Correlation immunity

In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs.

Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in

,

is statistically independent of the value of

A function

-th order correlation immune if for any independent

binary random variables

, the random variable

is independent from any random vector

When used in a stream cipher as a combining function for linear feedback shift registers, a Boolean function with low-order correlation-immunity is more susceptible to a correlation attack than a function with correlation immunity of high order.

Siegenthaler showed that the correlation immunity m of a Boolean function of algebraic degree d of n variables satisfies m + d ≤ n; for a given set of input variables, this means that a high algebraic degree will restrict the maximum possible correlation immunity.

Furthermore, if the function is balanced then m + d ≤ n − 1.

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