Non linearite des fonctions booleennes donnees par des traces de polyn^omes de degre binaire 3

Nous 'etudions la non lin'earit'e des fonctions d'efinies sur F_{2^m} o`u $m$ est un entier impair, associ'ees aux polyn^omes de degr'e 7 ou `a des polyn^omes plus g'en'eraux. —– We study the nonlinearity of the functions defined on F_{2^m} where $m$ is an odd integer, associated to the polynomials of degree 7 or more general polynomials.

💡 Research Summary

The paper investigates the non‑linearity of Boolean functions defined on the finite field F₂^m, where m is an odd integer, that are constructed as absolute trace of low‑degree polynomials. Specifically, the authors focus on polynomials of binary degree 3 – i.e., polynomials whose monomials have exponents that are powers of 2 – and give special attention to the case where the algebraic degree is 7 (the smallest non‑trivial binary‑degree‑3 polynomial). The function under study has the form

  f(x) = Tr(P(x)), x ∈ F₂^m,

where Tr denotes the absolute trace from F₂^m to F₂ and P(x) ∈ F₂^m