## BlockCipherSelfStudy.jl

Https://www.schneier.com/paper-self-study.html
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November 2013

# BlockCipherSelfStudy

## Introduction / Apology

Modern cryptography has moved on from block ciphers. It's now all about formal proofs of more complex systems. So this is just me pootling around in my free time, following Bruce Schneier's self-study course.

## RC5 Without Rotation

Defined here, RC5 is a block cipher that uses addition, xor, and plaintext-dependent rotations (although the amount of rotation cannot be determined from the plaintext alone).

It is very configurable - the size of half-blocks, the number of rounds, and the key size can all be varied. Here, in addition, to reduce strength, we disable rotations.

### State - 0 Rounds

Well, this is very easy. A plaintext of 0 gives you the state.

### State - 1 Round

An adaptive, chosen plaintext attack that reverses the maths, step-by-step, to retrieve the internal state. Getting the value of "the xor state" was tricky - I eventually realised that comparing the results from encypting two values, differing only in one bit, would (often) given the necessary information.

### Plaintext - Any Rounds, Lowest Bits

The lowest bits in each half-block can be tabulated independently of the rest of the bits (taking the two halves as a single pair). A single byte is very easy, giving rapid decryption of two bytes per block.

### Plaintext - Any Rounds

Extending the above, an adaptive attack (requiring about two blocks per bit) can search for the plaintext, bit by bit. This works because the only mixing between bits (without rotation) is via carry in additions. So there are only 4 combinations of lowest bit (for the two half-blocks) that affect the lowest bit. Then four more for the next, and so on.

### State - 8 bits 2 Rounds / 32 bits 1 Round

A GA search that finds the state. This weights scoring of successfully translated half-blocks to build the state from the lsb and targets mutations at the least significant incomplete bit. So, for example, if all half-blocks have the first 3 bits of a plaintext encrypted correctly, scoring and mutation target the fourth bit, with some mutation at lower bits for carry.

### State - 32 bits 4 Rounds

A DFS that finds the state. This searches from least to most significant bit.

Back-tracking for the first 2-4 bits dominates processing time. Once those bits are OK, typically, no further backtracking at that level is necessary and more significant bits are found rapidly and (relatively) independently. I do not understand why - perhaps it is a bug, or perhaps it is simply that those bits cascade more (so there is some kind of geometric of exponential dependency on their values). Adding a "beamwidth" limit to the search, or inverting or reversing the bits tried, does not help.

### State - 32 bits 5 Rounds

As above, but using tabulated results.

Efficient pruning of the search is critical. This is why so much time is spent on the first few bits (see above) - because it is difficult to discriminate good and bad answers at this level. The approach here uses an adaptive set of filters, updated every few seconds.

A solution for the 16-byte zero key, found after 1/4 of the state space is explored, takes ~100 minutes.

### Linearity

RC5 without rotation is not linear. In other words, whether `.` is addition or xor,

``````p, q = encrypt(a.c, b.d)
r1, s1 = encrypt(a, b)
r2, s2 = encrypt(c, d)
p != r1.r2 && q != s1.s2
``````

The source of the non-linearity is the combination of addition (with carry) and xor. If you describe a round as addition over words then the xor is non-linear; if you describe it as addition over bits (ie xor) then the carry operations are non-linear.

Yet various places assert that RC5 without rotation "is linear".

If linearity is taken to mean, loosely, that a solution can be composed from smaller parts, then the only way that RC5 without rotation is linear, as far as I can tell, is that the lowest bit is independent of other bits. This leads to attacks which progressively solve "upwards" from the least significant bit, as described above.

The diagram below shows how carries ripple out when the least significant bit of one half-block is changed (only half the encryption process is shown):

``````b a+s axb a+s axb a+s axb a+s a
0       2       4       6

0 0+0 0x0 0+1 0x0 0+0 0x1 1+0 1
0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1
0 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0
0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0
0 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0
0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0
0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0

0
0 1
1 1   0
1 1   1
1   1   1
1 1   0 1
1 0
1   1   1   0 0       1 1   1
``````

At the same time, my limited understanding of linear and differential attacks suggests that RC5 without rotation is, in a sense, "too linear". I can't find a way to relate "distant" bits without also considering key expansion. But this may be my inexperience, or simply laziness (perhaps key expansion must be included).

## RC5 With Rotation By Round

As above, but the first round rotation is 1 bit; the second round 2 bits; etc.

### Plaintext - 5 Rounds

An adaptive attacck, searching for the plaintext.

The influence of the first 4 bits on a randomly chosen key, with 5 rounds, is shown below:

`````` 0 444433211111000>^<8766667666665665
1 5444332211100000>^<866666667766565
2 55545543322210010>^<87666666667666
3 665654433332111000>^<8666766667666
``````

The output bit marked with `> <` is at `r(r+1)/2` - the cumulative shift position - and most influenced (`^`; digits are 10% units relative to the peak) by changing the input.

Clearly the influence of each bit is restricted to a a range of output bits at and "above" the rotation. So for each ciphertext character we can try using a search over a limited number of bits.

In practice many character / key combinations can be found at 5 rounds, but not at 6. 