This
post describes the mathematical principles behind the Freakazoid
cipher such that someone could recreate it with software if they
wished. For a targeted guide to some of the terminology used here,
please see my previous post discussing some fundamentals of encryption.

It is not meant to be a serious attempt at a cipher, and is presented as food for thought only. It was originally developed more than ten years ago as part of a semester of research as a math undergrad.

It is not meant to be a serious attempt at a cipher, and is presented as food for thought only. It was originally developed more than ten years ago as part of a semester of research as a math undergrad.

Freakazoid
is a cipher based on DES, an old cipher that was used as an industry standard for a time, but with the added step that a new key is generated each
block of data to be encrypted. The generation of this key is governed
by a master key. For clarity, we call a key applied to a particular
block of data the block key.

**Generation of Block Keys****Step**

**A**

**1**– Select a set of 'generating numbers' using the master key. In the implementation that was tested, a master key of 192 bits generated 12 such numbers by computing

2^x_1
* 3^x_2
* 5^x_3
* … * 47^x_15 53^x_16 , where x_k is
the kth bit out of 16 bits from the master key dedicated to this
particular generating number, and where the sequence (2,3,5,7,11, …
, 47,53) is the first 16 primes.

See Figure 1 for an illustration of this for two generating numbers.

Figure 1 - Step A1 |

**Step****A****2**– For i=1,2,...,12. Compute the square root of the ith generating number to y_i bits of precision beyond the whole number. In this case, y_i is the ith prime number greater than 100.
Note
that in most systems, a floating point sqrt() operating cannot
provide this much precision, so an array based method adapted from
the iterated subtraction method shown in this NIST document was used
instead.

The
non-integral bits from each of the generating numbers are used to
produce the block keys.

Figure 2 - Step A2 |

**Step**

**A**

**3**– To generate a block key of 64 bits, take a consecutive sequence of 64 bits from each of the 12 square root sequences, wrapping around to the beginning of each sequence if the end is reached.

After
12 sequences of 64 bits each are collected, update the starting
points in the sequences for the next block. For each of the 12
starting points s in the sequence, for each new block, apply (s + 64)
mod y_i to update the ith starting point.

For
the first block, take the first 64 bits from each of the square root
sequences. This is the only time in billions of blocks that the
starting points will be aligned so. Every starting point is updated
to bit #65.

For
the second block, the bits takes longer to describe.

We
would attempt to take bits 65-128 from the first sequence, but since
y_1 = 101, that sequence is only 101 bits long. So we take bits
65-101 and 1-27 instead. The first starting point is updated to bit
28.

Likewise,
we would take bits 65-103 and 1-25 from the second sequence. Starter
s_2 would be updated to be 26.

We would take bits 65-107 and 1-21 from the third sequence. S_3 would be updated to 22.

We would take bits 65-107 and 1-21 from the third sequence. S_3 would be updated to 22.

After
two blocks, the starting points would be 28, 26, 22, 16, 10, 2, 65,
65, 65, 65, 65 and 65.

After
four or more blocks, the starting points would be completely chaotic.

See Figure 3 for an illustration of Steps A3 and A4 for a simpler case of two sequences of length 3 and 5, respectively.

**Step A4**– Addition Step

The
nth bit of the block key is the XOR sum of the nth bits of
each of the 64-bit sequences.

Figure 3 - Steps A3 and A4 |

**Use of Block Keys for Encryption**These block keys are used in two encryption methods similar to those in DES.

**Step B1**– Addition Step

The
block key is added to the plaintext bit-by-bit with the XOR
operation.

**Step B2**– Rearrangement Step

There
are 8! = 5040 ways to permute the 8 bytes that make up a block. One
of these 5040 arrangements is selected by taking the first 16 bits of
the block key as P, and finding P mod 5040.

The
bytes of the block are rearranged according to that permutation.

**Step B3**– Substitution Step

The
64-bit block is now broken up into 4-bit nibbles. Each of these
nibbles can only take on 1 of 16 possible values (0000, 0001, 0010, …
,1110, 1111).

A
substitution table is created to replace every nibble of one value
with a nibble of another value. For example, if the permutation of 1-16 had '8' in the 3rd entry, then
every 3 or 0011 would be replaced by an 8 or 1000.

Every
nibble is substituted according to the table exactly once.

There
are 16! ~ 2.1 * 10^13 ways to set up such a table with a permutation of the integers 0 to 15, and we determine
which one is used by taking the modulo 16! of the remaining 48 bits
that were not used in selecting the rearrangement permutation.

The
block output from step B3 is our ciphertext.

Figure 4 shows a 256x256, 256-colour bitmap of plaintext and corresponding simulated ciphertext from DES and from Freakazoid. The vertical striping pattern is due to the same plaintext 64-bit (8-byte, 8-pixel) block being encrypted the same way every time. This behaviour shows up as stripes because the number of pixels in a row is a multiple of 8.

Figure 4 shows a 256x256, 256-colour bitmap of plaintext and corresponding simulated ciphertext from DES and from Freakazoid. The vertical striping pattern is due to the same plaintext 64-bit (8-byte, 8-pixel) block being encrypted the same way every time. This behaviour shows up as stripes because the number of pixels in a row is a multiple of 8.

Figure 4 - Plaintext, DES ciphertext, and Freakazoid ciphertext |

To
decrypt the block, the same block key needs to be generated.

From
there...

- The
table in step B3 is generated and the inverse substitution is applied
(e.g. turning all nibbles of 1000 or 8 back into 0011 or 3).

- The
permutation in step B2 is generated, and the inverse rearrangement is
applied.

- The
block key is added again with XOR to the block, as it was in step B1,
thus revealing the plaintext. (addition and subtraction are the same
with XOR)

**Hypothetical Questions and Answers****Q1:**Is repetition of blocks a big problem?

**A1:**No, not in a practical sense. Typically, large amounts of data would be compressed along with being encrypted before being sent through an public channel, like the internet. Any reasonable compression method should be able to minimize repetitions in the data.

Presumably that compressed-encrypted data would also be encoded with some error-identifying or error-correcting code, which would add some redundancy. Perhaps an attacker could find something about your encoding method, but it would be a bizarre case where the encoding method is worth keeping secret.

Furthermore, there are methods of 'padding' the data, such as adding a byte of random data to the end of every block, that drastically reduce the rate of ciphertext repetition in cases of plaintext repetition. So at best, Freakazoid's block key generation provides an alternative or additional layer to already existing padding schemes.

**Q2:**Are these square root sequences sufficiently random?

**A2:**Yes. Although in the presentation above, I show the first bits after the decimal point being used, in a real implementation, the first 100 such bits would not be used in a sequence. A similar burn-in is used for the blocks so that no block simply uses the first 64 bits from each sequence.

Furthermore, the choice of generating numbers is deliberate. Each of the 2^16 possible generating numbers is square-free, meaning none of them can be evenly divided by a square number. This feature prevents any square-root from being an integer multiple of any other.

**Q3:**If a new block key generated for every block? Won't they repeat?

**A3:**They will only repeat after trillions of blocks are used. The set of sequence lengths are all prime. More importantly, they are co-prime. When A and B are coprime, the sum of repeated sequences of length A and B only repeats after A*B bits are used. This is also true for three or more primes, so the sequence produced in Step A4 by the square roots only repeats after 101*103*107*...*251 > 10^24 bits, or about 1 trillion terabytes. If this is still not long enough, additional generating numbers can be used, and each number will increase the repeat time by a factor of at least 250.

**Q4:**Do you still have this? Can I have it?

**A4:**I might still have it, but it's been 10 years and 2 or 3 hard drives. My old external hard drive and maybe some cd-roms have it, but both of those media have been historically unreliable. If I find it, I will update this question with the link to the C++ code and executable. However, the code is such an awful primitive mess that it might be faster to recode it from the description given in this post instead of from the source code itself.

The pictures in Figure 4 were made with a simulation of encryption in R. These pictures look essentially the same as the ones in my original test case.

**Q5:**Was it secure?

**A5:**I think so, but I could be fooling myself.

At the absolute minimum, ciphertext should pass statistical tests for randomness, and it does.

Freakazoid has been thoroughly tested for randomness properties using
the Diehard Battery of tests (pun intended), and patterns could not
be found in many megabytes of encrypted data.

Furthermore,
compression programs like WinRAR identify and use patterns and
redundancies to make files smaller were utterly unable to make
Freakazoid ciphertext smaller. In my deliberately unfair test case of
a simple MS-paint image, the original image could be compressed to
about 2% of its size, the DES-encrypted image could be compressed to
about 4%, and the Freakazoid-encrypted image was compressed to about
101% of the original image.

Because
of its similarities to DES, such as the iterated summation and the
substitution boxes, it is expected that a single block of
Freakazoid-encrypted data is nearly as secure as a single block of
DES. For large files that may include repeated blocks of plaintext
data, Freakazoid outperforms DES in the sense that it will not reveal
the repetition in the ciphertext.

Freakazoid's substitution tables are procedurally generated, whereas the
substitution tables in DES are carefully selected to be particularly
strong against an attack called differential cryptanalysis. However,
since differential cryptanalysis depends upon an attacker being able
to encrypt a large number of blocks with the same key and the same
substitution table, the relative weakness of any one table is
presumably covered up by the fact that any given table is almost
never reused.

Finally, thanks to Reddit user /u/qhcf for the valuable feedback on this post.

Finally, thanks to Reddit user /u/qhcf for the valuable feedback on this post.