Minimum Distance between Bent and 1-resilient Boolean Functions

In this paper we study the minimum distance between the set of bent functions and the set of 1-resilient Boolean functions and present a lower bound on that. The bound is proved to be tight for functions up to 10 input variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of 1-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied upto 12-variable functions and we show that the construction provides a large class of 1-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of 8-variable, 1-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from 10 variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.

[1]  Susan Stepney,et al.  Evolving Boolean Functions Satisfying Multiple Criteria , 2002, INDOCRYPT.

[2]  Subhamoy Maitra,et al.  Further constructions of resilient Boolean functions with very high nonlinearity , 2002, IEEE Trans. Inf. Theory.

[3]  Soumen Maity,et al.  Construction of Cryptographically Important Boolean Functions , 2002, INDOCRYPT.

[4]  Claude Carlet,et al.  A Larger Class of Cryptographic Boolean Functions via a Study of the Maiorana-McFarland Construction , 2002, CRYPTO.

[5]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[6]  Yuriy Tarannikov.  New Constructions of Resilient Boolean Functions with Maximal Nonlinearity , 2001, FSE.

[7]  James L. Massey,et al.  A spectral characterization of correlation-immune combining functions , 1988, IEEE Trans. Inf. Theory.

[8]  Subhamoy Maitra.  Autocorrelation Properties of Correlation Immune Boolean Functions , 2001, INDOCRYPT.

[9]  Hans Dobbertin,et al.  Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity , 1994, FSE.

[10]  Claude Carlet.  On the Coset Weight Divisibility and Nonlinearity of Resilient and Correlation-Immune Functions , 2001, SETA.

[11]  Joos Vandewalle,et al.  Propagation Characteristics of Boolean Functions , 1991, EUROCRYPT.

[12]  Yuriy Tarannikov,et al.  Autocorrelation Coefficients and Correlation Immunity of Boolean Functions , 2001, ASIACRYPT.

[13]  Yuliang Zheng,et al.  On Relationships among Avalanche, Nonlinearity, and Correlation Immunity , 2000, ASIACRYPT.

[14]  Yuliang Zheng,et al.  GAC - the Criterion for Global Avalance Characteristics of Cryptographic Functions , 1995, J. Univers. Comput. Sci..

[15]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[16]  Yuliang Zheng,et al.  Improved Upper Bound on the Nonlinearity of High Order Correlation Immune Functions , 2000, Selected Areas in Cryptography.

[17]  Yuriy Tarannikov,et al.  On Resilient Boolean Functions with Maximal Possible Nonlinearity , 2000, INDOCRYPT.

[18]  Yuriy Tarannikov,et al.  On the Constructing of Highly Nonlinear Resilient Boolean Functions by Means of Special Matrices , 2001, INDOCRYPT.

[19]  Palash Sarkar,et al.  Construction of Nonlinear Boolean Functions with Important Cryptographic Properties , 2000, EUROCRYPT.

[20]  Jennifer Seberry,et al.  Nonlinearly Balanced Boolean Functions and Their Propagation Characteristics (Extended Abstract) , 1993, CRYPTO.

[21]  Palash Sarkar,et al.  Nonlinearity Bounds and Constructions of Resilient Boolean Functions , 2000, CRYPTO.

[22]  Palash Sarkar,et al.  Spectral Domain Analysis of Correlation Immune and Resilient Boolean Functions , 2000, IACR Cryptol. ePrint Arch..

[23]  Pascale Charpin,et al.  On Propagation Characteristics of Resilient Functions , 2002, Selected Areas in Cryptography.