Paper constructs new APN functions with boomerang uniformity one
Original title: New Classes of CCZ-Inequivalent APN Functions with Boomerang Uniformity One in Odd Characteristic: Differential and Boomerang Spectra
The constructions are CCZ-inequivalent to listed APN monomial and Ness-Helleseth binomial families for q greater than 7.
A cryptography paper constructs almost perfect nonlinear functions in odd characteristic by one-point modifications of even perfect nonlinear functions. The functions have boomerang uniformity one, and the authors give complete differential and boomerang spectrum formulas. For q greater than 7, differential spectra separate them from listed APN monomial families and the original Ness-Helleseth binomials and their extension. They also bound when CCZ-equivalence and EA-equivalence coincide near a PN function.
- The functions have maximal algebraic degree n(p-1).
- Three pairwise CCZ-inequivalent classes exist over F with exponent 15 times ell.
- Six such classes exist over F with exponent 120 times r.
- Characteristic three yields unbounded numbers of such classes.
- Some APN pairs at Hamming distance one are CCZ-equivalent and some are not.
From the original article
1 min read at arxiv.orgLet $q=p^n>3$ be an odd prime power and let $g:\F_q\to\F_q$ be an even perfect nonlinear (PN) function with $g(0)=0$. We construct almost perfect nonlinear (APN) functions $G_{λ,M}(x)=g(x)+M(x)+λ(x^{q-1}-1)$, where $λ\in\F_q^*$, $-λ\notin g(\F_q)$, and $M$ is an $\F_p$-linearized permutation polynomial.
By Namhun Koo · arxiv.org
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