A family of symmetric counterexamples to the bent sum decomposition conjecture

@misc{kepes2026family,
  title = {{A family of symmetric counterexamples to the bent sum decomposition conjecture}},
  author = {Máté Róbert Kepes and Anita Kercsó-Molnár and Viczián András},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06744},
  eprint = {2611.06744},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06744},
}

TY  - UNPB
TI  - A family of symmetric counterexamples to the bent sum decomposition conjecture
AU  - Máté Róbert Kepes
AU  - Anita Kercsó-Molnár
AU  - Viczián András
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06744
UR  - https://arxiv.org/abs/2611.06744
AB  - In this paper, we provide an infinite family of counterexamples to Tokareva's conjecture on sums of two bent functions. The conjecture states that every Boolean function in an even number of variables n of algebraic degree at most n/2 can be written as the sum of two bent functions. We show that for every n=2^k≥ 8, the symmetric function F_n=1+σ_2+σ_4+σ_6+⋯+σ_n/2 has degree n/2 and cannot be written as a sum of at most two bent functions. Here σ_j is the elementary symmetric Boolean polynomial of degree j. Furthermore, we show that the eight-variable member of this family can be represented as the sum of exactly three bent functions. Using the connection between bent-sum decompositions and derivatives of bent functions, we also obtain an infinite family of balanced Boolean functions that cannot be derivatives of bent functions.
ER  -