Servicios Personalizados
Revista
Articulo
Indicadores
-
Citado por SciELO
-
Accesos
Links relacionados
-
Citado por Google
-
Similares en SciELO
-
Similares en Google
Compartir
Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales
versión impresa ISSN 0370-3908
Rev. acad. colomb. cienc. exact. fis. nat. vol.36 no.139 Bogotá abr./jun. 2012
MATEMÁTICAS
* Universidad Distrital Francisco José de Caldas, Bogotá, Colombia. Email: paacostas@udistrital.edu.co
**Universidad Nacional de Colombia, Bogotá, Colombia. Email: valbis@accefyn.org.co
AMS Classification 2010: 13B25, 13F25, 11T55.
Abstract
Known results on orthogonal systems and permutation polynomials vectors over finite fields are extended to modular algebras of the form , where K is a finite field,
is an irreducible polynomial,
= 1, 2, . . ., and to the algebra of formal power series
, where L1 = K[X]/(p(X)) = L.
Key words: Permutation polynomial, orthogonal systems, permutation polynomial vectors
Resumen
Resultados sobre sistemas ortogonales y vectores de polinomios de permutación se extienden a las álgebras modulares de la forma , donde K es un cuerpo finito,
un polinomio irreducible,
= 1, 2, . . . y al álgebra de las series potenciales formales
, donde L1 = K[X]/(p(X)) = L.
Palabras clave: Polinomio de permutación, sistemas ortogonales, vectores de polinomios de permutación.
1. Introduction
Let be a finite field,
its ring of polynomials and
an irreducible monic polynomial. It is known that
is a finite field an that
,
= 1,2,..., are L-algebras (see infra for details). In previous papers ([1] and [2]) the authors obtained results about permutation polynomial over the L-algebras
(formal series over L) and
, analogos to some known results over finite fields, Galois rings
, and the rings
(see, for example, [5],[7],[8],[10],[13] and [14]). Permutation polynomial, find applications currently in cryptography and coding theory (see [4] for more references).
In this paper we deal with systems of polynomials in and
, obtaining results than in some cases lead to construct new permutation polynomials. The systems we are interested in are know as orthogonal systems has being studied by NIEDERREITER in [6] when the coefficients of the polynomials are in finite fields. Moreover, WEI & ZHANG in [12] and SHIUE, SUN & ZHANG in [8] extended some of these results to certain finite rings.
2. Preliminaries.
In this section we recall some properties of and
needed for the best undertading of what follows (see [3],[9]. Here the elements of
will be denoted by
![](img/revistas/racefn/v36n139/v36n139a07eq3.jpg)
wher is the class of equivalence
modulo
The elements of L will simply be denote by α It is know that
![](img/revistas/racefn/v36n139/v36n139a07eq4.jpg)
is a local ring with maximal ideal (Z), and
![](img/revistas/racefn/v36n139/v36n139a07e0.jpg)
are the only ideals of
![](img/revistas/racefn/v36n139/v36n139a07s6.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s7.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s8.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s1.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s6.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s9.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq5.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq6.jpg)
If
, its reduction
modulo
is the polynomial in
whose coefficients are the classes modulo
of the coefficients of
. Clearly, if
≤ µ,
If
![](img/revistas/racefn/v36n139/v36n139a07eq7.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s15.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s1.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s16.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s17.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s18.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s19.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s17.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s16.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s17.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s20.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s11.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s20.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s1.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s17.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s21.jpg)
A zero ∈
of
is said to be non-singular if
![](img/revistas/racefn/v36n139/v36n139a07eq8.jpg)
for some j= 1,...,n. Otherwise
![](img/revistas/racefn/v36n139/v36n139a07s17.jpg)
3. Orthogonal systems and permutation polynomial vectors.
In this section we introduce definitions and some results on the systems we are interested in. For a given commutative ring![](img/revistas/racefn/v36n139/v36n139a07s22.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s23.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s12.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s24.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s25.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s26.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s27.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s28.jpg)
Accordingly to ZHANG ([11],[12]) this means that the polynomial induces a permutation polynomial over
if, and only if,
is a uniform map.
The system of polynomials in
is said to be an orthogonal system over
if the map
given by
for
is a uniform map over
, i.e., if the system of equations
![](img/revistas/racefn/v36n139/v36n139a07e1.jpg)
has solutions in
where
is the reduction of
in
. If n = k the system is called permutation polynomial vector (PPV) over
Is clear that when k = 1, an orthogonal system is a permutation polynomial.
Proposition 3.1. Let . If
is an orthogonal system over
the it is and orthogonal system over
In particular, it is an orthogonal system over L.
Proof. Let be an orthogonal system over
. Then the system
![](img/revistas/racefn/v36n139/v36n139a07e2.jpg)
has solution in
. Let N be the number of these solutions. Each of them has
descendants. On the other hand, from (2) we see that there are
systems of the form
![](img/revistas/racefn/v36n139/v36n139a07e3.jpg)
each of which has, by hypothesis, different solutions, i.e., taken altogether all the above
systems will have
different solutions. Since each solution descends from
then
![](img/revistas/racefn/v36n139/v36n139a07eq9.jpg)
therefore, . So,
is an orthogonal system over
.
Corollary 3.1. Let be a permutation polynomial vector over
, then
is a permutation polynomial vector over
. In particular is a permutation polynomial vector over L.
Proposition 3.2. let be an orthogonal system over L1 = L, and such that the zeroes of
are nonsingular for all
. Then
is an orthogonal system over
(
= 1,2,...).
![](img/revistas/racefn/v36n139/v36n139a07s50.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e4.jpg)
which has solutions. Since the zeroes of
are non singular, each of the polynomial in (3) has exactly
descendants in
([1, lem. 2.2 ]). All of them are not different, since otherwise each zero of (4) would have
descendants and since each element of L can be viewed in
ways in
, then for k > 1, a zero of (4) would have more than
ways to be viewed in
![](img/revistas/racefn/v36n139/v36n139a07s56.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq11.jpg)
different solutions, thus for > 1, k >1. But this contradicts, the cardinality of
.
Therefore, the number of descendants, let us say D, contributed, by each polynomial in the system (4) to the solutions of system (3) is such that
![](img/revistas/racefn/v36n139/v36n139a07eq12.jpg)
Thus, and the total number of solutions of (3) is
is an orthogonal system over
Corollary 3.2. Let be a PPV over L such that the zeroes of
are nonsingular for all
. Then
is a PPV over
.
Proposition 3.3. Let > 1 be,
and
be polynomials such that
is a PPV over
then,
the projection of
in
is a permutation polynomial.
Proof. If is a PPV over
then, for all
the system
![](img/revistas/racefn/v36n139/v36n139a07e5.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s67.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s68.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e6.jpg)
has again a unique solution, different to the solution of (5), because, otherwise,
![](img/revistas/racefn/v36n139/v36n139a07s69.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s27.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s70.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s27.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s71.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e7.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq13.jpg)
system that necessarily is one of previous system (6). Therefore
![](img/revistas/racefn/v36n139/v36n139a07s67.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s27.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s73.jpg)
Corollary 3.3. Every polynomial in a PPV is a permutation polynomial.
Proposition 3.4. Let be an orthogonal system over
and
and at least one them a unit.
Then the polynomial
![](img/revistas/racefn/v36n139/v36n139a07eq14.jpg)
is a permutation polynomial over
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
Proof. Let be . We see that the number of solutions of
![](img/revistas/racefn/v36n139/v36n139a07eq15.jpg)
is
![](img/revistas/racefn/v36n139/v36n139a07s27.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e8.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s34.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e9.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s76.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s77.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s34.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq16.jpg)
has at least solutions. Now, the polynomial
is a permutation polynomial, and for
in
, the equation
=
has qv(k-1) solutions.
So, the equation
![](img/revistas/racefn/v36n139/v36n139a07eq19.jpg)
has
![](img/revistas/racefn/v36n139/v36n139a07eq20.jpg)
Corollary 3.4. Every polynomial an orthogonal system is a permutation polynomial.
Proposition 3.5. If
![](img/revistas/racefn/v36n139/v36n139a07eq21.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq22.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq23.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq24.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq25.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq26.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
Proof. If
is a permutation polynomial then is also a permutation polynomial ([1, lem. 3.3]). By hypothesis, the zeroes of
are non singular; then by the corollary to theorem 2 in [6], the system
is an orthogonal system over L and by proposition 3.2
is an orthogonal system over
.
Proposition 3.6. Let . If
is an orthogonal system over
then for all permutation polynomial
over
, the polynomial
![](img/revistas/racefn/v36n139/v36n139a07eq33.jpg)
is a permutation polynomial.
Proof. Let be . Since
is a permutation polynomial,
has
solutions
in
and the system
![](img/revistas/racefn/v36n139/v36n139a07eq38.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s34.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq30.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq39.jpg)
has
![](img/revistas/racefn/v36n139/v36n139a07eq40.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq41.jpg)
is a permutation polynomial over
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
Proposition 3.7. Let
![](img/revistas/racefn/v36n139/v36n139a07eq42.jpg)
If
![](img/revistas/racefn/v36n139/v36n139a07eq43.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq44.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq45.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq46.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq47.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq30.jpg)
![](img/revistas/racefn/v36n139/v36n139a07s2.jpg)
Proof. Since is a permutation polynomial, for all permutation polynomial, for all permutation polynomial
, in particular it is a permutation polynomial for
, where at least one
is a unit. Then by proposition 3.5, the system
is an orthogonal system over
.
Proposition 3.8. Let be a polynomials system. Then there exist coefficients
, where at least one of them is a unit, such that
![](img/revistas/racefn/v36n139/v36n139a07eq54.jpg)
is not a permutation polynomial.
Proof. Let where at least one of them is a unit. If the polynomial
![](img/revistas/racefn/v36n139/v36n139a07eq56.jpg)
were a permutation polynomial, then the polynomial
![](img/revistas/racefn/v36n139/v36n139a07eq57.jpg)
is also a permutation polynomial with
![](img/revistas/racefn/v36n139/v36n139a07eq58.jpg)
Proposition 3.9. If is an orthogonal system, then any of its nonempty subsystem is again an orthogonal system.
Proof. If an orthogonal system then
![](img/revistas/racefn/v36n139/v36n139a07e10.jpg)
has
![](img/revistas/racefn/v36n139/v36n139a07eq59.jpg)
![](img/revistas/racefn/v36n139/v36n139a07e11.jpg)
has
![](img/revistas/racefn/v36n139/v36n139a07eq60.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq61.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq59.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq62.jpg)
![](img/revistas/racefn/v36n139/v36n139a07sk.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq59.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq63.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq59.jpg)
![](img/revistas/racefn/v36n139/v36n139a07eq64.jpg)
Corollary 3.5. If is a PPV, then any of its nonempty subsystems is an orthogonal system.
Proof. It is clear from proposition 3.9 and the definition of PPV.
Acknowledgements
We wish to express our thanks to Yuguang Lu for his help in the reading and understanding of [8], [10] and [14].
References
[1] Acosta, S. P. A. & Albis, V.S. Characterization of multivariate permutation polynomials in positive characteristic, Sao Paulo J. Math. Sci. 3 N. 1 (2009), 1-12. [ Links ]
[2] Acosta, S. P. A. & Albis, V.S. Permutation polynomials in one indeterminate over modular algebras, Rev. Acad. Colomb. Cienc. 30 Nº. 117 (2006), 541-548. [MR:2334082] [ Links ]
[3] Albis, V. S. & Chaparro, R. On a conjecture of Borevich and Shafarevich ,Rev. Acad. Colomb. Cienc. 21 (1997), 313-319. [MR:98g:11130] [ Links ].
[4] Laigle-Chapuy, Y. Permutations polynomials and applications to coding theory., Finite Fields Appl. 13 (2007), 58-70. [ Links ]
[5] Niederreiter, H. Permutation polynomials in finite fields, Proc. Japan Acad. 46 Nº. 9 (1970), 1001-1005. [MR: 44#5298] [ Links ].
[6] Niederreiter, H. Orthogonal system of polynomials in finite fields, Proc. Amer. Math. Soc. 28 (1971), 415-422. [MR: 45#230] [ Links ].
[7] Nöbauer, Wilfriend. Zur Theorie der polynomialsformationen und permutations polynome, Math. Annalen 157 (1964), 332-342. [ Links ]
[8] Shiue, P. J. S; Sun, Q. & Zhang, Q. Multivariate permutation polynomials and orthogonal systems over residue class rings, Chinese. Ann. Math. Ser. A. 17 Nº. 1 (1996), 43-46. [in Chinese] [MR: 97e:11152] [ Links ].
[9] Smits, T. H. . On the group of units of . Indag. Math. 44 (1982), 355-358. [ Links ]
[10] Sun, Q. A note on permutation polynomials vectors over Adv. Math. (china) 25 Nº. 1 (1996), 311-314. . [in Chinese] [MR: 98h:11157] [ Links ].
[11] Zhang, Q Polynomials functions and permutation polynomials over some finite commutative rings. J. Number Theory 105 (2004), 192- 202. [ Links ]
[12] Wei, Q. & Zhang, Q. On strong orthogonal systems and weak permutation polynomials over finite commutative rings, Finite Fields Appl. 13 (2007), 113-120. [ Links ]
[13] Zhang, Q. On the polynomials in several indeterminates which can be extended to permutation polynomial vector over , Adv, Math.22 Nº.5 (1993), 456- 457. [ Links ]
[14] Zhang, Q. Permutation polynomial in several indeterminates over , Chinese An. Math. Ser. A. 16 Nº.2 (1995), 168-172. . [in Chinese] [MR: 96g:11143] [ Links ].
Recibido: 5 de marzo de 2012
Aceptado para publicación: 19 de abril de 2012