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Computing Walsh coefficients from the algebraic normal form of a Boolean function

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Abstract

We study the relationship between the Walsh Transform of a Boolean function and its Algebraic Normal Form(ANF), and present algorithms that compute the Walsh coefficients at a small set of points in terms of certain parameters derived from the ANF of a Boolean function. In the first part of this paper, based on the previous result by Gupta and Sarkar, we investigate the formula in Gupta-Sarkar’s algorithm in a novel iterative method and obtain a recurrence relation for the Walsh Transform of a Boolean function. The second part is devoted to applying this formula to algorithms to evaluate it. Experimental result shows that for the specified classes of Boolean functions, our algorithms can perform better than Gupta-Sarkar’s algorithm. For example, the proposed algorithm “ComputeWalsh” is able to compute the Walsh coefficients of the functions for which the complexity of Gupta-Sarkar’s algorithm is impractical. Besides, for functions acting on high number of variables (m>30) and having low number of monomials, the proposed algorithms are advantageous over the Fast Walsh Transform which is a standard method of computing the Walsh Transform with a complexity of O(m2m) operations.

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Notes

  1. By convention, all α (j)’s are pairwise unequal.

  2. Properly speaking, we can use the term of component function only for v≠0, so we make an abuse here.

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Acknowledgment

We thank all the anonymous reviewers for their helpful suggestions on the quality improvement of our paper. This work is supported by the National 973 Program of China (No.2011CB302400), the Strategic Priority Research Program of Chinese Academy of Sciences (No.XDA06010701, No.XDA06010702), the National Natural Science Foundation of China (No.61303257) and Institute of Information Engineering’s Research Project on Cryptography (No.Y3Z0023103, No.Y3Z0011102).

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Authors and Affiliations

  1. Trusted Computing and Information Assurance Laboratory, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Xinxin Gong

  2. Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, People’s Republic of China

    Bin Zhang

  3. Institute of Software, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Wenling Wu & Dengguo Feng

Authors
  1. Xinxin Gong
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  2. Bin Zhang
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  3. Wenling Wu
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  4. Dengguo Feng
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Corresponding author

Correspondence to Xinxin Gong.

Appendices

Appendix A

The randomly generated Boolean function f 1(x) with m=128, p=64 in Example 5 is f 1(x 1,...,x 128)=x I1⊕...⊕x I64, where all I i s are as follows:

figure c

Appendix B

The randomly generated homogeneous function f(x) with m=64, p=64 and d=18 in Example 6 is f(x 1,...,x 64)=x I1⊕...⊕x I64, where.

figure d
figure e
figure f

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Gong, X., Zhang, B., Wu, W. et al. Computing Walsh coefficients from the algebraic normal form of a Boolean function. Cryptogr. Commun. 6, 335–358 (2014). https://doi.org/10.1007/s12095-014-0103-8

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  • DOI: https://doi.org/10.1007/s12095-014-0103-8

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