Threefish

Threefish uses ${\displaystyle {\frac {N_{r}}{4}}+1}$ different round keys (${\displaystyle N_{r}}$: Number of rounds). To calculate these keys an additional key word ${\displaystyle k_{N_{w}}}$ is appended to the original key words ${\displaystyle k_{0},k_{1},\dots ,k_{N_{w}-1}}$. An additional tweak word ${\displaystyle t_{2}}$ is also appended to the tweak words ${\displaystyle t_{0},t_{1}}$.

${\displaystyle k_{N_{w}}=C_{240}\oplus k_{0}\oplus k_{1}\oplus \dots \oplus k_{N_{w}-1};\quad C_{240}={\text{0x1BD11BDAA9FC1A22}}}$

${\displaystyle t_{2}=t_{0}\oplus t_{1}}$

The purpose of the seemingly arbitrary constant ${\displaystyle C_{240}}$ is to frustrate some attacks that take advantage of the relationship between ${\displaystyle k_{N_{w}}}$ and the other keywords.

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

${\displaystyle k_{s,i}={\begin{cases}k_{(s+i){\bmod {(}}N_{w}+1)}&i=0,\dots ,N_{w}-4\\k_{(s+i){\bmod {(}}N_{w}+1)}+t_{s{\bmod {3}}}&i=N_{w}-3\\k_{(s+i){\bmod {(}}N_{w}+1)}+t_{(s+1){\bmod {3}}}&i=N_{w}-2\\k_{(s+i){\bmod {(}}N_{w}+1)}+s&i=N_{w}-1\end{cases}}}$

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

Threefish Mix Function

The mix function takes a tuple of words ${\displaystyle (x_{0},x_{1})}$ and returns another tuple of words ${\displaystyle (y_{0},y_{1})}$. The function is defined like this:

${\displaystyle y_{0}=x_{0}+x_{1}\mod 2^{64}}$

${\displaystyle y_{1}=(x_{1}\lll R_{(d{\bmod {8}}),j})\oplus y_{0}}$

${\displaystyle R_{d,j}}$ 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. 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.

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.

• if ${\displaystyle d\;{\bmod {\;}}4=0}$ the round key ${\displaystyle k_{d/4,i}}$ is added to word ${\displaystyle i}$
• the mix function is applied to consecutive words, the rotation widths depend on ${\displaystyle d}$ and the word number
• the words are permutated using a permutation independent from the round number

Threefish256 and Threefish512 apply this round 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 is added to the words and the words are converted back to a string of bytes.