Welcome to the resource topic for
**2024/1688**

**Title:**

Revisiting Products of the Form X Times a Linearized Polynomial L(X)

**Authors:**
Christof Beierle

**Abstract:**

For a q-polynomial L over a finite field \mathbb{F}_{q^n}, we characterize the differential spectrum of the function f_L\colon \mathbb{F}_{q^n} \rightarrow \mathbb{F}_{q^n}, x \mapsto x \cdot L(x) and show that, for n \leq 5, it is completely determined by the image of the rational function r_L \colon \mathbb{F}_{q^n}^* \rightarrow \mathbb{F}_{q^n}, x \mapsto L(x)/x. This result follows from the classification of the pairs (L,M) of q-polynomials in \mathbb{F}_{q^n}[X], n \leq 5, for which r_L and r_M have the same image, obtained in [B. Csajbok, G. Marino, and O. Polverino. A Carlitz type result for linearized polynomials. Ars Math. Contemp., 16(2):585–608, 2019]. For the case of n>5, we pose an open question on the dimensions of the kernels of x \mapsto L(x) - ax for a \in \mathbb{F}_{q^n}.

We further present a link between functions 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 with bounded differential uniformity from a given function f_L fulfilling certain properties.

**ePrint:**
https://eprint.iacr.org/2024/1688

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