plain text is XORed by the pseudorandom byte stream, then it can be decrypted by XORing it by the same byte steam.

The project includes 2 variants: 1) A simple table with a method to visualize the permutation state of the table and the avalanche effect as a bitmap 2) A more serious attempt at a secure implementation.



Source code

Here is the GitHub page to the project (master branch).
Or just directly download the Zip file (experimental branch).