466 research outputs found
Linear complexity and trace representation of quaternary sequences over based on generalized cyclotomic classes modulo
We define a family of quaternary sequences over the residue class ring modulo
of length , a product of two distinct odd primes, using the generalized
cyclotomic classes modulo and calculate the discrete Fourier transform
(DFT) of the sequences. The DFT helps us to determine the exact values of
linear complexity and the trace representation of the sequences.Comment: 16 page
Linear complexity of Legendre-polynomial quotients
We continue to investigate binary sequence over defined by
for integers , where
is the Legendre symbol and we restrict
. In an earlier work, the linear complexity of
was determined for under the assumption of . In this work, we give possible values on the linear complexity of
for all under the same conditions. We also state that the
case of larger can be reduced to that of .Comment: 11 page
On the -error linear complexity of binary sequences derived from the discrete logarithm in finite fields
Let be a power of an odd prime . We study binary sequences
with entries in defined by using
the quadratic character of the finite field : for the
ordered elements . The
is Legendre sequence if .
Our first contribution is to prove a lower bound on the linear complexity of
for .
The bound improves some results of Meidl and Winterhof. Our second
contribution is to study the -error linear complexity of for .
It seems that we cannot settle the case when and leave it open
Polynomial quotients: Interpolation, value sets and Waring's problem
For an odd prime and an integer , polynomial quotients
are defined by which are
generalizations of Fermat quotients .
First, we estimate the number of elements for which
for a given polynomial over the finite
field . In particular, for the case we get bounds on the
number of fixed points of polynomial quotients.
Second, before we study the problem of estimating the smallest number (called
the Waring number) of summands needed to express each element of
as sum of values of polynomial quotients, we prove some lower bounds on the
size of their value sets, and then we apply these lower bounds to prove some
bounds on the Waring number using results from bounds on additive character
sums and additive number theory
On -nearly bent Boolean functions
For each non-constant Boolean function , Klapper introduced the notion of
-transforms of Boolean functions. The {\em -transform} of a Boolean
function is related to the Hamming distances from to the functions
obtainable from by nonsingular linear change of basis.
In this work we discuss the existence of -nearly bent functions, a new
family of Boolean functions characterized by the -transform. Let be a
non-affine Boolean function. We prove that any balanced Boolean functions
(linear or non-linear) are -nearly bent if has weight one, which gives a
positive answer to an open question (whether there exist non-affine -nearly
bent functions) proposed by Klapper. We also prove a necessary condition for
checking when a function isn't -nearly bent
Linear complexity of quaternary sequences over Z_4 derived from generalized cyclotomic classes modulo 2p
We determine the exact values of the linear complexity of 2p-periodic
quaternary sequences over Z_4 (the residue class ring modulo 4) defined from
the generalized cyclotomic classes modulo 2p in terms of the theory of of
Galois rings of characteristic 4, where p is an odd prime. Compared to the case
of quaternary sequences over the finite field of order 4, it is more dificult
and complicated to consider the roots of polynomials in Z_4[X] due to the zero
divisors in Z_4 and hence brings some interesting twists. We answer an open
problem proposed by Kim, Hong and Song
A General Construction of Binary Sequences with Optimal Autocorrelation
A general construction of binary sequences with low autocorrelation are
considered in the paper. Based on recent progresses about this topic and this
construction, several classes of binary sequences with optimal autocorrelation
and other low autocorrelation are presented
On the -error linear complexity of binary sequences derived from polynomial quotients
We investigate the -error linear complexity of -periodic binary
sequences defined from the polynomial quotients (including the well-studied
Fermat quotients), which is defined by where is an
odd prime and . Indeed, first for all integers , we determine
exact values of the -error linear complexity over the finite field \F_2
for these binary sequences under the assumption of f2 being a primitive root
modulo , and then we determine their -error linear complexity over the
finite field \F_p for either when or when . Theoretical results obtained indicate that such sequences possess `good'
error linear complexity.Comment: 2 figure
Linear complexity problems of level sequences of Euler quotients and their related binary sequences
The Euler quotient modulo an odd-prime power can be uniquely
decomposed as a -adic number of the form where for and we set all
if . We firstly study certain arithmetic properties of
the level sequences over via introducing a
new quotient. Then we determine the exact values of linear complexity of
and values of -error linear complexity for binary
sequences defined by .Comment: 16 page
Trace representation of pseudorandom binary sequences derived from Euler quotients
We give the trace representation of a family of binary sequences derived from
Euler quotients by determining the corresponding defining polynomials. Trace
representation can help us producing the sequences efficiently and analyzing
their cryptographic properties, such as linear complexity.Comment: 16 page
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