Further Results on Sum-Freedom of Binary and q-ary Functions

Abstract

The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the q-ary version of the notion is a natural extension. For each integer k with 0≤ k≤ n, there is a kth order sum-free function on Bbb F_2ⁿ. It is also known that when k/n is not close to 0 or 1, the multiplicative inverse function on Bbb F_2ⁿ is not kth order sum-free. We generalize these two results to q-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.

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BibTeX RIS

BibTeX

@misc{hou2026further,
  title = {{Further Results on Sum-Freedom of Binary and $q$-ary Functions}},
  author = {Xiang-dong Hou and Shujun Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.31489},
  eprint = {2609.31489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31489},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Further Results on Sum-Freedom of Binary and q-ary Functions
AU  - Xiang-dong Hou
AU  - Shujun Zhao
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.31489
UR  - https://arxiv.org/abs/2609.31489
AB  - The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the q-ary version of the notion is a natural extension. For each integer k with 0≤ k≤ n, there is a kth order sum-free function on Bbb F_2^n. It is also known that when k/n is not close to 0 or 1, the multiplicative inverse function on Bbb F_2^n is not kth order sum-free. We generalize these two results to q-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.
ER  -

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