Nonlinearity Bounds and Constructions of Resilient Boolean Functions

In this paper we investigate the relationship between the nonlinearity and the order of resiliency of a Boolean function. We first prove a sharper version of McEliece theorem for Reed-Muller codes as applied to resilient functions, which also generalizes the well known Xiao-Massey characterization. As a consequence, a nontrivial upper bound on the nonlinearity of resilient functions is obtained. This result coupled with Siegenthaler's inequality leads to the notion of best possible trade-off among the parameters: number of variables, order of resiliency, nonlinearity and algebraic degree. We further show that functions achieving the best possible trade-off can be constructed by the Maiorana-McFarland like technique. Also we provide constructions of some previously unknown functions.

[1]  Thomas Johansson,et al.  Fast Correlation Attacks Based on Turbo Code Techniques , 1999, CRYPTO.

[2]  Xiang-dong Hou.  On the covering radius of R(1, m) in R(3, m) , 1996, IEEE Trans. Inf. Theory.

[3]  Sarbani Palit,et al.  Cryptanalysis of LFSR-Encrypted Codes with Unknown Combining Function , 1999, ASIACRYPT.

[4]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[5]  Jennifer Seberry,et al.  On Constructions and Nonlinearity of Correlation Immune Functions (Extended Abstract) , 1994, EUROCRYPT.

[6]  Thomas Siegenthaler,et al.  Correlation-immunity of nonlinear combining functions for cryptographic applications , 1984, IEEE Trans. Inf. Theory.

[7]  Xiang-dong Hou.  On the norm and covering radius of the first-order Reed-Muller codes , 1997, IEEE Trans. Inf. Theory.

[8]  Palash Sarkar,et al.  Highly Nonlinear Resilient Functions Optimizing Siegenthaler's Inequality , 1999, CRYPTO.

[9]  Claude Carlet,et al.  More Correlation-Immune and Resilient Functions over Galois Fields and Galois Rings , 1997, EUROCRYPT.

[10]  Rainer A. Rueppel,et al.  Products of linear recurring sequences with maximum complexity , 1987, IEEE Trans. Inf. Theory.

[11]  Eric Filiol,et al.  Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity , 1998, EUROCRYPT.

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

[13]  Claude Carlet,et al.  On Correlation-Immune Functions , 1991, CRYPTO.

[14]  Thomas Johansson,et al.  Improved Fast Correlation Attacks on Stream Ciphers via Convolutional Codes , 1999, EUROCRYPT.

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

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

[17]  Sangjin Lee,et al.  On the Correlation Immune Functions and Their Nonlinearity , 1996, ASIACRYPT.

[18]  Enes Pasalic,et al.  Further Results on the Relation Between Nonlinearity and Resiliency for Boolean Functions , 1999, IMACC.

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

[20]  Thomas Siegenthaler,et al.  Decrypting a Class of Stream Ciphers Using Ciphertext Only , 1985, IEEE Transactions on Computers.

[21]  Willi Meier,et al.  Fast Correlation Attacks on Stream Ciphers (Extended Abstract) , 1988, EUROCRYPT.

[22]  Xiang-dong Hou.  Covering Radius of the Reed-Muller CodeR(1, 7) - A Simpler Proof , 1996, J. Comb. Theory, Ser. A.

[23]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[24]  Cunsheng Ding,et al.  The Stability Theory of Stream Ciphers , 1991, Lecture Notes in Computer Science.