Threefish

Threefish works on words of 64 bits (unsigned Little endian integers). $w\; \backslash in\; \backslash \{4,8,16\backslash \}$ is the number of plaintext words and also of key words. The tweak consists of two words. All additions and subtractions are defined modulo $2^\{64\}$.

Threefish encrypts in $r$ rounds and uses $\backslash frac\{r\}\{4\}\; +\; 1$ different round keys. After every four rounds, and before the first, $w$ round key words are added to the $w$ data words. To calculate the round keys an additional key word $k\_w$ is appended to the original key words $k\_0,\; k\_1,\; \backslash dots,\; k\_\{w\; -\; 1\}$. Also, an additional tweak word $t\_2$ is appended to the tweak words $t\_0,\; t\_1$.

:$k\_w\; =\; C\; \backslash oplus\; k\_0\; \backslash oplus\; k\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; k\_\{w\; -\; 1\};\; \backslash quad\; C\; =\; \backslash text\{0x1BD11BDAA9FC1A22\}$

:$t\_2\; =\; t\_0\; \backslash oplus\; t\_1$

The purpose of the seemingly arbitrary constant $C$ is to frustrate some attacks that take advantage of the relationship between $k\_w$ and the other keywords.

The round key words $k\_\{s,\; i\}$ are now defined like this:

k_{s, i} =

\begin{cases}

k_{(s+i) \bmod (w + 1)} & i = 0, \dots, w - 4 \\

k_{(s+i) \bmod (w + 1)} + t_{s \bmod 3} & i = w - 3 \\

k_{(s+i) \bmod (w + 1)} + t_{(s + 1) \bmod 3} & i = w - 2 \\

k_{(s+i) \bmod (w + 1)} + s & i = w - 1

\end{cases}

Here $s\; =\; 0,\; 1,\; \backslash dots\; ,\; r/4$, where $4s$ is the number of the round in which the round key word $k\_\{s,i\}$ is used.

File:Skein Mix Function.png

The mix function takes a tuple of words $(x\_0,\; x\_1)$ and returns another tuple of words $(y\_0,\; y\_1)$. The function is defined like this:

y_0 = (x_0 + x_1) \bmod 2^{64}

y_1 = (x_1 \lll R_{(d \bmod 8), j}) \oplus y_0

$R\_\{d,\; j\}$ is a fixed set of rotation constants chosen to achieve quick diffusion.

The permutation step swaps the positions of the words according to a constant pattern.

Bit-level permutation is not achieved in this step, but this is not necessary since the MIX functions

provides bit-level permutations in the form of bitwise rotations.{{Citation needed|date=November 2015}} The Permute step and rotation constants

in the MIX functions are chosen in such a way that the overall effect is complete diffusion of all the bits

in a data block.{{Citation needed|date=November 2015}}

Because this permutation is fixed and independent of the key, the time needed to compute it does not provide

information about the key or plaintext. This is important because on most modern microprocessors performance

optimisations can make the time taken to compute an array operation dependent on where the data is stored in memory.

In ciphers where array lookup depends on either the key or plaintext (as is the case for the substitution step in AES),

it can make the cipher vulnerable to timing attacks by examining the time required

for encryption. The permutation is therefore deliberately designed to ensure that it should execute in the same fashion

independent of the key being used or the data encrypted.{{Citation needed|date=November 2015}}

• if $d\backslash ;\backslash bmod\backslash ;4\; =\; 0$ the round key $k\_\{d/4,i\}$ is added to word $i$
• the mix function is applied to pairs of words, the rotation widths $R\_\{d\; \backslash bmod\; 8,j\}$ depend on round number $d$ and word pair $j\backslash in\backslash \{0,\backslash cdots,w/2-1\backslash \}$
• the words are permutated using a permutation independent from the round number

Threefish256 and Threefish512 apply this round $r=72$ times ($d=0,1,\; \backslash dots,71$). Threefish1024 applies it 80 times ($d=0,1,\; \backslash dots,79$).

After all rounds are applied, the last round key words $k\_\{r/4,i\}$ are added to the words and the words are converted back to a string of bytes.

In October 2010, an attack that combines rotational cryptanalysis with the rebound attack was published. The attack mounts a known-key distinguisher against 53 of 72 rounds in Threefish-256, and 57 of 72 rounds in Threefish-512. It also affects the Skein hash function.{{cite journal

| last1 = Khovratovich | first1 = Dmitry

| last2 = Nikolic | first2 = Ivica

| last3 = Rechberger | first3 = Christian

| doi = 10.1007/S00145-013-9150-0

| issue = 3

| journal = Journal of Cryptology

| pages = 452–479

| title = Rotational Rebound Attacks on Reduced Skein

| url = https://eprint.iacr.org/2010/538

| volume = 27

| year = 2014}} This is a follow-up to the earlier attack published in February, which breaks 39 and 42 rounds respectively.{{cite conference

| last1 = Khovratovich | first1 = Dmitry

| last2 = Nikolic | first2 = Ivica

| editor1-last = Hong | editor1-first = Seokhie

| editor2-last = Iwata | editor2-first = Tetsu

| contribution = Rotational Cryptanalysis of ARX

| contribution-url = https://scholar.archive.org/work/mq7kxdfdmvh53numspafm2of6a

| doi = 10.1007/978-3-642-13858-4_19

| pages = 333–346

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Fast Software Encryption, 17th International Workshop, FSE 2010, Seoul, Korea, February 7–10, 2010, Revised Selected Papers

| volume = 6147

| year = 2010}} In response to this attack, the Skein team tweaked the rotation constants used in Threefish and thereby the key schedule constants for round 3 of the NIST hash function competition.

In 2009, a related key boomerang attack against a reduced round Threefish version was published. For the 32-round version, the time complexity is $2^\{226\}$ and the memory complexity is $2^\{12\}$; for the 33-round version, the time complexity is $2^\{352.17\}$ with a negligible memory usage. The attacks also work against the tweaked version of Threefish: for the 32-round version, the time complexity is $2^\{222\}$ and the memory complexity is $2^\{12\}$; for the 33-round version, the time complexity is $2^\{355.5\}$ with a negligible memory usage.{{cite conference

| last1 = Chen | first1 = Jiazhe

| last2 = Jia | first2 = Keting

| editor1-last = Kwak | editor1-first = Jin

| editor2-last = Deng | editor2-first = Robert H.

| editor3-last = Won | editor3-first = Yoojae

| editor4-last = Wang | editor4-first = Guilin

| contribution = Improved Related-Key Boomerang Attacks on Round-Reduced Threefish-512

| contribution-url = https://eprint.iacr.org/2009/526

| doi = 10.1007/978-3-642-12827-1_1

| pages = 1–18

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Information Security, Practice and Experience, 6th International Conference, ISPEC 2010, Seoul, Korea, May 12–13, 2010. Proceedings

| volume = 6047

| year = 2010}}

• Twofish
• Blowfish (cipher)

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