JP Jour. Algebra, Number Theory & Appl. 4(1) (2004), 79-87
THE THREE FIXED COEFFICIENT PRIMITIVE POLYNOMIAL THEOREM STEPHEN D. COHEN Department of Mathematics, University of Glasgow Glasgow G12 8QW, Scotland e-mail:
[email protected] CHARLES KING Department of Mathematics, University of Glasgow Glasgow G12 8QW, Scotland e-m.ail:
[email protected]. uk Abstract The final computational step is provided that secures the theorem that, for any integer n ~ 7, there exists a primitive polynomial over any finite field Fq whose first three coefficients are arbitrarily
prescribed
elements of Fq. This builds on theoretical work of Fan Shuqin and Han Wenbao, especially in the case when
l'q
of D. Mills when the characteristic conjecture of Mills is established.
has characteristic
2 or 3, and
is at least 5. In particular,
a
1. Introduction
Let IFq be the finite field of order q, a power of the (prime) characteristic
p. The first
In
coefficients of a monic polynomial (of degree
n) refer
XIL
+
to the
2000 Mathematics Subject Classification: 11T06. Key words and phrases: primitive polynomial, finite field, fixed coefficients. 'This work was supported by a Nuffield Foundation Grant URB-03. Received September 18, 2003 © 2004 Pushpa Publishing House
STEPHEN D. COHEN and CHARLES KING
80
coefficients
aI, ..., am'
For many purposes, it would be useful to be
guaranteed the existence of a primitive polynomial of degree n with some coefficients arbitrarily prescribed. For example, it could be that the first In
coefficients are those specified. A complete existence theorem has been
established for m = 1 (see [1]) and, provided n ~ 5 and p is odd, for m
=
2 (see, for example, [2]). (Note that a natural difficulty arises in
tackling this question whenever m existence problem for. nz,
=
s p.) Recent work has focused on the
3 with n ~ 7, and, save for a finite number of
possible exceptional values of q when n = 7, the following result has been proved. Theorem primitive
1. For every finite field
polynomial
of degree
ii
}q
and any n ~ 7, there exists a
over Fq with its first three coefficients
arbitrarily prescribed members of IFq. Specifically, using a new p-adic method that deals effectively with the awkward cases when p = 2 or 3, Fan and Han [3] established Theorem 1 whenever n ~ 8; a number of cases required direct verification through computation in the field. For n
=
7, their estimates would suffice to show
that there are only finitely many possible exceptions, but the magnitude of this number made completion through computation prohibitive. Independently, using a superior estimate but assuming p ;:::5, Mills proved the assertion of Theorem 1 for n ~ 7 provided q > 361 and conjectured ([5, Conjecture 5.2]) that the result holds generally (for p ~ 5). The estimates used in [5] are better by a factor of order
those of l3] except when
al
=
a2
=
a3
.Jq
than
= 0 are the prescribed values, So
in this work, the "all-zero" case is distinguished from all others. In fact, Mills used computation to eliminate (as possible exceptions) a number of prime powers q
S
179. Other values (comprising 9 primes and 6 non-
primes) up to 361 , ere unable to be dealt with through computation at the time. Finally, Fan and Han [4] refined their estimates for use when
p
= 2
or 3 and completed the proof of Theorem 1 for these characteristics.
·.. PRIMITIVE Again,
some
additionally,
computations
POLYNOMIAL THEOREM were
required
in the all-zero case for q
The purpose
for values
81 of q
computations
by means of using Magma, Version 2.10) which eliminate Conjecture
complete
81
(obtained
those cases left
the proof of Theorem
1. In particular,
5.2 of [5] is shown to hold. We also verify independently
cases established
computationally
these,
n is 7. A total of 20 values
the degree
arbitrary
triples
(aI, a2, a3)'
and,
= 256.
of this note is to describe further
open in [5] and thereby
s:
in [4] for values
Altogether
those
of q ~ 16. In all of
of q were considered
for
13 values needed to be checked
in the case of the all-zero triple; of these only 4 ones were not already covered by the general case. 2. Method ':"he proportion . .. prnrutive
.
IS
of monic polynomials
q)( qll := -----.
P q ()ti
-
over
Fq of degree n that are
I-ience J a "random" ran om se arc h on po 1ynomia. 1s
1)
nqll
with the first three coefficients average,
a
example,
£361 (7) > 26.4. To improve
extensive
searching
that
if
XII
an = (-ltw,
primitive
prescribed
+
al
polynomial
is required),
xn-1 + ... + an
can be expected
after
£q(n)
=
liP q(n)
(-l)"w.
triple
(aI, a2, a3)
E
Fq [x]
is
primitive,
element
for the
f, i.e., begin by setting
polynomial except an-l primitive
Subject to this constraint search
For
because
we improve on this as follows. Observe then
necessarily
of lFq itself. So, for given
values of q and n fix, once and for all, a specific primitive =
trials.
the "strike rate" (important
where w is a primitive
set an
to deliver, on
element
(fixed constant
lexicographically
wand
term), given a
earliest
to zero all non-specified
primitive coefficients
(that of z), which is allowed to vary. If that fails to produce a
polynomial,
take
all-2
= 1 and vary
an-l
again, and so on (in
the obvious manner). With respect when
using
to the above, a qualification
Magma
is that
the
system
to the term "lexicographic" orders
the
elements
of
'1~q
82
STEPHEN
differently
according
prime, then
)I~
D. COHEN and CHARLES KING
as to whether
purposes
polynomial"
the expected
is
over the prime
subfield:
the latter
is
0 < -1 < 1 < -2 < ...
= 2. "strike rate" we first give a formula
may not be new) for the number degree n with prescribed
n.= lq(n)mq(n),
p
the easiest value of w to use is a
first with respect to the ordering
p odd, and 0 < 1, p
To calculate
=
q
root of the "Conway
p -1
q
is 1 < w < w2 < ... < w -1 < 0, where w is a primitive
element of Fq. For calculating
< -2-'
or not. When
is ordered as 0 < 1 < 2 < ... < p - 1. When q is composite,
then the ordering
lexicographically
q is prime
of merely
irreducible
coefficient as described
(which
polynomials
of
above. To do this write
lq(n) is defined as the largest divisor of n whose
where
of q - 1. Thus mq (n) is the part of n none of
prime factors are divisors
whose prime factors are divisors of q -1. Of course, lq(n) and mq(n) are eo-prime. Lernrna 2. Let w be a given primitive
of irreducible polynomials
element of Fq. Then the number
of degree n over Fq with constant term (- Itw
is given by
where if mq(n)
=
1,
if mq(n) > 1.
Proof. By expressing have that
the total
the value as a character
number
of elements
sum (or otherwise),
~ of .~ n
that
q
have
we
[~-norm
qll._l
Nn(~) (= ~ elements ~ E
)I~n
q-l ) = w
of a subfield
.
qn_l --1-. This q-
IS
F
q
a, din
count,
however,
with the property.
is the root of an irreducible
polynomial
may
Suppose
of degree
include
din d over
and l~.
·.. PRIMITIVE POLYNOMIALTHEOREM
83
n
= [N d(~)]d
and this cannot be a
primitive element of i~q unless gCd(~ , q - 1)
= 1. The latter is the case
(Thus
if q. Of course,
these values depend on the choice of w made. The value of
tq
is to be
q.
In every
compared with Eq. Table 1 summarises the results for the 9 prime values of case Nq
=
O.
Table 1. Prime values of q q
q3
Iq
181
5929741
1
22.32.5
191
6967871
1
193
7189057
199
7880599
Lq
Eq
w
Tq
tq
29·281·4338835663
7.236
2
94
7.197
2·5·19
127·197·10627·183569
7.055
19
97
7.020
1
263
43·1208179545301
7.130
5
97
7.091
1
2·3 2 .]1
29·211·883·11552213
7.256
3
104
7.222
2.3.57
307189·41233879
6.967
2
92
6.933
23·37
29·491·1709·5076443
7.236
3
106
7.205
Mq
2
211
9393931
7
223
11089567
1
241
13997521
1
24.3.5
63113·3117376319
6.971
7
101
6.943
29·62273·160\74771
7.222
3
110
7.193
2180921·604842197
6.979
3
114
6.957
I
257
16974593
1
28
331
36264691
1
2·3·5·11
Next, Table 2 contains the data relevant to the 6 composite values of q for which sometimes both a5 and a6 had to be varied.
Table 2. Composite values of q with N q > 0 q
q3
Iq
Lq
Mq
Eq
Cq(x)
Nq
Tq
tq
16
4096
1
3·5
29 443·113·127
7.077
X4 + x + 1
275
64
6.676
25
15625
1
23.3
29·449 ·19531
6.976
x2+4x+2
239
59
6.816
210
61
6.624
19683
1
213
1093 .368089
6.747
x3 +2x+1
32 32768
1
31
71· 127 ·122921
6.933
x5 + x2 + 1
157
62
6.740
49 117649
1
24·3
7.175
X2 + 6x+ 3
42
76
7.031
81 531441
1
24·5
7.181
X4 + 2x3 + 2
2
97
7.086
27
29·113·911
.4733
29·547·1093·16493
STEPHEN D. COHEN and CHARLES KING
86
Finally, Table 3 relates to the 5 composite values for which only
a6
needed variation. Table 3. Composite values of q with N q q
q3
Iq
Lq
64
262144
7
32.72
121 1771561
=
0
Eq
Cq(x)
Tq
tq
43·127·337·5419
7.133
x6 +x4 +x3 +x+l
63
7.024
1
23.3.5
43·45319·1623931
7.108
X2 +7x + 2
94
7.047
125 1953125
1
22.·31
379·19531·519499
6.963
x2 +3x+3
89
6.905
169 4826809
7
23.3.72
29 .22079 .5229043
7.207
x2 +12x+2
97
7.164
361 4704588
1 23.32.5 197·701·70841·226871
7.026
x2 +18x+2
108 7.007
Mq
We observe that the
tq
average number of trials required was always
comparable with the expected value Eq. In fact, in every case
tq
< Eq.
We do not know why this should be so. In addition to the above calculations there were 4 values of q (3 prime values together with 256) for which a primitive polynomial was sought and found in the all-zero case. In every case we obtained the first primitive trinomial x 7 + X + b as displayed in Table 4. Table 4. The all-zero case q
b
197
31
239
4
256
w
269
31
In Table 4, the entry corresponding to b for the Conway polynomial
x8
q
= 256 is the root w of
+ x4 + x3 + x2 + 1.
The calculations were performed using systems running Microsoft Windows 2000, and employing Pentium 4 processors running at 2.8 Ghz. As an example it took approximately 9 hours 36 minutes of system time to complete the q
=
331 case. This compares with 22 hours 17 minutes of
STEPHEN D. COHEN and CHARLES KING
86
Finally, Table 3 relates to the 5 composite values for which only
a6
needed variation. Table 3. Composite values of q with N q q
q3
Iq
Lq
64
262144
7
32.72
121 1771561
=
0
Eq
Cq(x)
Tq
tq
43·127·337·5419
7.133
x6 +x4 +x3 +x+l
63
7.024
1
23.3.5
43·45319·1623931
7.108
X2 +7x + 2
94
7.047
125 1953125
1
22.·31
379·19531·519499
6.963
x2 +3x+3
89
6.905
169 4826809
7
23.3.72
29 .22079 .5229043
7.207
x2 +12x+2
97
7.164
361 4704588
1 23.32.5 197·701·70841·226871
7.026
x2 +18x+2
108 7.007
Mq
We observe that the
tq
average number of trials required was always
comparable with the expected value Eq. In fact, in every case
tq
< Eq.
We do not know why this should be so. In addition to the above calculations there were 4 values of q (3 prime values together with 256) for which a primitive polynomial was sought and found in the all-zero case. In every case we obtained the first primitive trinomial x 7 + X + b as displayed in Table 4. Table 4. The all-zero case q
b
197
31
239
4
256
w
269
31
In Table 4, the entry corresponding to b for the Conway polynomial
x8
q
= 256 is the root w of
+ x4 + x3 + x2 + 1.
The calculations were performed using systems running Microsoft Windows 2000, and employing Pentium 4 processors running at 2.8 Ghz. As an example it took approximately 9 hours 36 minutes of system time to complete the q
=
331 case. This compares with 22 hours 17 minutes of
... PRIMITIVE POLYNOMIAL THEOREM
87
system time using similar systems with Pentium 3 processors running at
1 Ghz. The authors hold a disk containing all the data obtained. References [1]
S. D. Cohen, Primitive elements and polynomials with arbitrary Math. 83 (1990), 1-7.
(2]
S. D. Cohen and D. Mills, Primitive polynomials with first and second coefficients prescribed, Finite Fields Appl. 9 (2003), 334-350.
[3]
S.·Q. Fan and W.-B. Han, Character sums over Galois rings and primitive polynomials over finite fields, Finite Fields Appl. 10 (2004), 36-52.
[4]
trace, Discrete
S.-Q. Fan and W.-B. Han, Primitive polynomials with three coefficients prescribed, Finite Fields Appl., to appear.
[5]
D. Mills, Existence of primitive polynomials with three coefficients prescribed, JP J. Algebra, Number Theory Appl. 4(1) (2004), 1-22.
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