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 $x_1,x_2,\dots ,x_n$ is statistically independent of the value of $f(x_1,x_2,\dots ,x_n)$.

Definition

A function $f:{\mathbb{𝔽}}_2^n\to {\mathbb{𝔽}}_2$ is $k$-th order correlation immune if for any independent $n$ binary random variables $X_0\dots X_{n-1}$, the random variable $Z=f(X_0,\dots ,X_{n-1})$ is independent from any random vector $(X_{i_1}\dots X_{i_k})$ with $0\le i_1<\dots <i_k<n$.

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.