Analysis of an 8 bit version of the cipher at
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October 2013

The 8 Bit "Stupid" Cipher

An informal analysis of an 8 bit version of this cipher, as discussed on HN.

The cipher implementation can be seen in Cipher.jl.


Motivation - it seems that the only way to learn about implementing ciphers is to break them. At the same time, the ciphers that I know are ridiculously hard to break. So breaking a "known bad" design might be a good first step.

Structure - this page contains only a basic summary of the code and results. There are links to source files, which may contain more information in the comments.

Language - Julia combines the speed of C (or close) with the flexibility of Python and has ambitions to replace statistical analysis packages like R.

Why 8 bit? - other stream ciphers work with 8 bit characters, so I can re-use any tools I develop. In some ways it may also simplify analysis (less state for a given key size, but for a given sized state, less information is exposed when a character is encrypted).

Non-Uniform Bytes And Bits

All results below are for a plaintext size of 659408 bytes (the size of the Little Brother text). The tests used Pearson's Chi-Squared test to assess whether bytes and bits were uniformly distributed (bytes from 0 to 255; bits individually over 0 and 1).

Ciphertext is significantly different from random bytes when the plaintext is constant (0x00, 0x55 or 0xff) for key lengths of 3 and 8 bytes. The least significant bits are particularly unreliable. 16 byte keys give better output, but some keys are still problematic.

When encrypting random data the output does appear to be (consistent with the hypothesis that it is) random, at all key sizes.

When encrypting text the data for 8 and 16 byte keys also appeared to be random.

The analysis can be seen in Statistics.jl.

Correlated Bits

The small state and xor-logic of the PRNG suggests that ciphertext for constant plaintext will vary less than expected. This can be measured as the mean number of bits common to successive ciphertext characters. A random stream will give 4 bits.

For 3 byte keys, all constant plaintexts give anomalous values.

For Little Brother plaintext, 3 and 8 byte keys give anomalous values.

The analysis can be seen in Statistics.jl.

Plaintext Injection Attack

In some cases, injecting a pre-calculated fragment in the plaintext can force the internal state of the cipher to a known point (excluding the internal counter, which is known anyway). Following text can then be decrypted directly.

A practical example where this can be used is the encryption of a web page that displays user-supplied data (like a name or comment).

The fragment is a counter (modulo 0xff) that mirrors the counter in the cipher state. The probable mechanism is a combination of the counter plaintext canceling some (or all) counter bits and zeroing of state when a byte is xored with a similar value (the cipher character is already xored with state[A], it is then xored again with state[A] when mixed into the state).

For 3 byte keys, a 32 byte fragment affects 4% of keys. For 4 byte keys a longer fragment (150 bytes) is necessary to affect a similar percentage.

Even when the known unique state is not achieved (including larger key sizes), counter fragments significantly reduce the cipher state. In a random sample (size 100) of 8 byte keys encrypting a counter all had state (excluding the internal counter) that repeated over unexpectedly short periods. The longest period was 26 characters and 71% achieved stationary state (period 1) after 1500 characters or less.

The analysis can be seen in Prefix.jl.

Ciphertext Collisions (1)

When Little Brother is encrypted with 100 distinct, random keys, of length 3 bytes, the endings of ~30% of the files are not unique.

A practical example where this can be used is against disk encryption, where the same file can be identified for multiple users.

The probable mechanism is that initial state is slowly eroded by similar-byte xor.

The analysis can be seen in SelfEncrypt.jl.

Distinguishing Attack

Ciphertext bitwise cross-correlated with a counter (modulo 0xff) generally shows clear structure for 3 byte keys. Typically a peak or trough at ~0 offset with other features at 64 and 128 bytes offset. Similar structures also appear to be visible (varying by key; statistical significance unclear) with 8 byte keys.

A practical example where this can be used is detecting the use of this (weak) cipher in data of unknown origin.

The probable mechanism is matching the internal counter.

Here is the correlation for Little Brother, encrypted with key 0xacb89d:

counter correlation for 3 byte key

The analysis can be seen in BitCorrelation.jl.

Statistical Plaintext Matching

There is a small but consistent bitwise (anti-)correlation between the lowest two bits of plaintext and ciphertext for 8 byte keys at zero offset.

A practical example where this can be used is when a user is suspected of encrypting a particular document.

The probable mechanism is that these bits control permutation when key length is a power of 2 (the calculation is modulo key length).

Here is the correlation for 5 random 8 byte keys, encrypting Little Brother, with the zero offset points marked in a distinct colour.

plaintext correlation for lowest 2 bits with 8 byte key

The analysis can be seen in BitCorrelation.jl.

Partial Known Plaintext Attack

If the initial character of the plaintext is known then the key space is reduced by 8 bits.

A practical example where this can be used is when encrypting UTF-8 text, which has a known header.

The attack works because the ciphertext is xored with the plaintext and the internal state; knowing the plaintext constrains the internal state.

The analysis for 3 byte keys can be seen in BruteForce.jl.

Ciphertext Collisions (2)

Around 1% of 3 byte keys have siblings that give matching encryption results. For example, both 0x25730e and 0x92ef1c give the same ciphertext for 100 random 16 byte strings.

This implies that the cipher is less secure than would be expected from the key length.

I do not know the mechanism, but this may be related to the "missing logic" in the code used in the partial plaintext attack (the routine is less symmetric than expected because some combinations of pos values cannot occur).

The analysis can be seen in BruteForce.jl.

Limited State

The Cipher can be viewed as a stream cipher using "xor with a PRNG" whose state is augmented with entropy from the plaintext. One way to measure the diversity of the PRNG (and hence the security of the cipher) is to look for repeated internal state.

In Statistics.jl the internal state (excluding the counter, which is known to an extrernal attacker) is monitored during encryption. If a state is detected a second time then encryption continues and the number of characters between the second and third occurence of that state is recorded. This is repeated for 100 random keys.

A good quality PRNG would not repeat internal state until all available states had been exhausted (ie period = pow(2, N) where N is the size of the state in bits). But a generator that incorporates external input might not be able to achieve that. Birthday collisions might limit the period to pow(2, N/2). For a 3 byte key that would be 4096; for a 16 byte key ~2E19.

In practice, for 3 byte keys, the state repeats every 400 characters or so. This is true even for random plaintext.

8 and 16 byte keys are better - in 100 tests only one 8 byte and no 16 byte keys had repeating state over 10000 plaintext characters.


Code copyright Andrew Cooke 2013, licensed under the GPLv3.

Little Brother copyright Cory Doctorow, licensed under CC ANCS3 (see file) (it's used as example English text for analysis, but it's actually a great read - probably more interesting than this project).