6,108 research outputs found
Linear complexity over F_q and over F_{q^m} for linear recurring sequences
Since the \F_q-linear spaces \F_q^m and \F_{q^m} are isomorphic, an -fold multisequence
over the finite field \F_q with a given characteristic polynomial f \in \F_q[x], can be identified
with a single sequence over \F_{q^m} with characteristic polynomial .
The linear complexity of , which we call the generalized joint linear complexity of
, can be significantly smaller than the conventional joint linear complexity of
. We determine the expected value and the variance of the generalized joint linear complexity of
a random -fold multisequence with given minimal polynomial. The result on the expected
value generalizes a previous result on periodic -fold multisequences. Finally we determine the expected
drop of linear complexity of a random -fold multisequence with given characteristic polynomial ,
when one switches from conventional joint linear complexity to generalized joint linear complexity
Tame Decompositions and Collisions
A univariate polynomial f over a field is decomposable if f = g o h = g(h)
for nonlinear polynomials g and h. It is intuitively clear that the
decomposable polynomials form a small minority among all polynomials over a
finite field. The tame case, where the characteristic p of Fq does not divide n
= deg f, is fairly well-understood, and we have reasonable bounds on the number
of decomposables of degree n. Nevertheless, no exact formula is known if
has more than two prime factors. In order to count the decomposables, one wants
to know, under a suitable normalization, the number of collisions, where
essentially different (g, h) yield the same f. In the tame case, Ritt's Second
Theorem classifies all 2-collisions.
We introduce a normal form for multi-collisions of decompositions of
arbitrary length with exact description of the (non)uniqueness of the
parameters. We obtain an efficiently computable formula for the exact number of
such collisions at degree n over a finite field of characteristic coprime to p.
This leads to an algorithm for the exact number of decomposable polynomials at
degree n over a finite field Fq in the tame case
Sequences of irreducible polynomials without prescribed coefficients over odd prime fields
In this paper we construct infinite sequences of monic irreducible
polynomials with coefficients in odd prime fields by means of a transformation
introduced by Cohen in 1992. We make no assumptions on the coefficients of the
first polynomial of the sequence, which belongs to \F_p [x], for some
odd prime , and has positive degree . If for
some odd integer and non-negative integer , then, after an initial
segment with , the degree of the polynomial
is twice the degree of for any .Comment: 10 pages. Fixed a typo in the reference
Efficient linear feedback shift registers with maximal period
We introduce and analyze an efficient family of linear feedback shift
registers (LFSR's) with maximal period. This family is word-oriented and is
suitable for implementation in software, thus provides a solution to a recent
challenge posed in FSE '94. The classical theory of LFSR's is extended to
provide efficient algorithms for generation of irreducible and primitive LFSR's
of this new type
Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings
We study linear difference equations with variable coefficients in a ring
using a new nonlinear method. In a ring with identity, if the homogeneous part
of the linear equation has a solution in the unit group of the ring (i.e., a
unitary solution) then we show that the equation decomposes into two linear
equations of lower orders. This decomposition, known as a semiconjugate
factorization in the nonlinear theory, generalizes the classical operator
factorization in the linear context. Sequences of ratios of consecutive terms
of a unitary solution are used to obtain the semiconjugate factorization. Such
sequences, known as eigensequences are well-suited to variable coefficients;
for instance, they provide a natural context for the expression of the
classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear
difference equations with periodic coefficients and also derive formulas for
the general solutions of linear functional recurrences satisfied by the
classical special functions such as the modified Bessel and Chebyshev.Comment: Application of nonlinear semiconjugate factorization theory to linear
difference equations with variable coefficients in rings; 29 pages,
containing the main theory and more than 8 examples worked out in detai
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