This paper studies the properties and constructions of non linear functions, which are a core component of cryptographic primi- tives including data encryption algorithms and one-way hash function. A main contribution of this paper is to reveal the relationship between nonlinearity and propagation characteristic, two critical indicators of the cryptographic strength of a Boolean function. In particular, (we prove that (i) if f, a Boolean function on Vn, satisfies the propagation criterion with respect to all but a subset R of vectors in Vn, then the nonlinearity of f satisfies Nf ≥ 2 n-1 - 21/2(n+t)-1, where t is the rank of R, and
(ii) When │R│ > 2, the nonzero vectors in R are linearly dependent. Furthermore we show that
(iii) if │R│= 2 then n must be odd, the nonlinearity of f satisfies Nf = 2 n-1 - 2 1/2(n-1), and the nonzero vector in R must be a linear struc- ture of f.
(iv) there exists no function on Vn such that │R│ = 3
(v) if │R│ = 4 then n must be even, the nonlinearity of f satisfies Nf = 2 n-1 - 2 1/2 n, and the nonzero vectors in R must be linear structures of f.
(vi) if │R│ = 5 then n must be odd, the nonlinearity of f is Nf = 2 n-1 - 2 1/2(n-1), the four nonzero vectors denoted by β1, β2, β3 and β4 are related by the equation β1⊕ β2⊕ β3⊕ β4⊕ = 0 and none of the four vectors is a linear structure of f.
(vii) there exists no function on Vn such that │R│ = 6.
We also discuss the structures of functions with │R│ = 2,4,5. In particular we show that these functions have close relationships with bent functions, and can be easily constructed from the latter.