## 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 \(2^{n-1} - 2^{\frac{n}{2}}\).

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## 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).

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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

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

### Keywords

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

### Mathematics Subject Classification

- 06E30
- 94A60
- 11T71