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 [math]\displaystyle{ x_1,x_2,\ldots,x_n }[/math] is statistically independent of the value of [math]\displaystyle{ f(x_1,x_2,\ldots,x_n) }[/math].

Definition

A function [math]\displaystyle{ f:\mathbb{F}_2^n\rightarrow\mathbb{F}_2 }[/math] is [math]\displaystyle{ k }[/math]-th order correlation immune if for any independent [math]\displaystyle{ n }[/math] binary random variables [math]\displaystyle{ X_0\ldots X_{n-1} }[/math], the random variable [math]\displaystyle{ Z=f(X_0,\ldots,X_{n-1}) }[/math] is independent from any random vector [math]\displaystyle{ (X_{i_1}\ldots X_{i_k}) }[/math] with [math]\displaystyle{ 0\leq i_1\lt \ldots\lt i_k\lt n }[/math].

Results in cryptography

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.

References

Further reading

1. Cusick, Thomas W. & Stanica, Pantelimon (2009). "Cryptographic Boolean functions and applications". Academic Press. ISBN:9780123748904.