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Monomial bent functions and Stickelberger’s theorem. (English) Zbl 1159.11052
The authors use results on Gauss sums, mainly Stickelberger’s theorem, to obtain a necessary and sufficient criterion for certain monomial functions from \(L = \mathbb F_{2^n}\) into \(\{-1,1\}\) of the form \(\mu_L(\alpha x^d)\) with \(\mu_L(x) = (-1)^{\text{tr}(x)}\), \(\text{tr}(x)\) denotes the absolute trace function, to be bent:
For integers \(d,k,n\) with \(n = 2k\), consider the mapping \(V_d: \mathbb Z/(2^n-1)\mathbb Z \rightarrow \{0,1,\ldots,2n\}\) given by \(V_d(j) = \text{wt}(j) + \text{wt}(-jd)\), where \(\text{wt}(z)\) is the weight of the binary representation of \(z\) modulo \(2^n-1\). Suppose that the integer \(d\) satisfies the conditions \[ \min_{0 < j < 2^n-1}V_d(j) = k, \quad\text{and}\quad V_d(j) = k \Rightarrow jd = 0, \] then \(\mu_L(\alpha x^d)\) is bent if and only if \(\sum_{j \in \mathcal{I}_d}\alpha^j = 1\), with \(\mathcal{I}_d = \{j\;|\;V_d(j) = k\}\). Furthermore the dual function is then described.
Using this result the authors give short alternative proofs for the bentness of the Gold function and the Kasami function. In the latter case the authors do not need the restriction \(\gcd(n,3) = 1\) as it is needed in previous proofs [see J. F. Dillon and H. Dobbertin [Finite Fields Appl. 10, 342–389 (2004; Zbl 1043.05024)]. Finally the authors show that the dual of the Kasami function as not monomial.

MSC:
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
06E30 Boolean functions
11T24 Other character sums and Gauss sums
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