The ternary function f(x) mapping F34kto F3and given by f(x) = Tr4k(a1x2(3k+1)+ a2x(3k+1)2), where a1is a nonsquare in F34kand a2is defined explicitly by a1, is proven to be a regular bent function of degree four belonging to the completed Maiorana-McFarland class. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives.

In this paper, we discover that univariate form of a Niho bent function is a sum of functions having the form of a Leander-Kholosha bent function taken with particular coefficients from F*(2n) for every term. We know that the Niho bent functions are related to o-polynomials. The power terms in the univariate Niho bent function can be derived by working, in a first step, on each monomial of the corresponding o-polynomial separately, and in a second step, adding them to obtain the global expression. This allows, knowing the monomials in an o-polynomial, to obtain the power terms of the polynomial representing corresponding bent function. However, the coefficients are not calculated explicitly. The explicit form is given for the bent functions obtained from quadratic and cubic o-polynomials. We also calculate the algebraic degree of any bent function in the Leander-Kholosha class.

In this paper, we extend the class of Niho bent function consisting of 2rterms discovered by Leander and Kholosha. The extension is achieved by inserting coefficients of the power terms in the original function. Doing this, we obtain relation to all the existing quadratic o-monomials. We also calculate the algebraic degree of any function in the extended class.

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander-Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided.

Several new classes of binary andp-ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums.

In this paper, most of the known infinite classes of generalized bent functions are analyzed for their relation to the completed Maiorana-McFarland class. This is done using the criterion based on second-order derivatives of a function. In particular, it is shown that, unlike in the binary case, not all quadratic bent functions are EA-equivalent to a function of the Maiorana-McFarland type. This is the first attempt to rise this problem for the generalized bent functions.

This paper consists of two main contributions. First, the Niho bent function consisting of 2rexponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland classM. In particular, it is proven that two families do not belong to the completed classM. The latter result gives a positive answer to an open problem whether the classHof bent functions introduced by Dillon in his thesis of 1974 differs from the completed classM.

In 2006, Helleseth and Kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value. The bentness was finally proved later in 2009 using a complicated technique that employs Stickelberger's theorem. Extensively using the previously found results and approaches, in this paper, a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity. Furthermore, we resolve the sign by finding the trace representation of the dual function.

Computed is the dual of the Niho bent function consisting of 2rexponents that was found by Leander and Kholosha. The algebraic degree of the dual is calculated and it is shown that this new bent function is not of the Niho type. This note is a follow-up of the recent paper by Carlet and Mesnager.

The p-ary binomial function f(x) mapping GF(p4k) to GF(p) given by f(x) = Tr4k(xp3k+p2k-pk+1 + x2)equations is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems. Finally, we characterize the size of a cyclotomic coset containing the exponent of a monomial bent functions (the same result holds in a binomial case) and provide numerical data.

Thep-ary functionf(x) mappingGF(p4k) toGF(p) and given byf(x)=Tr4k(xp3k+p2k-pk+1+x2) is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.

In this paper, we study ternary monomial functions of the formf(x) = Trn(axd), wherexisin \BBF3nandTrn: \BBF3nrarr \BBF3is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006IEEE Transactions on Information Theorypaper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular.

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m -sequence of period 2m- 1 and the d-decimation of an m-sequence of a shorter period 2n- 1 for an even number m = 2n. Assuming that d satisfies d(2l+ 1) = 2i(mod 2n- 1) for some l > 0 and i > 0, it is proved that the cross correlation takes on either exactly three or four values depending on whether I and n are coprime or not. The distribution of the cross-correlation values is also completely determined. Our results theoretically confirm the numerical data by Ness and Helleseth. It is conjectured that there are no other decimations that give at most four-valued cross correlation apart from the ones proved here.

Recently, the complete five-valued cross-correlation distribution has been determined between twom-sequences{st} and{sdt} of periods2m-1 that differ by the decimationd= [(22k+1)/(2k+1)] wheremis odd andk= 1, i.e.,d= 5/3 . In this paper, the correlation distribution is given in terms of some exponential sums for anykwhengcd(k,m) = 1 andmis odd. Furthermore, two new conjectures on exponential sums are presented that are of interest in their own right. Proving these conjectures would imply that the correlation distribution is independent ofkunder the conditions above and thus the same as for the casek= 1. The conjectures are proven formodd andk= 2 and the paper gives a new result that the correlation distribution ford=17/5 is the same as ford= 5/3.

Considered is the distribution of the cross correlation between m-sequences of length 2m-1, where m is even, and m-sequences of shorter length 2m/2-1. The infinite family of pairs of m-sequences with four-valued cross correlation is constructed and the complete correlation distribution of this family is determined.

Considered is the distribution of the cross correlation between in-sequences of length 22k-1, where m = 2k, and m-sequences of shorter length 2k-1. New pairs of m -sequences with three-valued cross correlation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued cross correlation apart from the ones proven here.

Considered are p-ary bent functions having the form f(x)=Tr/sub n/(/spl sigma//sub i=0//sup s/a/sub i/x/sup di/). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. The (weak) regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given. Finally, some criteria for an arbitrary quadratic function to be bent are proven.

In this note, a new primary construction of bent functions consisting of a linear combination of $2^r$ Niho exponents is presented. This construction generalizes one of the constructions proven by H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke, and P. Gaborit.

The distribution of the cross correlation between the ternary m-sequence {s/sub t/} of period n=3/sup m/-1 and the decimated sequences {s/sub dt/} and {s/sub dt+1/} of period (3/sup m/-1)/2, where d=3/sup k/+1/2 with k odd and gcd(k,m)=1 is determined. The method to find this distribution is related to the result by Coulter and Matthews that f(x)=x/sup d/ is a planar function over GF(3/sup m/).

We consider p-ary bent functions having the form f(x) = Tr/sub n/ (ax/sup d/). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. We also discuss the (weak) regularity of our new as well as of known monomial bent functions and give the first example of a not weakly regular bent function.