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**2022/610**

**Title:**

On the Differential Spectrum of a Differentially 3-Uniform Power Function

**Authors:**
Tingting Pang, Nian Li, Xiangyong Zeng

**Abstract:**

In this paper, we investigate the cardinality, denoted by (j_1,j_2,j_3,j_4)_2, of the intersection of (\mathcal{C}^{(2)}_{j_1}-1)\cap(\mathcal{C}^{(2)}_{j_2}-2)\cap(\mathcal{C}^{(2)}_{j_3}-3) \cap(\mathcal{C}^{(2)}_{j_4}-4) for j_1,j_2,j_3,j_4\in\{0,1\}, where \mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1 are the cyclotomic classes of order two over the finite field \mathbb{F}_{p^n}, p is an odd prime and n is a positive integer. By making most use of the results on cyclotomic classes of orders two and four as well as the cardinality of the intersection (\mathcal{C}^{(2)}_{i_1}-1)\cap(\mathcal{C}^{(2)}_{i_2}-2)\cap(\mathcal{C}^{(2)}_{i_3}-3), we compute the values of (j_1,j_2,j_3,j_4)_2 in the case of p=5, where i_1,i_2,i_3\in\{0,1\}. As a consequence, the power function x^{\frac{5^n-1}{2}+2} over \mathbb{F}_{5^n} is shown to be differentially 3-uniform and its differential spectrum is also completely determined.

**ePrint:**
https://eprint.iacr.org/2022/610

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