Structures of cryptographic functions with strong avalanche characteristics

Abstract

This paper studies the properties and constructions of nonlinear functions, which are a core component of cryptographic primitives including data encryption algorithms and one-way hash functions. 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 V _n , satisfies the propagation criterion with respect to all but a subset ℜ of vectors in V _n , then the nonlinearity of f satisfies N _f ≥2ⁿ⁻¹ −2^1/2(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 Ninf = 2ⁿ⁻¹−2^1/2(n−1), and the nonzero vector in ℜ must be a linear structure of f.

• (iv) there exists no function on V _n such that ¦ℜ¦=3.

• (v) if ¦ℜ¦=4 then n must be even, the nonlinearity of f satisfies N _f = 2ⁿ⁻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 N _f =2ⁿ⁻¹²^1/2(n−1), the four nonzero vectors in ℜ, denoted by β ₁, β ₂, β ₃ and β ₄, are related by the equation β ₁ ⊕ β ₂ ⊕ β ₃ ⊕ β ₄=0, and none of the four vectors is a linear structure of f.

• (vii) there exists no function on V _n 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.