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 ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is statistically independent of the value of ${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})}$.

Definition

A function ${\displaystyle f:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F} _{2}}$ is ${\displaystyle k}$-th order correlation immune if for any independent ${\displaystyle n}$ binary random variables ${\displaystyle X_{0}\ldots X_{n-1}}$, the random variable ${\displaystyle Z=f(X_{0},\ldots ,X_{n-1})}$ is independent from any random vector ${\displaystyle (X_{i_{1}}\ldots X_{i_{k}})}$ with ${\displaystyle 0\leq i_{1}<\ldots <i_{k}<n}$.

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.[1]

References

1. T. Siegenthaler (September 1984). "Correlation-Immunity of Nonlinear Combining Functions for Cryptographic Applications". IEEE Transactions on Information Theory. 30 (5): 776–780. doi:10.1109/TIT.1984.1056949.

Further reading

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