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Peer-reviewedCC BY 4.0Finite Fields and Their Applications

New GAPN functions with the lowest possible algebraic degree

  • Noureddine El-Asri
  • Valentin Suder
Open at sourcePDFDOICite

Abstract

Most of the known (Almost) Perfect Nonlinear functions over an arbitrary finite field Fp have the lowest possible algebraic degree (2) and can be represented by Dembowski-Ostrom (DO) polynomials. We define Generalized DO (GDO) polynomials as homogeneous polynomials of algebraic degree p over finite fields with characteristic p . We illustrate the similarities between DO and GDO polynomials when studied through generalized derivatives. We also discuss the differential uniformity and Walsh coefficients of GDO functions. Our main contribution is to use this framework in order to propose several new classes of Generalized Almost Perfect Nonlinear (GAPN) functions of algebraic degree p , the lowest possible over the finite field Fp.

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Cite

BibTeXRIS
BibTeX
@article{elasri2026gapn,
  title = {{New GAPN functions with the lowest possible algebraic degree}},
  author = {Noureddine El-Asri and Valentin Suder},
  journal = {Finite Fields and Their Applications},
  year = {2026},
  month = sep,
  doi = {10.1016/j.ffa.2026.102885},
  url = {https://hal.science/hal-05758976v1},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - New GAPN functions with the lowest possible algebraic degree
AU  - Noureddine El-Asri
AU  - Valentin Suder
PY  - 2026
DA  - 2026/09/22
JO  - Finite Fields and Their Applications
DO  - 10.1016/j.ffa.2026.102885
UR  - https://hal.science/hal-05758976v1
AB  - Most of the known (Almost) Perfect Nonlinear functions over an arbitrary finite field Fp have the lowest possible algebraic degree (2) and can be represented by Dembowski-Ostrom (DO) polynomials. We define Generalized DO (GDO) polynomials as homogeneous polynomials of algebraic degree p over finite fields with characteristic p . We illustrate the similarities between DO and GDO polynomials when studied through generalized derivatives. We also discuss the differential uniformity and Walsh coefficients of GDO functions. Our main contribution is to use this framework in order to propose several new classes of Generalized Almost Perfect Nonlinear (GAPN) functions of algebraic degree p , the lowest possible over the finite field Fp.
SN  - 1071-5797
SN  - 1090-2465
ER  -