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 −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 Ninf = 2n−1−21/2(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−1−21/2 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−1²1/2(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.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Adams, C. M., Tavares, S. E.: Generating and counting binary bent sequences. IEEE Transactions on Information Theory IT-36 No. 5 (1990) 1170–1173
Dillon, J. F.: A survey of bent functions. The NSA Technical Journal (1972) 191–215
Evertse, J.-H.: Linear structures in blockciphers. In Advances in Cryptology — EUROCRYPT'87 (1988) vol. 304, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 249–266
Meier, W., Staffelbach, O.: Nonlinearity criteria for cryptographic functions. In Advances in Cryptology — EUROCRYPT'89 (1990) vol. 434, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 549–562
Nyberg, K.: On the construction of highly nonlinear permutations. In Advances in Cryptology — EUROCRYPT'92 (1993) vol. 658, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 92–98
Preneel, B., Govaerts, R., Vandewalle, J.: Boolean functions satisfying higher order propagation criteria. In Advances in Cryptology — EUROCRYPT'91 (1991) vol. 547, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 141–152
Preneel, B., Leekwijck, W. V., Linden, L. V., Govaerts, R., Vandewalle, J.: Propagation characteristics of boolean functions. In Advances in Cryptology — EUROCRYPT'90 (1991) vol. 437, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 155–165
Seberry, J., Zhang, X. M., Zheng, Y.: Nonlinearity and propagation characteristics of balanced boolean functions. To appear in Information and Computation 1994
Seberry, J., Zhang, X. M., Zheng, Y.: Relationships among nonlinearity criteria. Presented at EUROCRYPT'94 1994
Webster, A. F.: Plain text/ciphertext bit dependencies in cryptographic system. Master's Thesis, Department of Electrical Engineering, Queen's University, Ontario, Cannada 1985
Webster, A. F., Tavares, S. E.: On the design of S-boxes. In Advances in Cryptology — CRYPTO'85 (1986) vol. 219, Lecture Notes in Computer Science Springer-Verlag, Berlin, Heidelberg, New York pp. 523–534
Yarlagadda, R., Hershey, J. E.: Analysis and synthesis of bent sequences. IEE Proceedings (Part E) 136 (1989) 112–123
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Seberry, J., Zhang, X.M., Zheng, Y. (1995). Structures of cryptographic functions with strong avalanche characteristics. In: Pieprzyk, J., Safavi-Naini, R. (eds) Advances in Cryptology — ASIACRYPT'94. ASIACRYPT 1994. Lecture Notes in Computer Science, vol 917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0000429
Download citation
DOI: https://doi.org/10.1007/BFb0000429
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59339-3
Online ISBN: 978-3-540-49236-8
eBook Packages: Springer Book Archive