# Construction and search of balanced Boolean functions on even number of variables towards excellent autocorrelation profile

## Abstract

In a very recent work by Tang and Maitra (IEEE Ttans Inf Theory 64(1):393–402, 2018], a theoretical construction of balanced functions f on n-variables ($$n\equiv 2 \bmod 4$$) with very good autocorrelation and Walsh spectra values ($$\varDelta _f < 2^{\frac{n}{2}}$$ and $$nl(f) > 2^{n-1} - 2^{\frac{n}{2}} + 2^{\frac{n}{2}-3} - 5\cdot 2^{\frac{n-2}{4}}$$) has been presented. The theoretical bounds could be satisfied for all such $$n \ge 46$$. The case for $$n \equiv 0 \bmod 4$$ could not be solved in the said paper and it has also been pointed out that though theoretically not proved, such constructions may provide further interesting examples of Boolean functions. In this follow-up work, we concentrate in two directions. First we present a construction method for balanced functions f on n-variables ($$n\equiv 0 \bmod 4$$ and $$n \ge 52$$) with $$\varDelta _f < 2^{\frac{n}{2}}$$ and $$nl(f) > 2^{n-1} - 2^{\frac{n}{2}}$$). Secondly, we apply search methods in suitable places to obtain balanced functions on even variables in the interval $$[10, \ldots , 26]$$ with improved parameters that could never be achieved before. As a consequence, for the first time we could provide examples of balanced Boolean functions f having $$\varDelta _f < 2^{\frac{n}{2}}$$ for $$n \equiv 0 \bmod 4$$, where $$n = 12, 16, 20,$$ and 24. Whatever functions we present in this paper have nonlinearity greater than $$2^{n-1} - 2^{\frac{n}{2}}$$.

This is a preview of subscription content, access via your institution.

## References

1. Bartholomew-Biggs, M.: The steepest descent method. In: Nonlinear Optimization with Financial Applications, pp. 51–64 (2005).

2. Burnett L., Millan W., Dawson E., Clark A.: Simpler methods for generating better Boolean functions with good cryptographic properties. Aust. J. Comb. 29, 231–248 (2004).

3. Carlet C.: Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications. IEEE Trans. Inf. Theory 54(3), 1262–1272 (2008).

4. Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974).

5. Dobbertin, H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Proceedings of the Second International Workshop on Fast Software Encryption. Lecture Notes in Computer Science, vol. 1008, pp. 61–74. Springer, Berlin (1995).

6. Kavut S., Maitra S., Yucel M.D.: Search for Boolean functions with excellent profiles in the rotation symmetric class. IEEE Trans. Inf. Theory 53(5), 1743–1751 (2007).

7. Kavut, S., Maitra, S., Tang, D.: C code for our results (2017). https://drive.google.com/open?id=0B1s_TxsFtjSPTjJ3cWE4VjV3OGs

8. Lachaud G., Wolfmann J.: The weights of the orthogonals of the extended quadratic binary goppa codes. IEEE Trans. Inf. Theory 36(3), 686–692 (1990).

9. Mesnager S.: Bent functions: fundamentals and results. Springer, Cham (2016).

10. Rothaus O.S.: On “bent” functions. J. Comb. Theory Ser. A 20(3), 300–305 (1976).

11. Seberry, J., Zhang, X.M., Zheng, Y.: On constructions and nonlinearity of correlation immune functions. In: EUROCRYPT 1993. Lecture Notes in Computer Science, Vol. 765, pp. 181–199. Springer, Berlin (1994).

12. Shannon C.E.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28(4), 656–715 (1949).

13. Tang D., Maitra S.: Construction of $$n$$-variable ($$n\equiv 2 \text{ mod } 4$$) balanced Boolean functions with maximum absolute value in autocorrelation spectra $$< 2^{\frac{n}{2}}$$. IEEE Trans. Inf. Theory 64(1), 393–402 (2018).

14. Tang D., Carlet C., Tang X.: Differentially 4-uniform bijections by permuting the inverse function. Des. Codes Cryptogr. 77(1), 117–141 (2015).

15. Zhang, X.M., Zheng, Y.: GAC—the criterion for global avalanche characteristics of cryptographic functions. J. Univers. Comput. Sci., 320–337 (1996).

## Acknowledgements

The authors wish to thank the anonymous reviewers for their very helpful comments. The work of Deng Tang was supported by the National Natural Science Foundation of China (Grant No. 61602394) and the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2682018ZT25 and 2682016CX113).

## Author information

Authors

### Corresponding author

Correspondence to Deng Tang.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

## Rights and permissions

Reprints and Permissions

Kavut, S., Maitra, S. & Tang, D. Construction and search of balanced Boolean functions on even number of variables towards excellent autocorrelation profile. Des. Codes Cryptogr. 87, 261–276 (2019). https://doi.org/10.1007/s10623-018-0522-1

• Revised:

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1007/s10623-018-0522-1

### Keywords

• Absolute Indicator
• Autocorrelation Spectrum
• Balancedness
• Boolean Function
• Nonlinearity

• 06E30
• 94A60
• 11T71