The explicit construction of irreducible polynomials over finite fields
Designs, Codes and Cryptography, 2, 169-174 (1992) © 1992 Kluwer Academic Publishers. Manufactured in The Netherlands.
The Explicit Construction of Irreducible Polynomials Over Finite Fields STEPHEN D. COHEN
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW,, Scotland Communicated by S.A. Vanstone Received November 1, 1991. Revised January 14, 1992.
Abstract. For a finite field GF(q) of odd prime power order q, and n > 1, we construct explicitly a sequence of monic irreducible reciprocal polynomials o f degree n2 m (m = 1, 2, 3 . . . . ) over GF(q). It is the analog for fields of odd order of constructions of Wiedemann and of Meyn over GF(2). We also deduce iterated presentations of GF(q n2**).
I. The Construction Given a polynomial f ( x ) of degree n over a field F,, the reciprocal operator Q associates with f the reciprocal polynomial f Q defined by
f Q ( x ) = xnf(x +
x-l).
Wiedemann [7] showed that successive Q-iterates of the q u a d r a t i c f ( x ) = x 2 + x + 1 form a sequence o f irreducible polynomials of degree 2 m, m = 1, 2, 3 . . . . over GF(2). Then Meyn [4] observed that the process can similarly be aplied to a suitable irreducible polynomial f of degree n over GF(2). We may c o m b i n e their results as follows. q'~-IEOREM 1. Let fo(x) be a monic irreducible polynomial o f degree n >>_ 1 over GF(2) whose coefficients o f x n-1 and o f x are both 1. For each m >- 1 define fm by fro(X) • f Q _ I ( X ) ,
m >- 1.
Then, f o r each m = 1, 2, 3, . "d hfm is an irreducible polynomial over G F ( 2 ) o f degree n2 m and order a divisor o f 2 n2 - + 1. Notes. (i) To recover Wiedemann's result, take fo(x) = x + 1. In this case, as noted in [1], the order of fro is exactly 2 2m-1 + 1 for m --- 9. Whether or not this extends to all values of m is unknown. It would b e most interesting if it were true, although I cannot see any compelling reason why it ought to be so.
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s.~ COHEN
(ii) Niederreiter [5] has given an exact expression for the number of monic irreducible polynomials of degree n over GF(2) satisfying the hypotheses of Theorem 1. In particular, it yields the existence of such a polynomial except when n = 3. Thus if we assume that GF(q), where q = 2 n (n : 3), is defined explicitly by such a polynomial 3~ and we let ct E GF(q) be any root off0 then the sequence of polynomials
go(x) = x + a,
gin(x) = g~-l(X), m _> 1,
is an explicit sequence of irreducible polynomials of degree 2 m over GF(q) having order • . m-1 a &visor of q2 + 1 (m _ 1). (While it is not essential we usually have in mind here than n is odd). Our purpose is to provide an analog of Theorem 1 over finite fields GF(q) of odd order• Meyn [4] attempted this using the Q-operator but with limited success. We modify that approach. To this end over any field of characteristic ;a 2, introduce the operator R whose effect is to associate with a monic polynomial f of degree n over F the monic reciprocal polynomial f R defined by
f R ( x ) = (2x)n f(V2(x +
x-l))
= 2" fQ(1/2 x).
(1)
We also abbreviate the product f ( 1 ) f ( - 1 ) ( E F ) to Mf)• THEOREM 2• Let fo(x) be a monic irreducible polynomial of degree n >_ 1 over GF(q), q odd, where n is even if q -= 3 (mod 4). Suppose also that X ff o) is a non-square in GF(q). For each m >- 1 define fm by
fro(X)
-~- f R _ l ( X
).
(2)
Then, for each m = 1, 2, 3 . . . . . fm is an irreducible polynomial over GF(q) of degree n2 m and order a divisor of qn2m-~ + 1. We shall comment further on Theorem 2 after its proof which is based on a result of Meyn [4], Theorem 8. LEMMA 3. Let f be a monk: irreducible polynomial over GF(q), q odd. Then f Q is irreducible over GF(q) if and only if f ( 2 ) f ( - 2 ) is a non-square in GF(q).
Proof of Theorem 2. By induction X(fm) = ( - 1 ) n Cm 2 X(j~), = d 2 X(fo),
Cm E GF(q), m >_ 1,
dm E GF(q),
THE EXPLICIT CONSTRUCTIONOF IRREDUCIBLE POLYNOMIALSOVER FINITE FIELDS
171
because either - 1 is a square in GF(q) (when q - 1 (mod 4)) or n is even. Hence, to prove the result by induction on m, it suffices to replace 3~ by fm and prove the result for m=l. Put go(x) = 2~3~(1A x). Note that, by (1), fl = f0R = g~2. Thus by Lemma 3, fl is irreducible if and only if v = g0(2)g0(-2) is a non-square. But v = 22~ h(f0). Hence fl is irreducible. Further, sincej~ is a reciprocal polynomialj~(0) = 1 and so its order divides q~ + 1 (see e.g., [3]). This completes the proof. Again it would be of interest to have results or even computational data on the orders of the polynomials in the sequence {fm } produced by Theorem 2. In particular, any such for which the order of fm is always its maximal value would be especially significant. That for any odd prime power q and any n _> 1 (with n even if q - 3 (mode 4)) there always exists an irreducible polynomial of degree n satisfying the hypotheses of Theorem 2 is a consequence of the following fact. LEMMA 4. For any odd q and integer n >_ 1, there exists a monic irreducible polynomial of degree n over GF(q) with X(f) a non-square in GF(q).
Proof. By Lemma 3 and (1), every irreducible reciprocal polynomial of degree 2n over GF(q) arises by applying the R-operator to a polynomial of the type whose existence we are seeking. But the total number of irreducible reciprocal polynomials of degree 2n over GF(q) is known, see [2], [3] or [4]. Specifically, if n = 2Ur, u _ 0, r odd, it is given by the expression 1/2n (qn _ 1), i f r = 1, and otherwise by (1/2u+l)Tr(r, qn/r), where ~r(r, q) denotes the total number of monic irreducible polynomials of degree r over GF(q). In every case it is positive and the result holds. As a consequence of Lemma 4, except when q --- 3 (mod 4) and n is odd and granted GF(q n) is represented by a suitable irreducible polynomial 3~ of degree n over GF(q), we have by means of Theorem 2 an explicit construction of a family of irreducible polynomials of degree 2 m, m >_ 1 over GF(q ~) (as in Note (ii) following Theorem 1). In the excluded case we need to find a suitable quadratic for j~. In fact, in this particular situation when q -- 3 and n is odd this can generally be achieved starting from any irreducible polynomial f o f degree n over GF(3) provided the values f ( 0 ) , f ( + l ) are not all 1 or all - 1 as follows. One o f f ( x ) , - f ( - x ) , xnf(1/x) - x ~ f ( - 1 / x ) attains the values 1 and - 1 at x = 1 and -1, respectively. Replace f by this polynomial. Then fR is irreducible (by Lemma 3 ) a n d f R ( 1 ) = fR(--1) = --1 while fR(0) = 1, since f R is reciprocal. Finally take f0(x) = fR(x + 1) and apply Theorem 2. For a construction just as explicit as Wiedemann's over GF(2), take q = 3 and3~+(x) either o f x 2 5: x - 1, respectively, over GF(3). (This also occurs as an example in [4]). Then {fro~, m _ 0} is a sequence of irreducible polynomials of degree 2 m+l over GF(3); j~ has order 8 and for m _ 1, fm~ has order a divisor of 32m + 1. It would be interesting to know when equality holds (for some choice of +). When q = 5, we may take3~(x) = x + 2 whencefl(x) = x 2 + x + 1 and the process proceeds. Irl~this case f0 has order 4 and fl has order 3 or 6. On the other hand we can also begin with 3~(x) as any of the primitive polynomials x 2 4- x + 2 or x 2 5: 2x - 2 each having order 24.
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S.D. COHEN
As further comments on Theorem 1 and 2, we observe that while these theorems define a unique sequence of irreducible polynomials fm from a given f0, there is usually (when 3~ is not itself essentially a reciprocal polynomial) an alternative sequence. In general replace~(x) by c - l x " f o ( 1 / x ) where c = 3~(0). In this way the quadratic character of k(3~), say, is preserved and a second sequence of irreducible polynomials with similar properties is defined, for example, by (2). As far as Theorem 1 is concerned, this process produces a sequence equivalent to that of Varshamov [6] who took fl(x) = (x 2 + 1)nfo(x(xz 2 + 1) -1)
(3)
and proceeded inductively. In particular, because j~, 3~ • • • are reciprocal polynomials, it makes no difference for m _> 2 to determine 3~, 3~ • • • as Q-iterates or by repetition of the operator (3). We also note incidentally that in Theorem 1 (and the above variation) we may replace GF(2) by any field of even order provided the absolute trace of the coefficients of x n-1 and x are both 1. One final minor observation when q is odd is that in Theorem 2 we may replace anyfm(x ) (m _> 1) b y f m ( - x ) without altering the conclusions of the theorem. 2. Iterated Presentations The concept of an iterated presentation of an infinite extension of a finite field is discussed fully in Chapter 3 of [1]. The extensions with which we are concerned here have the shape
GF(q
2-) = 6
G F ( q 2m)
m=0
(where q itself may, in the most general situation, be regarded as a power q~). The most natural iterated presentation of GF (q20*) comprises an inductively described sequence of quadratic polynomials {gin, m = 0, 1, 2, . . . }, where gm is defined and irreducible over GF(q2~). Ideally everything should be as explicit as possible; of course, some knowledge of the base field G F ( q ) (or of GF(q2), say) may have to be allowed. Iterated presentations of GF(22~*) can be totally explicit. Two such are discussed in [1]; the one associated with Wiedemann's construction (Theorem 1) is as follows. THEOREM 5. The following sequence is an iterated presentation f o r GF(2
2 oo
). Take go(x) = x 2 + x + 1 and gin(x) = x 2 + ~xmx + 1, m >_ 1, where czm is a root o f gm-1.
Note that in Theorem 5, gm is a consequence of the Q-operator applied to gin-1. Further 2m let Am = al • • • am E G F ( q ). The order of Am is simply the product of the orders of CXl. . . . ~Xm-It should generally be large: if it is actually 2 2m - - 1 then we would have constructed a primitive element of GF(22m). Now with q a power of 2 and allowing the appropriate knowledge of G F ( q ) we can construct (as noted in [4]) an iterated presentation of G F ( q 2**) by means of the general case of Theorem 1.
THE EXPLICIT CONSTRUCTION OF IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS
173
For odd values of q the only type of iterated presentation of GF(q 2®) appearing in [1] is a basic one achieved by iterates of the square-root operator. We give a version of this and follow it with our analogs of Theorem 5. T~EOREM 6. Suppose q - 1 (mod 4) and let c be a non-square in G F ( q ) . The following sequence is then an iterated presentation o f GF(q2**). Take go(x) = x 2 - c and, f o r each m >_ 1, gin(x) = x 2 - 13m, where [3m is a root ofgm-1. Moreover, the order of{3 m is 2me, where e is the order o f c. If, on the other hand, q m 3 (mod 4), modify the above presentation by setting c = - 1 , :31 a non-square in G F ( q 2) and re-defining e as the order of/31. For simplicity we state our analogs of Theorem 5 for the cases q = 3 and q -- 1 (mod 4) only; one could easily write down extensions to q - 3 (mod 4) or, more generally to GF(qn2**). In every case the constructions derive from Theorem 2. We remark that the construction for GF(32.*) is as explicit as Theorem 5. THEOREM 7. The following sequence is an iterated presentation o f GF(32~*). Take go(x) = x 2 + x - 1, and f o r m >__ 1, gin(x) = x 2 + tamX + 1, w h e r e o t m i s a r o o t o f g m _ 1. Notes. (i) gm results b y applying the R-operator to gm-1. (ii) Let A m = ot 1 . . . Olm. Since. cq has order 8 and each o/i has order 2 e i , say, where e i is odd (i _> 2), then A m has order 8e2 • • • em, a divisor of (3 2~' - 1)/2 m-1 . But now let ~m be a root of hm_l, where {hm, m = 0, 1, 2 . . . . } is the following particular iterated presentation of GF(3 2.*) occurring in Theorem 6. Take h0(x ) = x 2 + 1, hi(x) = x 2 (~1 -at- 1), h m ( x ) = X 2 - - [3 m (m >_ 2). Then ~m has order exactly 2 m+2, the even part of 32m - 1 and we propose (without much optimism) A m ~m for a primitive element of GF(32m), m >_ 2.
TrIEOREM 8. Let q -- 1 (mod 4) and c be an element o f G F ( q ) f o r which C 2 - - 1 is a non-square in GF(q). The following sequence is then an iterated presentation f o r GF(q ). Take go(x) = x 2 + 2cx + 1, and f o r m >- 1, gin(x) = x 2 - 2CtmX + 1, where ctm is a root o f gm-1. Again we may hope to derive from Theorem 8 (along with Theorem 6) elements of G F ( q 2m) of large order and, perhaps, even a primitive element.
References 1. Brawley, J.V. and Schnibben, G.E. 1989. Infinite Algebraic Extensions o f Finite Fields, Contemporary Mathematics, vol. 95, Providence, RI: American Math. Society. 2. Carlitz, L. 1967. Some theorems on irreducible reciprocal polynomials over a finite field. J. Reine Angew. Math. 227:212-220. 3. Cohen, S.D. 1969. On irreducible polynomials of certain types in finite fields. Proc. Camb. Ph/l. Soc. 66:335-244. 4. Meyn, H. 1990. On the construction of irreducible self-receiprocal polynomials over finite fields. Appl. Algebra Eng. Comm. Comp. 1:43-53.
174
S.D. COHEN
5. Niederreiter, H. 1990. An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field. Appl. Algebra Eng. Comm. Comp. 1:119-124. 6. Varshamov, R.R. 1984. A general method of synthesis for irreducible polynomials over Galois fields. Soviet Math. Dokl. 29:334-336. 7. Wiedemann, D. 1988. An iterated quadratic extension of GF(2). Fibonacci Quart. 26:290-295.
The Explicit Construction of Irreducible Polynomials Over Finite Fields STEPHEN D. COHEN
Department of Mathematics, University of Glasgow, Glasgow, G12 8QW,, Scotland Communicated by S.A. Vanstone Received November 1, 1991. Revised January 14, 1992.
Abstract. For a finite field GF(q) of odd prime power order q, and n > 1, we construct explicitly a sequence of monic irreducible reciprocal polynomials o f degree n2 m (m = 1, 2, 3 . . . . ) over GF(q). It is the analog for fields of odd order of constructions of Wiedemann and of Meyn over GF(2). We also deduce iterated presentations of GF(q n2**).
I. The Construction Given a polynomial f ( x ) of degree n over a field F,, the reciprocal operator Q associates with f the reciprocal polynomial f Q defined by
f Q ( x ) = xnf(x +
x-l).
Wiedemann [7] showed that successive Q-iterates of the q u a d r a t i c f ( x ) = x 2 + x + 1 form a sequence o f irreducible polynomials of degree 2 m, m = 1, 2, 3 . . . . over GF(2). Then Meyn [4] observed that the process can similarly be aplied to a suitable irreducible polynomial f of degree n over GF(2). We may c o m b i n e their results as follows. q'~-IEOREM 1. Let fo(x) be a monic irreducible polynomial o f degree n >>_ 1 over GF(2) whose coefficients o f x n-1 and o f x are both 1. For each m >- 1 define fm by fro(X) • f Q _ I ( X ) ,
m >- 1.
Then, f o r each m = 1, 2, 3, . "d hfm is an irreducible polynomial over G F ( 2 ) o f degree n2 m and order a divisor o f 2 n2 - + 1. Notes. (i) To recover Wiedemann's result, take fo(x) = x + 1. In this case, as noted in [1], the order of fro is exactly 2 2m-1 + 1 for m --- 9. Whether or not this extends to all values of m is unknown. It would b e most interesting if it were true, although I cannot see any compelling reason why it ought to be so.
170
s.~ COHEN
(ii) Niederreiter [5] has given an exact expression for the number of monic irreducible polynomials of degree n over GF(2) satisfying the hypotheses of Theorem 1. In particular, it yields the existence of such a polynomial except when n = 3. Thus if we assume that GF(q), where q = 2 n (n : 3), is defined explicitly by such a polynomial 3~ and we let ct E GF(q) be any root off0 then the sequence of polynomials
go(x) = x + a,
gin(x) = g~-l(X), m _> 1,
is an explicit sequence of irreducible polynomials of degree 2 m over GF(q) having order • . m-1 a &visor of q2 + 1 (m _ 1). (While it is not essential we usually have in mind here than n is odd). Our purpose is to provide an analog of Theorem 1 over finite fields GF(q) of odd order• Meyn [4] attempted this using the Q-operator but with limited success. We modify that approach. To this end over any field of characteristic ;a 2, introduce the operator R whose effect is to associate with a monic polynomial f of degree n over F the monic reciprocal polynomial f R defined by
f R ( x ) = (2x)n f(V2(x +
x-l))
= 2" fQ(1/2 x).
(1)
We also abbreviate the product f ( 1 ) f ( - 1 ) ( E F ) to Mf)• THEOREM 2• Let fo(x) be a monic irreducible polynomial of degree n >_ 1 over GF(q), q odd, where n is even if q -= 3 (mod 4). Suppose also that X ff o) is a non-square in GF(q). For each m >- 1 define fm by
fro(X)
-~- f R _ l ( X
).
(2)
Then, for each m = 1, 2, 3 . . . . . fm is an irreducible polynomial over GF(q) of degree n2 m and order a divisor of qn2m-~ + 1. We shall comment further on Theorem 2 after its proof which is based on a result of Meyn [4], Theorem 8. LEMMA 3. Let f be a monk: irreducible polynomial over GF(q), q odd. Then f Q is irreducible over GF(q) if and only if f ( 2 ) f ( - 2 ) is a non-square in GF(q).
Proof of Theorem 2. By induction X(fm) = ( - 1 ) n Cm 2 X(j~), = d 2 X(fo),
Cm E GF(q), m >_ 1,
dm E GF(q),
THE EXPLICIT CONSTRUCTIONOF IRREDUCIBLE POLYNOMIALSOVER FINITE FIELDS
171
because either - 1 is a square in GF(q) (when q - 1 (mod 4)) or n is even. Hence, to prove the result by induction on m, it suffices to replace 3~ by fm and prove the result for m=l. Put go(x) = 2~3~(1A x). Note that, by (1), fl = f0R = g~2. Thus by Lemma 3, fl is irreducible if and only if v = g0(2)g0(-2) is a non-square. But v = 22~ h(f0). Hence fl is irreducible. Further, sincej~ is a reciprocal polynomialj~(0) = 1 and so its order divides q~ + 1 (see e.g., [3]). This completes the proof. Again it would be of interest to have results or even computational data on the orders of the polynomials in the sequence {fm } produced by Theorem 2. In particular, any such for which the order of fm is always its maximal value would be especially significant. That for any odd prime power q and any n _> 1 (with n even if q - 3 (mode 4)) there always exists an irreducible polynomial of degree n satisfying the hypotheses of Theorem 2 is a consequence of the following fact. LEMMA 4. For any odd q and integer n >_ 1, there exists a monic irreducible polynomial of degree n over GF(q) with X(f) a non-square in GF(q).
Proof. By Lemma 3 and (1), every irreducible reciprocal polynomial of degree 2n over GF(q) arises by applying the R-operator to a polynomial of the type whose existence we are seeking. But the total number of irreducible reciprocal polynomials of degree 2n over GF(q) is known, see [2], [3] or [4]. Specifically, if n = 2Ur, u _ 0, r odd, it is given by the expression 1/2n (qn _ 1), i f r = 1, and otherwise by (1/2u+l)Tr(r, qn/r), where ~r(r, q) denotes the total number of monic irreducible polynomials of degree r over GF(q). In every case it is positive and the result holds. As a consequence of Lemma 4, except when q --- 3 (mod 4) and n is odd and granted GF(q n) is represented by a suitable irreducible polynomial 3~ of degree n over GF(q), we have by means of Theorem 2 an explicit construction of a family of irreducible polynomials of degree 2 m, m >_ 1 over GF(q ~) (as in Note (ii) following Theorem 1). In the excluded case we need to find a suitable quadratic for j~. In fact, in this particular situation when q -- 3 and n is odd this can generally be achieved starting from any irreducible polynomial f o f degree n over GF(3) provided the values f ( 0 ) , f ( + l ) are not all 1 or all - 1 as follows. One o f f ( x ) , - f ( - x ) , xnf(1/x) - x ~ f ( - 1 / x ) attains the values 1 and - 1 at x = 1 and -1, respectively. Replace f by this polynomial. Then fR is irreducible (by Lemma 3 ) a n d f R ( 1 ) = fR(--1) = --1 while fR(0) = 1, since f R is reciprocal. Finally take f0(x) = fR(x + 1) and apply Theorem 2. For a construction just as explicit as Wiedemann's over GF(2), take q = 3 and3~+(x) either o f x 2 5: x - 1, respectively, over GF(3). (This also occurs as an example in [4]). Then {fro~, m _ 0} is a sequence of irreducible polynomials of degree 2 m+l over GF(3); j~ has order 8 and for m _ 1, fm~ has order a divisor of 32m + 1. It would be interesting to know when equality holds (for some choice of +). When q = 5, we may take3~(x) = x + 2 whencefl(x) = x 2 + x + 1 and the process proceeds. Irl~this case f0 has order 4 and fl has order 3 or 6. On the other hand we can also begin with 3~(x) as any of the primitive polynomials x 2 4- x + 2 or x 2 5: 2x - 2 each having order 24.
172
S.D. COHEN
As further comments on Theorem 1 and 2, we observe that while these theorems define a unique sequence of irreducible polynomials fm from a given f0, there is usually (when 3~ is not itself essentially a reciprocal polynomial) an alternative sequence. In general replace~(x) by c - l x " f o ( 1 / x ) where c = 3~(0). In this way the quadratic character of k(3~), say, is preserved and a second sequence of irreducible polynomials with similar properties is defined, for example, by (2). As far as Theorem 1 is concerned, this process produces a sequence equivalent to that of Varshamov [6] who took fl(x) = (x 2 + 1)nfo(x(xz 2 + 1) -1)
(3)
and proceeded inductively. In particular, because j~, 3~ • • • are reciprocal polynomials, it makes no difference for m _> 2 to determine 3~, 3~ • • • as Q-iterates or by repetition of the operator (3). We also note incidentally that in Theorem 1 (and the above variation) we may replace GF(2) by any field of even order provided the absolute trace of the coefficients of x n-1 and x are both 1. One final minor observation when q is odd is that in Theorem 2 we may replace anyfm(x ) (m _> 1) b y f m ( - x ) without altering the conclusions of the theorem. 2. Iterated Presentations The concept of an iterated presentation of an infinite extension of a finite field is discussed fully in Chapter 3 of [1]. The extensions with which we are concerned here have the shape
GF(q
2-) = 6
G F ( q 2m)
m=0
(where q itself may, in the most general situation, be regarded as a power q~). The most natural iterated presentation of GF (q20*) comprises an inductively described sequence of quadratic polynomials {gin, m = 0, 1, 2, . . . }, where gm is defined and irreducible over GF(q2~). Ideally everything should be as explicit as possible; of course, some knowledge of the base field G F ( q ) (or of GF(q2), say) may have to be allowed. Iterated presentations of GF(22~*) can be totally explicit. Two such are discussed in [1]; the one associated with Wiedemann's construction (Theorem 1) is as follows. THEOREM 5. The following sequence is an iterated presentation f o r GF(2
2 oo
). Take go(x) = x 2 + x + 1 and gin(x) = x 2 + ~xmx + 1, m >_ 1, where czm is a root o f gm-1.
Note that in Theorem 5, gm is a consequence of the Q-operator applied to gin-1. Further 2m let Am = al • • • am E G F ( q ). The order of Am is simply the product of the orders of CXl. . . . ~Xm-It should generally be large: if it is actually 2 2m - - 1 then we would have constructed a primitive element of GF(22m). Now with q a power of 2 and allowing the appropriate knowledge of G F ( q ) we can construct (as noted in [4]) an iterated presentation of G F ( q 2**) by means of the general case of Theorem 1.
THE EXPLICIT CONSTRUCTION OF IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS
173
For odd values of q the only type of iterated presentation of GF(q 2®) appearing in [1] is a basic one achieved by iterates of the square-root operator. We give a version of this and follow it with our analogs of Theorem 5. T~EOREM 6. Suppose q - 1 (mod 4) and let c be a non-square in G F ( q ) . The following sequence is then an iterated presentation o f GF(q2**). Take go(x) = x 2 - c and, f o r each m >_ 1, gin(x) = x 2 - 13m, where [3m is a root ofgm-1. Moreover, the order of{3 m is 2me, where e is the order o f c. If, on the other hand, q m 3 (mod 4), modify the above presentation by setting c = - 1 , :31 a non-square in G F ( q 2) and re-defining e as the order of/31. For simplicity we state our analogs of Theorem 5 for the cases q = 3 and q -- 1 (mod 4) only; one could easily write down extensions to q - 3 (mod 4) or, more generally to GF(qn2**). In every case the constructions derive from Theorem 2. We remark that the construction for GF(32.*) is as explicit as Theorem 5. THEOREM 7. The following sequence is an iterated presentation o f GF(32~*). Take go(x) = x 2 + x - 1, and f o r m >__ 1, gin(x) = x 2 + tamX + 1, w h e r e o t m i s a r o o t o f g m _ 1. Notes. (i) gm results b y applying the R-operator to gm-1. (ii) Let A m = ot 1 . . . Olm. Since. cq has order 8 and each o/i has order 2 e i , say, where e i is odd (i _> 2), then A m has order 8e2 • • • em, a divisor of (3 2~' - 1)/2 m-1 . But now let ~m be a root of hm_l, where {hm, m = 0, 1, 2 . . . . } is the following particular iterated presentation of GF(3 2.*) occurring in Theorem 6. Take h0(x ) = x 2 + 1, hi(x) = x 2 (~1 -at- 1), h m ( x ) = X 2 - - [3 m (m >_ 2). Then ~m has order exactly 2 m+2, the even part of 32m - 1 and we propose (without much optimism) A m ~m for a primitive element of GF(32m), m >_ 2.
TrIEOREM 8. Let q -- 1 (mod 4) and c be an element o f G F ( q ) f o r which C 2 - - 1 is a non-square in GF(q). The following sequence is then an iterated presentation f o r GF(q ). Take go(x) = x 2 + 2cx + 1, and f o r m >- 1, gin(x) = x 2 - 2CtmX + 1, where ctm is a root o f gm-1. Again we may hope to derive from Theorem 8 (along with Theorem 6) elements of G F ( q 2m) of large order and, perhaps, even a primitive element.
References 1. Brawley, J.V. and Schnibben, G.E. 1989. Infinite Algebraic Extensions o f Finite Fields, Contemporary Mathematics, vol. 95, Providence, RI: American Math. Society. 2. Carlitz, L. 1967. Some theorems on irreducible reciprocal polynomials over a finite field. J. Reine Angew. Math. 227:212-220. 3. Cohen, S.D. 1969. On irreducible polynomials of certain types in finite fields. Proc. Camb. Ph/l. Soc. 66:335-244. 4. Meyn, H. 1990. On the construction of irreducible self-receiprocal polynomials over finite fields. Appl. Algebra Eng. Comm. Comp. 1:43-53.
174
S.D. COHEN
5. Niederreiter, H. 1990. An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field. Appl. Algebra Eng. Comm. Comp. 1:119-124. 6. Varshamov, R.R. 1984. A general method of synthesis for irreducible polynomials over Galois fields. Soviet Math. Dokl. 29:334-336. 7. Wiedemann, D. 1988. An iterated quadratic extension of GF(2). Fibonacci Quart. 26:290-295.
