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MATHEMATICS OF COMPUTATIONVOLUME 63, NUMBER 208OCTOBER 1994, PAGES 759-765

PRIMITIVE NORMAL POLYNOMIALS OVER FINITE FIELDS

ILENE H. MORGAN AND GARY L. MULLEN

Abstract. In this note we significantly extend the range of published tables

of primitive normal polynomials over finite fields. For each p" < 1050 with

P < 97, we provide a primitive normal polynomial of degree n over Fp.

Moreover, each polynomial has the minimal number of nonzero coefficients

among all primitive normal polynomials of degree n over Fp. The roots of

such a polynomial generate a primitive normal basis of Fpn over Fp , and

so are of importance in many computational problems. We also raise several

conjectures concerning the distribution of such primitive normal polynomials,

including a refinement of the primitive normal basis theorem.

1. INTRODUCTION

For q a prime power and n > 2 an integer, let Fq denote the finite field oforder q. It is well known that Fqn can be viewed as a vector space of dimension

n over Fq. A basis of Fqn over Fq of the form a, cfl, ... , of is called a

normal basis, and if a is also a primitive element of Fqn, i.e., if o generates

the multiplicative group F*n, then the basis is said to be a primitive normal

basis.Normal and primitive normal bases are of great importance in many com-

putational problems involving finite fields, and so there has been considerableeffort directed to discussing various methods of obtaining such elements. It is

well known that for any prime power q and any integer n>2,Fqn contains a

normal basis over Fq ; see for example Lidl and Niederreiter [ 17, Theorem 3.73].

In [7], Davenport showed that if q = p is a prime, then for each « > 2, iycontains a primitive element a that generates a primitive normal basis of Fpn

over Fp , and in [4], Carlitz showed that for sufficiently large qn , Fqn contains

a primitive element that generates a primitive normal basis over Fq. Recently,Lenstra and Schoof [ 15] showed that for every prime power q and every integer

n >2, Fqn contains a primitive normal basis over Fq. This result completely

settled the question of the existence of primitive normal bases over finite fields,

although the above methods are all nonconstructive. Hachenberger [11] gives

an alternative proof of the primitive normal basis theorem, which at least the-

oretically provides a method to determine all primitive normal elements.

Received by the editor August 23, 1993.

1991 Mathematics Subject Classification. Primary 11T06; Secondary 11T30.Key words and phrases. Finite field, primitive normal basis.

We would like to thank the National Security Agency for partial support under the second

author's grant agreement #MDA904-92-H-3044.

©1994 American Mathematical Society0025-5718/94 $1.00+ $.25 per page

759

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

760 I. H. MORGAN AND G. L. MULLEN

For the more practical matter of finding a primitive normal element, Stepanov

and Shparlinski [26] give an upper bound N so that if ß e Fq« is a fixed prim-

itive element, then the sequence ß, ß2, ... , ßN must contain an element that

generates a primitive normal basis. In [9] von zur Gathen and Giesbrecht pro-

vide a probabilistic polynomial-time algorithm for the determination of a prim-itive normal element. Shoup [24] considers the problem of how to determin-

istically generate in polynomial time a subset of Fpn that contains a primitive

element. See also Menezes [18, Chapters 4-5] and Shparlinski [25, Chapters 2-

5] for discussions of various algorithms and theoretical results concerning the

distribution of primitive normal elements. For a discussion of various otherkinds of bases, see for example [8, 14, 19, 21-23]. We also refer to Lidl [16,§3] for a discussion of a number of recent results related to various types of

bases over finite fields, and we refer to [1] for a discussion of algorithmic com-

putations involving primitive and normal elements in the field F2m. In [20],

Niederreiter shows that normal bases are useful in the problem of factoring

polynomials over finite fields.A monic polynomial f(x) of degree n over Fq is called a primitive (resp.

normal) polynomial if it is the minimal polynomial of a primitive element of

Fqn (resp. it is the minimal polynomial of an element which generates a normal

basis of Fqn over Fq ). Alternatively, f(x) is primitive if q" - 1 is the small-

est positive integer s such that f(x) divides xs - 1, and it is normal if any

root of f(x) generates a normal basis of Fq* over Fq. Thus, f(x) is normal

if f(x) belongs to the linearized polynomial xq" -x so that the monic lin-

earized polynomial of least degree satisfied by a root of f(x) is the linearized

polynomial x9" - x. Recall that a linearized polynomial is a polynomial of the

form 2~3"~o' a¡x9' with a¡ e Fq«. In the terminology of Beard and West [3], aprimitive (resp. normal) polynomial is said to be of the first (resp. second) kind,and a polynomial which is both primitive and normal is said to be of the third

kind. We will however use the more natural terms of primitive (resp. normal)

polynomials and simply say that a polynomial is a primitive normal polynomial

if it is both primitive and normal.It is known that there are <j>(qn - l)/n primitive polynomials of degree n

over Fq, where <p denotes Euler's totient function from elementary number

theory; see [17, Theorem 3.5]. Moreover, there are &q(x" - l)/n normal poly-

nomials of degree n over Fq, where <f>q denotes the Euler function defined

on the ring Fq[x]; see [17, §3.4]. Very recently, Akbik [2] has obtained an

apparently different formula for the number of normal basis generators of Fp»

over Fp with p prime, although that formula is, in reality, just a different way

of rewriting the standard formula Q>q(x" - l)/n .While there is no known closed formula for the number of primitive normal

polynomials of degree n over Fq , Carlitz [4] obtained the asymptotic bound

that there are N'/n primitive normal polynomials over Fq of degree n , where

N' = <p(qn - i)%(xn - l)/qn + 0(q"(x'2+(ï),

the implied constant depending only upon e.For each p, d, and n satisfying p < 102, pd < 103, and pdn < 106, Beard

and West [3] provided a primitive normal polynomial of degree n over Fpi.

Very recently, Gulliver, Serra, and Bhargava [10] gave lists of primitive normal


PRIMITIVE NORMAL POLYNOMIALS OVER FINITE FIELDS 761

polynomials of small degrees over Fq for q < 19 except for q = 9. (It should

be pointed out that their polynomial 11000002 over FXi cannot be primitivesince -2 is not a primitive element in FX1; see Theorem 1 below. A primitivenormal polynomial is 11000007.)

The purpose of this paper is to significantly extend the range of publishedtables of primitive normal polynomials over prime fields. The range of our tables

is p" < 1050 with p < 97. Moreover, each of our polynomials has at most five

nonzero coefficients. Such polynomials with small Hamming weight provide

for easy implementation of the corresponding extension field arithmetic, andso our tables are intended to provide the practitioner with an easily accessible

collection of primitive normal polynomials for use in various applications.While extensive tables of primitive polynomials are available, see for example

Hansen and Mullen [12], most of those polynomials are not normal since theirtrace coefficients are 0, and thus they cannot be used to generate a normal basisof Fpn over Fp.

The following results from Lidl and Niederreiter [17, Theorem 3.18 and

Corollary 2.39] provide algorithms for testing whether a given polynomial f(x)

of degree n over Fq is primitive and normal.

Theorem 1. The monic polynomial f e Fq[x] of degree n > 1 is a primitive

polynomial over Fq if and only if (-1)" f(0) is a primitive element of Fq and

the least positive integer r for which xr is congruent modf(x) to some element

ofFq is r = (q»-l)/(q-l).

Theorem 2. For a e Fqn, {a, a9, ... , a9""1} is a normal basis of Fqn over Fq

if and only if the polynomials xn - 1 and axn~ ' + a9xn~2 H-\-a9" x + of

in Fqn[x] are relatively prime.

2. Tables

The methods of Hansen and Mullen [12] provide the starting point for our

search. Because of the availability of software programs used in [12], we haveused the following strategy in searching for a primitive normal polynomial over

Fp, even though some of the methods discussed in § 1 at least theoretically

provide faster algorithms. We first locate a primitive polynomial of degree nover Fp , as in [12], using Theorem 1. If a polynomial is primitive by Theorem

1, then the Euclidean algorithm is used to test a root a of the polynomialagainst Theorem 2 in order to determine whether the polynomial is a normal

polynomial, and hence whether its roots generate a normal basis of Fpn over

FP.

For ease of implementation of extension field arithmetic by a practitioner,

our search has focused on polynomials of low Hamming weight, i.e., on poly-

nomials with a small number of nonzero coefficients. In particular, for each

p and n , the polynomial which we have listed has the minimal weight among

all primitive normal polynomials of degree n over Fp . In addition, the givenpolynomial is the first primitive normal polynomial obtained among all polyno-

mials of that weight which were tested in the following natural order as in [12].

Consider f(x) = x"+ ¿~^"~0l a,jc' of degree n over Fp. Let Nf = Ph+Em a'P'be the corresponding number in base p. Thus, among polynomials of the same

weight, f(x) was tested before g(x) if Nf < Ng. Subject to this ordering, the



first primitive normal polynomial obtained is listed in the table. (Note that anynormal polynomial must have nonzero trace coefficient, i.e., the coefficient of

x"~x must be nonzero.) An asterisk denotes the fact that for the given poly-

nomial f(x), the reciprocal polynomial f(x)* - xnf(l/x) is also primitivenormal. The reciprocal of a primitive polynomial is of course always primitive,

but the reciprocal of a normal polynomial is not always normal.

In the Supplement section at the end of this issue we provide tables of the

primitive normal polynomials obtained using the procedure described above.

For each p" < 1050 with p < 97, we provide a primitive normal polynomialof degree n over Fp. As in [12], only the nonzero coefficients are listed so

that, for example, the polynomial x8 + x1 + 3 over F1 is listed as 8:1, 7:1,

0:3. Copies of the tables either in hardcopy or electronic form are available

from the authors.

3. Conjectures

In this section we raise several conjectures concerning the distribution ofprimitive normal polynomials over finite fields. We remind the reader that if

n > 2, the trace function is defined from Fqn to Fq by TR(y) = y + y9 + y92 +

-1- y9"' , and that, if y has degree n over Fq , the trace of y is of course

the negative of the coefficient of xn~x in the minimal polynomial of y over

Fq. We first recall that in [5] Cohen showed that, except for several necessary

exceptions, there is for every prime power q and every integer n > 2, aprimitive polynomial of degree n over Fq with an arbitrarily specified tracecoefficient. More specifically, he proved:

Theorem 3. Let n > 2 and let a e Fq with a ^ 0 if n = 2 or if n = 3 andq = 4. Then there exists a primitive polynomial of degree n over Fq with trace

a.

Based upon computer evidence and several special cases below, we propose

the following refinement of the primitive normal basis theorem of Lenstra and

Schoof [15] as well as the above result of Cohen:

Conjecture 1. Let n > 2 and let a e Fq with a # 0. Then there exists a

primitive normal polynomial of degree n over Fq with trace a.

We note that Cohen [6, p. 53] alludes to a special case of Conjecture 1 and also

briefly discusses a possible method of attack. If a denotes the trace coefficient

of a normal polynomial f(x) of degree n, then clearly a cannot be zero;otherwise, the roots of / would not be independent, and hence they could not

form a basis of Fq* over Fq. Conjecture 1 is thus clearly true for q = 2, and

it follows from Hachenberger [11, p. 146] and Theorem 3, that the conjecture

is also true for n = 2 and any prime power q.

Theorem 4. If every prime factor of q - 1 divides n, then for every nonzero

element b e Fq there is a primitive normal polynomial with trace coefficient b.

Proof. Let a be a primitive normal element in Fqn. We just need to checkthat ba is also a primitive normal element for every nonzero element b e Fq.

Clearly, ba is a normal element. Let w = (q" - l)/(q - 1), and let a = aw

be the norm of a over Fq. Then a is primitive in Fq and b = a' for some



0 < t < q - 2. Moreover, ba = aw,+l and ba is primitive if and only if

1 = gcd(wt +1, qn - 1) = gcd(wt + 1,0-1) = gcd(«/ +1,0-1).

Since every prime factor of q - 1 is a divisor of n, we have

gcd(«/+ 1, q- 1) = 1.

Thus, ba is primitive for every nonzero b e Fq and the proof is complete. D

Corollary 5. Conjecture 1 is true whenever n is a multiple of q - 1.

The following elementary argument provides a class of values of q for which

Conjecture 1 is also true. Assume that f(x) = xn + bx"~x + ■■ ■ + bxx + bo

with b ^ 0 is a primitive normal polynomial over Fq, a e Fqn is a root of

f(x), and that TR(q) = -b. If c / 0 € Fq, then clearly ca also generates

a normal basis of Fqn over Fq, and moreover, TR(ca) = -cb. Since a is a

primitive element of Fqn, we may write c e Fq as c = ak^9"' +9"~ +• +«+u ( o <

k < q - 1. Thus, the order of ca is ^{qn_x<k{qj"-qli+...+q+X)+X). As a result,

Conjecture 1 will be true if ca is a primitive element for each c ^0 e Fq, i.e.,

if gcd(<?-l,rv(0"-1+0''-2 + --- + (7 + l) + l) = 1, for fc = 0, 1.....0-2.If Conjecture 1 fails, it is most likely to fail for small values of q and n.

However, it has been verified by machine calculation to be true for all primepowers q < 97 with n < 6.

We also propose the following conjecture, which is a refinement of the anal-ogous conjecture of Hansen and Mullen [13, p. 434] for primitives over a field

with a prime number of elements.

Conjecture 2. (1) For each prime p and each n > 2 there exists a primitive

normal polynomial of degree n over Fp with at most five nonzero coefficients.

(2) If p> 11, then there is a primitive normal polynomial of degree n over

Fp with at most four nonzero coefficients.

It may be that (2) holds for p > 3 except for p = 5, n = 32 and p = 1, n =24.

The following algorithm can be used to construct a primitive (normal) poly-

nomial of degree n over a nonprime field Fpm of order pm. Let / be a primi-

tive polynomial of degree mn over Fp, taken for example from [12]. Choosean element ß e Fpmn whose order is equal to pmn - 1 so that ß is a prim-

itive element in Fpmn. Calculate the minimal polynomial Mß(x) of ß over

Fpm = Fq as Mß(x) = \~[n~o\x -ß9'), which according to [17, Corollary 2.19]

is a primitive polynomial of degree n over Fpm. In order to check whether

ß generates a normal basis of Fpmn over Fpm , we now simply apply Theorem

2. This technique was in fact used to verify Conjecture 1 for all prime powers

q < 97 with n < 6.As an illustration of this technique, as in Table A of [17], assume that F2\

is multiplicatively generated by a root a of the primitive polynomial x4 +x3 + 1 over F2. Let ß = a so that ß and ß4 form a primitive normalbasis of Fié over F4. Consequently, the minimal polynomial Mß(x) of ß

over F4 given by (x - ß)(x - ß4) = x2 + a5x + a5 is a primitive polynomial

of degree 2 over F4. Similarly, if y = a2, then y and y4 form a primitivenormal basis, and the minimal polynomial My(x) — x2+ax0x+ax0 is primitive



normal. Analogously, if Ô = a7 and e = axx, then Ms(x) = x2 + x + a5 and

Me(x) = x2+x+aX0 are also primitive normal polynomials over F4. Moreover,

x4 + x3 + 1 = Mß(x)My(x) and x4 + x + 1 = Ms(x)M((x).Alternatively, one could proceed as follows to construct a primitive (normal)

polynomial of degree n over a nonprime field Fpm. As above, let / be a

primitive polynomial of degree mn over Fp. Then by [17, Theorem 3.46], /factors over Fpm into m irreducibles, each of degree n. Moreover, from [17,

Corollary 2.19], each irreducible factor is a primitive polynomial of degree nover Fpm. As above, Theorem 2 can then be applied to determine whether a

root of such a primitive polynomial generates a normal basis.

Acknowledgment

We would like to thank the PSU Mathematics Department for use of its

network of SUN workstations. Special thanks are due Tom Hansen for use of

several programs from [12] and Gerry McKenna for use and technical support

of his finite field software package as well as a number of helpful comments.Without their assistance, this project would not have been undertaken. Finally,

we thank Shuhong Gao for his proof of Theorem 4, and the referee for pointing

out several misprints in an earlier version.

Bibliography

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J. Cryptology6(1993), 3-13.

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3. J. T. Beard and K. I. West, Some primitive polynomials of the third kind, Math. Comp. 28

(1974), 1166-1167.4. L. Carlitz, Primitive roots in a finite field, Trans. Amer. Math. Soc. 73 (1952), 373-382.

5. S. D. Cohen, Primitive elements and polynomials with arbitrary trace, Discrete Math. 83

(1990), 1-7.6. S. D. Cohen, Primitive elements and polynomials: existence results, Finite Fields, Coding

Theory, and Advances in Communications and Computing (G. L. Mullen and P. J.-S. Shiue,

eds.), Lecture Notes in Pure and Appl. Math., vol. 141, Marcel Dekker, New York, 1993,

pp. 43-55.

7. H. Davenport, Bases for finite fields, J. London Math. Soc. 43 (1968), 21-39; Addendum,ibid. 44(1969), 378.

8. S. Gao and H. W. Lenstra, Jr., Optimal normal bases, Des. Codes Cryptog. 2 (1992),

315-323.

9. J. von zur Gathen and M. Giesbrecht, Constructing normal bases in finite fields, J. Symbolic

Comput. 10 (1990), 547-570.

10. T. A. Gulliver, M. Serra, and V. K. Bhargava, The generation of primitive polynomials in

GF(q) with independent roots and their applications for power residue codes, VLSI testing

and finite field multipliers using normal basis, Internat. J. Electron. 71 (1991), 559-576.

11. D. Hachenberger, On primitive and free roots in a finite field, Appl. Alg. Eng., Comm.

Computing 3 (1992), 139-150.

12. T. Hansen and G L. Mullen, Primitive polynomials over finite fields, Math. Comp. 59

(1992), 639-643; Supplement: 59 (1992), S47-S50.

13. _, Primitive polynomials of low weight, Finite Fields, Coding Theory, and Advances in

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18. A. J. Menezes, ed., Applications of finite fields, Kluwer, Dordrecht, 1993.

19. R. C. Mullin, I. M. Onyszchuk, S. A. Vanstone, and R. M. Wilson, Optimal normal bases

in GF(pn), Discrete Appl. Math. 22 (1988/89), 149-161.

20. H. Niederreiter, Factoring polynomials over finite fields using differential equations and

normal bases, Math. Comp. 62 (1994), 819-830.

21. S. Schwarz, Irreducible polynomials over finite fields with linearly independent roots, Math.

Slovaca38(1988), 147-158.

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38 (1988), 291-312.

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pendent roots, Math. USSR-Sb. 63 (1989), 507-519.

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Kluwer, Dordrecht, 1992.

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Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla,

Missouri 65401-0249E-mail address : imorganflumr. edu

Department of Mathematics, The Pennsylvania State University, University Park,

Pennsylvania 16802E-mail address : mullenOmath. psu. edu


mathematics of computationvolume 63, number 208october 1994, pages s19-s23

Supplement to

PRIMITIVE NORMAL POLYNOMIALS OVER FINITE FIELDS

ILENE H. MORGAN AND GARY L. MULLEN

In the tables that follow, we have given, for each prime power pn < 1050 with p < 97.

a primitive normal polynomial of degree n over Fp. Each polynomial listed has mini-

mal weight, i.e.,the minimal number of nonzero coefficients, among all primitive normal

polynomials of that degree over Fp.

Only the nonzero terms are represented so that, for example, the polynomial x* + x' +3

over F7 is listed as 8:1, 7:1, 0:3. An asterisk denotes the fact that for the given polynomial

/(x), the reciprocal polynomial /(x)* = xnf(l/x) is also primitive normal. Copies of the

tables, either in electronic or hardcopy form, are available upon request from the authors.

For further details, we refer to the paper of the same title by the authors in this issue

of Mathematics of Computation.

© 1994 American Mathematical Society

0025-5718/94 $ 1.00 + $.25 per page

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primitive normal polynomials over finite fields · 2018-11-16 · primitive normal polynomials over...

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