Hans Dobbertin

New cyclic difference sets with Singer parameters

The main result in this paper is a general construction of @f(m)/2 pairwise inequivalent cyclic difference sets with Singer parameters (v,k,@l)=(2^m-1,2^m^-^1,2^m^-^2) for anym>=3. The construction was conjectured by the second author at Oberwolfach in 1998. We also give a complete proof of related conjectures made by No, Chung and Yun and by No, Golomb, Gong, Lee and Gaal which produce another difference set for each m>=7 not a multiple of 3. Our proofs exploit Fourier analysis on the additive group of GF(2^m) and draw heavily on the theory of quadratic forms in characteristic 2. By-products of our results are a new class of bent functions and a new short proof of the exceptionality of the Muller-Cohen-Matthews polynomials. Furthermore, following the results of this paper, there are today no sporadic examples of difference sets with these parameters; i.e. every known such difference set belongs to a series given by a constructive theorem.

Almost perfect nonlinear power functions on GF(2/sup n/): the Welch case

We summarize the state of the classification of almost perfect nonlinear (APN) power functions x/sup d/ on GF(2/sup n/) and contribute two new cases. To prove these cases we derive new permutation polynomials. The first case supports a well-known conjecture of Welch stating that for odd n=2m+1, the power function x/sup 2m+3/ is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/sup m+1/.

Construction of bent functions via Niho power functions

A Boolean function with an even number n = 2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x) = tr(α1xd1 + α2xd2), α1, α2, x ∈ F2n, are considered, where the exponents di (i = 1, 2) are of Niho type, i.e. the restriction of xdi on F2k is linear. We prove for several pairs of (d1, d2) that f is a bent function, when α1 and α2 fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F2n, are bijective.

Binary m-sequences with three-valued crosscorrelation: a proof of Welch's conjecture

We prove the long-standing conjecture of Welch stating that for odd n=2m+1, the power function x/sup d/ with d=2/sup m/+3 is maximally nonlinear on GF(2/sup n/) or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence of degree n and a decimation of that sequence by 2/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/sup m+1/.

Weight Divisibility of Cyclic Codes, Highly Nonlinear Functions on F 2 m , and Crosscorrelation of Maximum-Length Sequences

We study [2m-1,2m]-binary linear codes whose weights lie between w0 and 2m-w0, where w0 takes the highest possible value. Primitive cyclic codes with two zeros whose dual satisfies this property actually correspond to almost bent power functions and to pairs of maximum-length sequences with preferred crosscorrelation. We prove that, for odd m, these codes are completely characterized by their dual distance and by their weight divisibility. Using McElieces theorem we give some general results on the weight divisibility of duals of cyclic codes with two zeros; specifically, we exhibit some infinite families of pairs of maximum-length sequences which are not preferred.

Niho type cross-correlation functions via dickson polynomials and Kloosterman sums

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.

Some new three-valued crosscorrelation functions for binary m-sequences

In 1972 Niho gave various conjectures about the crosscorrelation between a binary maximum-length linear shift register sequence and a decimation of that sequence by an integer d. We prove that the crosscorrelation function for two new values of d takes on precisely three values and thereby confirm two of Nihos conjectures.

Ternary m-sequences with three-valued cross-correlation function: new decimations of Welch and Niho type

We show that the cross correlation between two ternary m-sequences of period 3/sup n/-1 that differ by the decimation d=2/spl middot/3/sup m/+1, where n=2m+1, takes on three different values. We conjecture the same result for the decimation d=2/spl middot/3/sup r/+1, where n is odd and r is defined by the condition 4r+1/spl equiv/0 mod n. These two new cases form in a sense ternary counterparts of two previously confirmed binary cases, the conjectures of Welch and Niho (1972).

Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.

Normal extensions of bent functions

In this paper, the notion of normal extension is introduced for bent functions, i.e., maximally nonlinear Boolean functions. We apply this concept to characterize when the direct sum of bent functions is normal, and we prove that the direct sum of a normal bent function and a nonnormal bent function is always nonnormal.