Characters, Weil sums and c-differential uniformity with an\n application to the perturbed Gold function

2020/09/16 by Pantelimon Stănică, Stanica, Pantelimon, Constanza Riera +3
Mathematics · #06E30 #11T06 #94A60 #94C10 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2009.07779

openalex publication_date 2020/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Building upon the observation that the newly defined~ citeEFRST20 concept\nof c-differential uniformity is not invariant under EA or\nCCZ-equivalence~ citeSPRS20, we showed in~ citeSG20 that adding some\nappropriate linearized monomials increases the c-differential uniformity of\nthe inverse function, significantly, for some~c. We continue that\ninvestigation here. First, by analyzing the involved equations, we find bounds\nfor the uniformity of the Gold function perturbed by a single monomial,\nexhibiting the discrepancy we previously observed on the inverse function.\nSecondly, to treat the general case of perturbations via any linearized\npolynomial, we use characters in the finite field to express all entries in the\nc-Differential Distribution Table (DDT) of an (n,n)-function on the finite\nfield F_pⁿ, and further, we use that method to find explicit expressions\nfor all entries of the c-DDT of the perturbed Gold function (via an arbitrary\nlinearized polynomial).\n

Related

• The c-differential behavior of the inverse function under the\n EA-equivalence
• Differential uniformity properties of some classes of permutation polynomials
• Power Functions over Finite Fields with Low c-Differential Uniformity
• Low c-differential uniformity for functions modified on subfields
• The differential properties of certain permutation polynomials over finite fields