An additive approach to nonlinearity degree of discrete functions on a primary cyclic group

An additive approach to the definition of nonlinearity degree for a discrete function on a cyclic group is proposed. For elementary abelian groups, this notion is equivalent to the ordinary “multiplicative” one. For polynomial functions on the ring of integers $\mod p^n$, this notion is equivalent to the minimal degree of a polynomial. It is proved that the nonlinearity degree on a cyclic group is a finite number if and only if the order of the group is a power of a prime. An upper bound for the nonlinearity degree of functions on a cyclic group of order $p^n$ is given.