ASIACRYPT 1994: Advances in Cryptology — ASIACRYPT'94 pp 119-132

# Structures of cryptographic functions with strong avalanche characteristics

Extended abstract
• Jennifer Seberry
• Xian -Mo Zhang
• Yuliang Zheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 917)

## 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
1. (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 −21/2(n+t)−1, where t is the rank of ℜ, and

2. (ii)

When ¦ℜ¦ > 2, the nonzero vectors in ℜ are linearly dependent. Furthermore we show that

3. (iii)

if¦ℜ¦=2 then n must be odd, the nonlinearity of f satisfies Ninf = 2n−1−21/2(n−1), and the nonzero vector in ℜ must be a linear structure of f.

4. (iv)

there exists no function on V n such that ¦ℜ¦=3.

5. (v)

if ¦ℜ¦=4 then n must be even, the nonlinearity of f satisfies N f = 2n−1−21/2 n, and the nonzero vectors in ℜ must be linear structures of f.

6. (vi)

if ¦ℜ¦=5 then n must be odd, the nonlinearity of f is N f =2n−1²1/2(n−1), the four nonzero vectors in ℜ, denoted by β1, β2, β3 and β4, are related by the equation β1β2β3β4=0, and none of the four vectors is a linear structure of f.

7. (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.

## Keywords

Boolean Function Linear Structure Propagation Criterion Truth Table Nonzero Vector
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Jennifer Seberry
• 1
• Xian -Mo Zhang
• 1
• Yuliang Zheng
• 1
1. 1.Department of Computer ScienceUniversity of WollongongWollongongAustralia