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Construction of bent functions from near-bent functions. (English) Zbl 1184.94240
A function $$f$$ from an $$n$$-dimensional vector space $$V_n$$ over $$\mathbb{F}_2$$ into $$\mathbb{F}_2$$ is called bent (near-bent) if its Walsh transform $$\hat{f}(u) = \sum_{x\in V_n}(-1)^{f(x)+\langle u,x\rangle}$$ where $$\langle\;,\;\rangle$$ denotes any inner product on $$V_n$$ takes values in $$\{\pm 2^{n/2}\}$$ ($$\{0,\pm 2^{(n+1)/2}\}$$) only. The authors suggest the construction of bent functions $$g(x,y) = yh(x)+(y+1)f(x)$$ from $$V_n\times V_1$$ into $$\mathbb{F}_2$$ from two near-bent functions $$f,h$$ on $$V_n$$. A necessary and sufficient condition for $$g$$ being bent is that $$\{u \in V_n\;|\;\hat{f}(u) \neq 0\;\text{and}\;\hat{h}(u) \neq 0\} = \emptyset$$.
The authors implement their construction using Gold near-bent functions and general quadratic near-bent functions obtaining bent functions of algebraic degree $$3$$, and Kasami-Welch near-bent functions to obtain further not quadratic bent functions.
A Boolean function $$f$$ of dimension $$n$$ is called normal (weakly-normal) if there exists an affine subspace of dimension $$n/2$$ on which $$f$$ is constant (affine). The authors point out the significance of their construction presenting non-weakly-normal bent functions in dimension $$12$$ and $$10$$. Until then the smallest dimension where a non-weakly-normal bent function has been found is $$14$$, [see A. Canteaut et al. Discrete Appl. Math. 154 (2), 202–218 (2006; Zbl 1091.94021)].

##### MSC:
 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 06E30 Boolean functions
##### Keywords:
bent function; weakly normal; Fourier transform
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##### References:
 [1] Braeken, An; Wolf, Christopher; Preneel, Bart, A randomised algorithm for checking the normality of cryptographic Boolean functions, (), 51-66 · Zbl 1088.68595 [2] Canteaut, A.; Carlet, C.; Charpin, P.; Fontaine, C., On cryptographic properties of the cosets of $$r(1, m)$$, IEEE trans. inform. theory, 47, 4, 1494-1513, (2001) · Zbl 1021.94014 [3] Canteaut, Anne; Charpin, Pascale, Decomposing bent functions, IEEE trans. inform. theory, 49, 8, 2004-2019, (2003) · Zbl 1184.94230 [4] Canteaut, Anne; Daum, Magnus; Dobbertin, Hans; Leander, Gregor, Finding nonnormal bent functions, Discrete appl. math., 154, 2, 202-218, (2006) · Zbl 1091.94021 [5] C. Carlet, On cryptographic complexity of boolean functions, in: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Proceedings of Fq6, 2002, pp. 53-69 · Zbl 1021.94524 [6] Carlet, Claude; Dobbertin, Hans; Leander, Gregor, Normal extensions of bent functions, IEEE trans. inform. theory, 50, 11, 2880-2885, (2004) · Zbl 1184.94232 [7] Dillon, J.; McGuire, G., Near-bent functions on a hyperplane, Finite fields appl., 14, 3, 715-720, (2008) · Zbl 1159.11051 [8] Hans Dobbertin, Construction of bent functions and balanced boolean functions with high nonlinearity, in: Preneel [13], pp. 61-74 · Zbl 0939.94563 [9] Xuejia Lai, Additive and linear structures of cryptographic functions, in: Preneel [13], pp. 75-85 · Zbl 0939.94508 [10] P. Langevin, G. Leander, and G. McGuire, Analysis of Kasami-Welch functions in odd dimension using Stickelberger’s theorem, submitted for publication, 2008 · Zbl 1245.11120 [11] MacWilliams, F.J.; Sloane, N.J., The theory of error-correcting codes, (1977), North-Holland Amsterdam · Zbl 0369.94008 [12] McGuire, Gary, Spectra of functions, subspaces of matrices, and going up versus going down, (), 51-66 · Zbl 1195.06010 [13] ()
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