Year: 2024
Type: article
Abstract: For a q-polynomial L over a finite field $$\mathbb {F}_{q^n}$$ F q n , we characterize the differential spectrum of the function $$f_L:\mathbb {F}_{q^n} \rightarrow \mathbb {F}_{q^n}, x \mapsto x \cdot L(x)$$ f L : F q n → F q n , x ↦ x · L ( x ) and show that, for $$n \le 5$$ n ≤ 5 , it is completely determined by the image of the rational function $$r_L :\mathbb {F}_{q^n}^* \rightarrow \mathbb {F}_{q^n}, x \mapsto L(x)/x$$ r L : F q n ∗ → F q n , x ↦ L ( x ) / x . This result follows from the classification of the pairs (L, M) of q-polynomials in $$\mathbb {F}_{q^n}[X]$$ F q n [ X ] , $$n \le 5$$ n ≤ 5 , for which $$r_L$$ r L and $$r_M$$ r M have the same image, obtained in Csajbók et al. (Ars Math Contemp 16(2):585–608, 2019). For the case of $$n>5$$ n > 5 , we pose an open question on the dimensions of the kernels of $$x \mapsto L(x) - ax$$ x ↦ L ( x ) - a x for $$a \in \mathbb {F}_{q^n}$$ a ∈ F q n . We further present a link between functions $$f_L$$ f L of differential uniformity bounded above by q and scattered q-polynomials and show that, for odd values of q, we can construct CCZ-inequivalent functions $$f_M$$ f M
Source: Designs Codes and Cryptography
Author: Christof Beierle
Institution: Ruhr University Bochum
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4403463552