^* 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

wher is the class of equivalence modulo The elements of L will simply be denote by α It is know that

is a local ring with maximal ideal (Z), and

If , its reduction modulo is the polynomial in whose coefficients are the classes modulo of the coefficients of . Clearly, if ≤ µ,

If

A zero ∈ of is said to be non-singular if

3. Orthogonal systems and permutation polynomial vectors.

induces a permutation polynomial over if, and only if, the equation has exactly

solutions for each

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

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

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

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

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,...).

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

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

.

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 .

is a permutation polynomial over and the zeroes of are non singular for all and where at least one them is a unit, then is an orthogonal system over .

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 .

is a permutation polynomial over for all permutation polynomial over , , and the zeroes of are non singular for all and then is an orthogonal system over .

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

Proof. Let where at least one of them is a unit. If the polynomial

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

Corollary 3.5. If is a PPV, then any of its nonempty subsystems is an orthogonal system.

References

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Recibido: 5 de marzo de 2012

Aceptado para publicación: 19 de abril de 2012