Branch number

In cryptography, the branch number is a numerical value that characterizes the amount of diffusion introduced by a vectorial Boolean function Template:Mvar that maps an input vector Template:Mvar to output vector $F(a)$. For the (usualTemplate:Sfn) case of a linear Template:Mvar the value of the differential branch number is produced by:

1. applying nonzero values of Template:Mvar (i.e., values that have at least one non-zero component of the vector) to the input of Template:Mvar;
2. calculating for each input value Template:Mvar the Hamming weight $W$ (number of nonzero components), and adding weights $W(a)$ and $W(F(a))$ together;
3. selecting the smallest combined weight across for all nonzero input values: $B_d(F)=\underset{a\ne 0}{\min }(W(a)+W(F(a)))$.

If both Template:Mvar and $F(a)$ have Template:Mvar components, the result is obviously limited on the high side by the value $s+1$ (this "perfect" result is achieved when any single nonzero component in Template:Mvar makes all components of $F(a)$ to be non-zero). A high branch number suggests higher resistance to the differential cryptanalysis: the small variations of input will produce large changes on the output and in order to obtain small variations of the output, large changes of the input value will be required.Template:Sfn

The term was introduced by Daemen and Rijmen in early 2000s and quickly became a typical tool to assess the diffusion properties of the transformations.Template:Sfn

Mathematics

The branch number concept is not limited to the linear transformations, Daemen and Rijmen provided two general metrics:Template:Sfn

• differential branch number, where the minimum is obtained over inputs of Template:Mvar that are constructed by independently sweeping all the values of two nonzero and unequal vectors Template:Mvar, Template:Mvar ($\oplus$ is a component-by-component exclusive-or): $B_d(F)=\underset{a\ne b}{\min }(W(a\oplus b)+W(F(a)\oplus F(b))$;
• for linear branch number, the independent candidates $\alpha$ and $\beta$ are independently swept; they should be nonzero and correlated with respect to Template:Mvar (the $LAT(\alpha ,\beta )$ coefficient of the linear approximation table of Template:Mvar should be nonzero): $B_l(F)=\underset{\alpha \ne 0,\beta ,LAT(\alpha ,\beta )\ne 0}{\min }(W(\alpha )+W(\beta ))$.

Sources

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