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A counterexample to Tokareva's conjecture on sums of two bent functions
Abstract
It is well known that the sum of two bent functions on 𝔽n has algebraic degree at most n/2 for even n≥ 4. Tokareva conjectured that the converse holds: every Boolean function with degree at most n/2 can be written as a sum of two bent functions. The conjecture has been verified for even n≤ 6 and for several special classes of Boolean functions. We disprove this conjecture by showing that x1x2(x3x4+x5x6+x7x8) is not a sum of two bent functions on 𝔽8. We extend this construction to obtain counterexamples in every even dimension n≥ 8. Using Shaporenko's reduction, we also obtain explicit counterexamples to Tokareva's derivative conjecture in every even dimension n≥ 10.
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BibTeX
@misc{feng2026counterexample,
title = {{A counterexample to Tokareva's conjecture on sums of two bent functions}},
author = {Yansong Feng and Hao Jiang and Guancheng Li and Tianyi Li and Lili Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32239},
eprint = {2610.32239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32239},
}
RIS
TY - UNPB
TI - A counterexample to Tokareva's conjecture on sums of two bent functions
AU - Yansong Feng
AU - Hao Jiang
AU - Guancheng Li
AU - Tianyi Li
AU - Lili Tang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32239
UR - https://arxiv.org/abs/2610.32239
AB - It is well known that the sum of two bent functions on 𝔽^n has algebraic degree at most n/2 for even n≥ 4. Tokareva conjectured that the converse holds: every Boolean function with degree at most n/2 can be written as a sum of two bent functions. The conjecture has been verified for even n≤ 6 and for several special classes of Boolean functions. We disprove this conjecture by showing that x_1x_2(x_3x_4+x_5x_6+x_7x_8) is not a sum of two bent functions on 𝔽^8. We extend this construction to obtain counterexamples in every even dimension n≥ 8. Using Shaporenko's reduction, we also obtain explicit counterexamples to Tokareva's derivative conjecture in every even dimension n≥ 10.
ER -