# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Canteaut, A.; Charpin, P.; Kyureghyan, G., A new class of monomial bent functions, () · Zbl 1162.94004 [2] Canteaut, A.; Daum, M.; Dobbertin, H.; Leander, G., Finding non-normal bent functions, Discrete appl. math., 154, 2, 202-218, (2006), in: Special Issue on Coding and Cryptography · Zbl 1091.94021 [3] P. Charpin, G. Kyureghyan, On cubic monomial bent functions in the class M, SIAM J. Discrete Math., in press · Zbl 1171.11062 [4] J.F. Dillon, Elementary Hadamard difference sets, PhD thesis, University of Maryland, 1974 · Zbl 0346.05003 [5] Dillon, J.F.; Dobbertin, H., New cyclic difference sets with Singer parameters, Finite fields appl., 342-389, (2004) · Zbl 1043.05024 [6] H. Dobbertin, private communication [7] Hollmann, H.; Xiang, Q., On binary cyclic codes with few weights, (), 251-275 · Zbl 1015.94548 [8] Hollmann, H.D.L.; Xiang, Q., A proof of the welch and niho conjectures on crosscorrelations of binary m-sequences, Finite fields appl., 7, 253-286, (2001) · Zbl 1027.94006 [9] Katz, N.M., Sommes exponentielles, Astérisque, 79, (1980) [10] Koblitz, N., p-adic analysis: A short course on recent work, London math. soc. lecture note ser., vol. 46, (1980), Cambridge Univ. Press · Zbl 0439.12011 [11] Lachaud, G.; Wolfmann, J., Kloosterman sums, elliptic curves and cyclic codes in characteristic 2, C. R. acad. sci. Paris Sér. I math., 305, 20, 881-883, (1987) · Zbl 0652.14009 [12] Leander, G., Monomial bent functions, IEEE trans. inform. theory, 52, 2, 738-743, (2006) · Zbl 1161.94414 [13] Lidl, R.; Niederreiter, N., Finite fields, Encyclopedia math. appl., vol. 20, (1983), Addison-Wesley [14] Robert, A., The gross – koblitz formula revisited, Rend. sem. mat. univ. Padova, 105, 157-170, (2001) · Zbl 1165.11302 [15] Stickelberger, J., Über eine verallgemeinerung der kreistheilung, Math. ann., 37, 321-367, (1890) · JFM 22.0100.01 [16] Weil, A., On some exponential sums, Proc. natl. acad. sci. USA, 34, 204-207, (1948) · Zbl 0032.26102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.