Rotation-symmetric Boolean function is a class of Boolean functions with good cryptographic properties, and researches on its weight and nonlinearity cryptographic properties have good theoretical value. Different from the conventional calculation method, in this paper, these problems are converted to the evaluation of exponential sum on finite fields with a specific normal basis. Some new results about the weight and nonlinearity of some rotation-symmetric Boolean functions of degree 2 with 4 ?? n and n=2s are obtained. Using the proposed method, the weight and nonlinearity of almost all Rotation-symmetric Boolean functions of degree 2 can be evaluated. This new method is also interesting for studies on the other Boolean functions.
张习勇, 祁应红,高光普,李玉娟. 一种计算旋转对称布尔函数的汉明重量和非线性度的新方法[J]. 电子与信息学报, 2015, 37(11): 2691-2696.
张习勇, 祁应红, 高光普, 李玉娟. A New Method for Evaluation of Hamming Weight and Nonlinearity of Rotation-symmetric Boolean Functions. JEIT, 2015, 37(11): 2691-2696.
Pieprzyk J and Qu C X. Fast hashing and rotation-symmetric functions[J]. Journal of Universal Computer Science, 1999, 5(1): 20-31.
[2]
Cusick T W and P. Fast evaluation, weights and nonlinearity of rotation-symmetric functions[J]. Discrete Mathematics, 2002, 258(1): 289-301.
[3]
Ciungu L C. Cryptographic Boolean functions: Thus-Morse sequences, weight and nonlinearity[D]. [Ph.D. dissertation], University at Buffalo, 2010.
[4]
Zhang X, Guo H, Feng R, et al.. Proof of a conjecture about rotation symmetric functions[J]. Discrete Mathematics, 2011, 311(14): 1281-1289.
[5]
Wang B, Zhang X, and Chen W. The hamming weight and nonlinearity of a type of rotation symmetric Boolean function [J]. Acta Mathematica Sinica, Chinese Series, 2012, 55(4): 613-626.
[6]
Cusick T W. Finding Hamming weights without looking at truth tables[J]. Cryptography and Communications, 2013, 5(1): 7-18.
[7]
Brown A and Cusick T W. Equivalence classes for cubic rotation symmetric functions[J]. Cryptography and Communications, 2013, 5(2): 85-118.
[8]
KV L, Sethumadhavan M, and Cusick T W. Counting rotation symmetric functions using Polya’s theorem[J]. Discrete Applied Mathematics, 2014, 169: 162-167.
[9]
Cusick T W and Cheon Y. Affine equivalence for cubic rotation symmetric Boolean functions with n=pq variables[J]. Discrete Mathematics, 2014, 327: 51-61.
[10]
Cusick T W and Cheon Y. Affine equivalence of quartic homogeneous rotation symmetric Boolean functions[J]. Information Sciences, 2014, 259: 192-211.
[11]
Kim H, Park S M, and Hahn S G. On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2[J]. Discrete Applied Mathematics, 2009, 157(2): 428-432.
[12]
Liu H. On the weight and nonlinearity of quadratic rotation symmetric function with two MRS functions[J]. General Mathematics Notes, 2013, 16(1): 12-19.
[13]
P and Maitra S. Rotation symmetric Boolean functions-count and cryptographic properties[J]. Discrete Applied Mathematics, 2008, 156(10): 1567-1580.
[14]
Hou X D. Explicit evaluation of certain exponential sums of binary quadratic functions[J]. Finite Fields and Their Applications, 2007, 13(4): 843-868.
[15]
Weinberger M J and Lempel A. Factorization of symmetric circulant matrices in finite fields[J]. Discrete Applied Mathematics, 1990, 28(3): 271-285.
[16]
Zhang X, Cao X, and Feng R. A method of evaluation of exponential sum of binary quadratic functions[J]. Finite Fields and Their Applications, 2012, 18(6): 1089-1103.
