110 research outputs found
Remarks on a cyclotomic sequence
We analyse a binary cyclotomic sequence constructed via generalized cyclotomic classes by Bai et al. (IEEE Trans Inforem Theory 51: 1849-1853, 2005). First we determine the linear complexity of a natural generalization of this binary sequence to arbitrary prime fields. Secondly we consider k-error linear complexity and autocorrelation of these sequences and point out certain drawbacks of this construction. The results show that the parameters for the sequence construction must be carefully chosen in view of the respective application
Continued fraction for formal laurent series and the lattice structure of sequences
Besides equidistribution properties and statistical independence the lattice profile, a generalized version of Marsaglia's lattice test, provides another quality measure for pseudorandom sequences over a (finite) field. It turned out that the lattice profile is closely related with the linear complexity profile. In this article we give a survey of several features of the linear complexity profile and the lattice profile, and we utilize relationships to completely describe the lattice profile of a sequence over a finite field in terms of the continued fraction expansion of its generating function. Finally we describe and construct sequences with a certain lattice profile, and introduce a further complexity measure
On the linear complexity of Sidel'nikov Sequences over Fd
We study the linear complexity of sequences over the prime field Fd introduced by Sidelânikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation
Multisequences with high joint nonlinear complexity
We introduce the new concept of joint nonlinear complexity for multisequences
over finite fields and we analyze the joint nonlinear complexity of two
families of explicit inversive multisequences. We also establish a
probabilistic result on the behavior of the joint nonlinear complexity of
random multisequences over a fixed finite field
On the normality of -ary bent functions
Depending on the parity of and the regularity of a bent function from
to , can be affine on a subspace of dimension
at most , or . We point out that many -ary bent
functions take on this bound, and it seems not easy to find examples for which
one can show a different behaviour. This resembles the situation for Boolean
bent functions of which many are (weakly) -normal, i.e. affine on a
-dimensional subspace. However applying an algorithm by Canteaut et.al.,
some Boolean bent functions were shown to be not - normal. We develop an
algorithm for testing normality for functions from to . Applying the algorithm, for some bent functions in small dimension we
show that they do not take on the bound on normality. Applying direct sum of
functions this yields bent functions with this property in infinitely many
dimensions.Comment: 13 page
Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension
The Walsh transform QËQ^ of a quadratic function Q:FpnâFpQ:FpnâFp satisfies |QË(b)|â{0,pn+s2}|Q^(b)|â{0,pn+s2} for all bâFpnbâFpn , where 0â€sâ€nâ10â€sâ€nâ1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(ââ(nâ1)/2âi=1aix2i+1):aiâF2}C1={Q(x)=Trn(âi=1â(nâ1)/2âaix2i+1):aiâF2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(â(n/2)â1i=1aix2i+1)+Trn/2(an/2x2n/2+1):aiâF2}C2={Q(x)=Trn(âi=1(n/2)â1aix2i+1)+Trn/2(an/2x2n/2+1):aiâF2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(âân/2âi=0aixpi+1):aiâFp}D={Q(x)=Trn(âi=0ân/2âaixpi+1):aiâFp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order ReedâMuller codes corresponding to C1C1 and C2C2 in terms of a generating function
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