Threefish

General
Designers	Bruce Schneier,Niels Ferguson,Stefan Lucks, Doug Whiting,Mihir Bellare, Tadayoshi Kohno,Jon Callas, Jesse Walker
First published	2008
Related to	Blowfish,Twofish
Cipher detail
Key sizes	256, 512 or 1024 bits (key size is equal to block size)
Block sizes	256, 512 or 1024 bits
Rounds	72 (80 for 1024-bit block size)
Speed	6.1cpb onCore 2.[1]
Best publiccryptanalysis
In October 2010, an attack that combinesrotational cryptanalysis with therebound attack was published. The attack mounts aknown-key distinguisher against 53 of 72 rounds in Threefish-256, and 57 of 72 rounds in Threefish-512. It also affects theSkein hash function.[2]

Threefish is asymmetric-keytweakable block cipher designed as part of theSkein hash function, an entry in theNIST hash function competition. Threefish uses noS-boxes or other table lookups in order to avoid cachetiming attacks;^[1] its nonlinearity comes from alternating additions withexclusive ORs. In that respect, it is similar toSalsa20,TEA, and the SHA-3 candidatesCubeHash andBLAKE.

Threefish and the Skein hash function were designed byBruce Schneier,Niels Ferguson,Stefan Lucks, Doug Whiting,Mihir Bellare, Tadayoshi Kohno,Jon Callas, and Jesse Walker. "Threefish is unpatented, and the source code is uncopyrighted and license-free; it is free for all uses."^[3]

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

Threefish encrypts in${\displaystyle r}$ rounds and uses${\displaystyle \frac{r}{4}+1}$ different round keys. After every four rounds, and before the first,${\displaystyle w}$ round key words are added to the${\displaystyle w}$ data words. To calculate the round keys an additional key word${\displaystyle k_w}$ is appended to the original key words${\displaystyle k_0,k_1,\dots ,k_{w-1}}$. Also, an additional tweak word${\displaystyle t_2}$ is appended to the tweak words${\displaystyle t_0,t_1}$.

${\displaystyle k_w=C\oplus k_0\oplus k_1\oplus \cdots \oplus k_{w-1};C=\text{0x1BD11BDAA9FC1A22}}$

${\displaystyle t_2=t_0\oplus t_1}$

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

The round key words${\displaystyle k_{s,i}}$ are now defined like this:

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

The mix function takes atuple of words${\displaystyle (x_0,x_1)}$ and returns another tuple of words${\displaystyle (y_0,y_1)}$. The function is defined like this:

${\displaystyle y_0=(x_0+x_1){mod2}^{64}}$

${\displaystyle y_1=(x_1⋘R_{(dmod8),j})\oplus y_0}$

${\displaystyle R_{d,j}}$ is a fixed set of rotation constants chosen to achieve quickdiffusion.

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 functionsprovides bit-level permutations in the form of bitwise rotations.^{[citation needed]} The Permute step and rotation constantsin the MIX functions are chosen in such a way that the overall effect is complete diffusion of all the bitsin a data block.^{[citation needed]}

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 totiming attacks by examining the time requiredfor encryption. The permutation is therefore deliberately designed to ensure that it should execute in the same fashionindependent of the key being used or the data encrypted.^{[citation needed]}

• if${\displaystyle dmod4=0}$ the round key${\displaystyle k_{d/4,i}}$ is added to word${\displaystyle i}$
• the mix function is applied to pairs of words, the rotation widths${\displaystyle R_{dmod8,j}}$ depend on round number${\displaystyle d}$ and word pair${\displaystyle j\in \{0,\cdots ,w/2-1\}}$
• the words are permutated using a permutation independent from the round number

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

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

In October 2010, an attack that combinesrotational cryptanalysis with therebound attack was published. The attack mounts aknown-key distinguisher against 53 of 72 rounds in Threefish-256, and 57 of 72 rounds in Threefish-512. It also affects theSkein hash function.^[2] This is a follow-up to the earlier attack published in February, which breaks 39 and 42 rounds respectively.^[4] In response to this attack, the Skein team tweaked the rotation constants used in Threefish and thereby thekey schedule constants for round 3 of the NIST hash function competition.^[1]

In 2009, a related keyboomerang attack against a reduced round Threefish version was published. For the 32-round version, the time complexity is${\displaystyle 2^{226}}$ and the memory complexity is${\displaystyle 2^{12}}$; for the 33-round version, the time complexity is${\displaystyle 2^{352.17}}$ with a negligible memory usage. The attacks also work against the tweaked version of Threefish: for the 32-round version, the time complexity is${\displaystyle 2^{222}}$ and the memory complexity is${\displaystyle 2^{12}}$; for the 33-round version, the time complexity is${\displaystyle 2^{355.5}}$ with a negligible memory usage.^[5]

• Twofish
• Blowfish (cipher)

• Block ciphers
• Free ciphers

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