Finite Fields and Their Applications

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1 Fnte Felds and Ther Applcatons Contents lsts avalable at ScenceDrect Fnte Felds and Ther Applcatons Typcal prtve polynoals over nteger resdue rngs Tan Tan a, Wen-Feng Q a,b, a Departent of Appled Matheatcs, Zhengzhou Inforaton Scence and Technology Insttute, Zhengzhou, PR Chna b State Key Laboratory of Inforaton Securty Insttute of Software, Chnese Acadey of Scences, Bejng, PR Chna artcle nfo abstract Artcle hstory: Receved Aprl 009 Revsed 3 August 009 Avalable onlne Septeber 009 Councated by Gary L. Mullen Keywords: Prtve polynoals Integer resdue rngs Fnte felds Let N be a product of dstnct pre nubers and Z/N be the nteger resdue rng odulo N. In ths paper, a prtve polynoal f x over Z/N such that f x dvdes x s c for soe postve nteger s and soe prtve eleent c n Z/N s called a typcal prtve polynoal. Recently typcal prtve polynoals over Z/N were shown to be very useful, but the exstence of typcal prtve polynoals has not been fully studed. In ths paper, for any nteger, a necessary and suffcent condton for the exstence of typcal prtve polynoals of degree over Z/N s proved. 009 Elsever Inc. All rghts reserved.. Introducton Let N be a postve nteger whch s a product of dstnct pre nubers and Z/N be the nteger resdue rng odulo N. It s a natural generalzaton of the defnton of prtve eleents and prtve polynoals over a fnte feld that an eleent ξ n Z/N s called a prtve eleent f ξ s a prtve root odulo p for every pre dvsor p of N, and a polynoal f x over Z/N s called a prtve polynoal f f x s onc and f x odulo p s a prtve polynoal over the pre feld Z/p for every pre dvsor p of N. Prtve eleents and prtve polynoals over Z/N are recently nvolved n [] whch studes the dstnctness of axal length sequences over Z/p p odulo, where p and p aredstnctoddprenubers.wenotethattheproof of dstnctness n [] reles on an portant condton whch states that there exsts a postve nteger s such that x s s congruent to soe prtve eleent ξ n Z/p p odulo f x and p p.it s easy to show that such nteger s does not always exst. However, over a fnte feld F q, where q s Ths work was supported by NSF of Chna under Grant Nos , * Correspondng author at: Departent of Appled Matheatcs, Zhengzhou Inforaton Scence and Technology Insttute, Zhengzhou, PR Chna. E-al addresses: tantan_d@6.co T. Tan, wenfeng.q@63.net W.-F. Q /$ see front atter 009 Elsever Inc. All rghts reserved. do:0.06/j.ffa

2 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons a pre nuber power, by Theore 3.8 n [], we know that for every prtve polynoal f x, there exsts an nteger s such that x s s congruent to soe prtve eleent of F q odulo f x. In ths sense, aong prtve polynoals over Z/N, the behavor of polynoals f x over Z/N for whch f x dvdes x s c for soe postve nteger s and soe prtve eleent c n Z/N s uch closer to the behavor of prtve polynoals over fnte felds. For convenence, such portant class of prtve polynoals over Z/N s called typcal prtve polynoals over Z/N. We note that the exstence of typcal prtve polynoals has not been fully studed n []. In ths paper, we copletely solve the exstence of typcal prtve polynoals over Z/N n theory, and soe necessary and suffcent condtons are presented, whch facltates the future applcatons of typcal prtve polynoals over Z/N.. Prelnares The concept of subexponent over fnte felds s proposed n [3] and [4] and entoned by [, pp. 3 33]. Defnton. Let q be a pre nuber power. If f x s a polynoal over F q wth f 0 0, then the least postve nteger e such that f x dvdes x e c for soe c F q s called the subexponent of f x. The subexponent has the followng sple property. Lea. Let q be a pre nuber power and f x be a polynoal over F q wth f 0 0. If fx dvdes x e a for soe postve nteger e and a F q, then e s dvsble by the subexponent of f x. Proof. Assue the subexponent of f x s s. Then by Defnton, there exsts an eleent c F q such that f x dvdes x s c over F q.snce f x also dvdes x e a over F q,weusthavee s. Thuswe can wrte e = u s + v where u 0 and 0 v < s. Now a x e x u s+v c u x v od f x holds over F q, and so f x dvdes x v a c u over F q. Because of the defnton of subexponent, ths s only possble f v = 0. Ths copletes the proof. As for prtve polynoals over fnte felds, ther subexponents are gven by Theore 3.8 n []. Lea 3. Let q be a pre nuber power. The onc polynoal f x F q [x] of degree s a prtve polynoal over F q f and only f f 0 s a prtve eleent of F q and ts subexponent s r = q / q. Moreover, f f x s prtve over F q,thenx r f 0 od f x. Lea 3 shows that for a prtve polynoal f x over a fnte feld, there always exsts an nteger s such that x s s congruent odulo f x to soe prtve eleent of the fnte feld and the least such s s just the subexponent of f x. 3. Man results Let N be a fxed postve nteger wth standard factorzaton N = k = p, where k and p, p,...,p k are dstnct pre nubers. The an topc of ths secton s the exstence of typcal prtve polynoals over Z/N, and so t s necessary for us to ake ts defnton explct here. Defnton 4. A polynoal f x over Z/N s called a typcal prtve polynoal f f x s a prtve polynoal and f x dvdes x s c for soe postve nteger s and soe prtve eleent c of Z/N.

3 798 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons Frst, t s easy to show the exstence of prtve eleents and prtve polynoals over Z/N wth the Chnese Reander Theore. For every k, t s clear that there exsts a prtve root ξ odulo p. Then by the Chnese Reander Theore, there exsts a unque eleent ξ n Z/N such that ξ ξ od p for k. Thus, ξ s a prtve eleent of Z/N. For any postve nteger, a prtve polynoal of degree over Z/N can be constructed analogously. For every k, t s clear that there exsts a prtve polynoal f x = x + j=0 c, jx j over Z/p. Then by the Chnese Reander Theore, for 0 j, there exsts a unque eleent c j n Z/N such that c j c, j od p, k. Thus, f x = x + j=0 c jx j s a prtve polynoal over Z/N. Second, we begn to dscuss the exstence of typcal prtve polynoals whch s the an task of ths secton. If f x s a prtve polynoal over Z/N and there exsts an nteger s such that f x dvdes x s c for soe eleent c Z/N,.e., x s c od f x, N, thensnce f x odulo p s a prtve polynoal over Z/p and x s c od f x, p for k, t follows fro Leas and 3 that s s dvsble by p /p for k. Thus,s ust be dvsble by θ N = lc p, p p,..., p k. p k On the other hand, snce Lea 3 ples that for k, wehave p p x f 0 od f x, p, where x θ N f 0 d od f x, p, d = θ N p p. By the Chnese Reander Theore, there exsts a unque eleent β Z/N such that Hence, t can be seen fro that β f 0 d od p, k. x θ N β od f x, N. Note that we have shown that s s dvsble by θ N, and so, the congruence yelds x s β s/θ N od f x, N. Obvously, f β s/θ N s a prtve eleent of Z/N, then t s necessary that β has to be a prtve eleent of Z/N. That s to say f x s s congruent od f x, N to soe prtve eleent of Z/N,

4 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons then t s necessary that x θ N has to be congruent od f x, N to soe prtve eleent of Z/N. The followng lea s an edate consequence of ths observaton. Lea 5. A prtve polynoal f x of degree s a typcal prtve polynoal over Z/N f and only f x θ N s congruent od f x, N to soe prtve eleent of Z/N. Moreover, for a typcal prtve polynoal f x of degree overz/n, θ N s also the least postve nteger r for whch f x dvdes x r c for soe eleent c Z/N. Wth all the above arguents, t can be seen that the role that θ N plays n the rng Z/N[x] suchlketherolethat the subexponent of prtve polynoals plays n the polynoal rngs over fnte felds ntroduced n Secton. It s easy to show that a prtve polynoal s not always a typcal prtve polynoal wth Lea 5. Please see a sple exaple. Exaple 6. It can be verfed that f x = x + x + 4 s a prtve polynoal over Z/ 3. Snce x θ 3 90 od f x, 3 and 90 s not a prtve eleent of Z/ 3, t follows fro Lea 5 that f x s not a typcal prtve polynoal over Z/ 3. Furtherore, based on Lea 5, we obtan the followng useful crteron. Lea 7. A prtve polynoal f x of degree over Z/N s a typcal prtve polynoal f and only f θ N, p = p p 3 for every k. Proof. If f x s a typcal prtve polynoal over Z/N, then Lea 5 ples that there exsts a prtve eleent ξ n Z/N such that x θ N ξ od f x, N. 4 Assue p s a pre dvsor of N. It edately follows fro 4 that Snce θ N s dvsble by p /p, we can wrte x θ N ξ od f x, p. 5 θ N, p = d p p where d s a postve dvsor of p. If d >, then we deduce fro 5 that x θ N p /d ξ p /d od f x, p, a contradcton to the fact that ξ s a prtve root odulo p. Thus,d = and 3 holds for p.

5 800 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons Conversely, suppose 3 holds for every k. Snce f x odulo p s a prtve polynoal over Z/p for k, t follows fro Lea 3 that x p /p ξ od f x, p for soe prtve root ξ odulo p, and so x θ N ξ t od f x, p 6 where t = θ N p p. Snce 3 ples that t, p =, k, t can be seen that ξ t s also a prtve root odulo p. By the Chnese Reander Theore, there exsts a unque eleent ξ n Z/N such that ξ ξ t od p, k. Then t follows fro 6 that x θ N ξ od f x, N. 7 Note that ξ s a prtve eleent of Z/N, and so 7 ples that f x s a typcal prtve polynoal over Z/N. It can be seen fro Lea 7 that gven a postve nteger, the exstence of a typcal prtve polynoal of degree over Z/N s copletely deterned by N and. Hence, ether all prtve polynoals of degree over Z/N are typcal prtve polynoals or none of the prtve polynoal of degree over Z/N s typcal prtve polynoal. Lea 8. Let f x be a onc polynoal of degree over Z/N.Ifk,then fx s a typcal prtve polynoal over Z/N f and only f f x odulo pq s a typcal prtve polynoal over Z/pq for any par of dstnct pre dvsors p and q of N. Proof. Frst of all, by the defnton of prtve polynoal over Z/N, t s clear that f x s a prtve polynoal over Z/N f and only f f x odulo pq s a prtve polynoal over Z/pq for any par of dstnct pre dvsors p and q of N. Thus, t suffces to consder prtve polynoals over Z/N. Assue f x s a prtve polynoal over Z/N. By Lea 7, f x s a typcal prtve polynoal over Z/N f and only f θ N, p = p p 8

6 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons for k. Note that and so 8 holds for p f and only f θ N = lc p, p p,..., p k, p k lc p, p j, p = p p j p 9 for any j. Ths together wth Lea 7 eans that f x s a typcal prtve polynoal over Z/N f and only f f x odulo p p j s a typcal prtve polynoal over Z/p p j for any j k. Thus, the lea s proved. Lea 8 ples that to nvestgate the exstence of typcal prtve polynoals over Z/N wth k, t suffces to dscuss the case k =. Therefore, n the followng, we focus on the exstence of typcal prtve polynoals over Z/pq, where p and q are two dstnct pre nubers. Before we start wth such dscussons, we establsh soe preparatory results concernng cyclotoc nubers. For a postve nteger a and a pre nuber p, f a s dvsble by p, then the greatest nteger r such that p r dvdes a s denoted by a. Ifa s not dvsble by p, then defne a = 0. Lea 9. Let p be a pre nuber and u be a postve nteger copre wth p. Suppose a = p e u + where e. Then for any nteger t 0, a p t = a + t f ether p, e, or t = 0,and a p t = u + + t + f p, e =, and t > 0. Proof. If t = 0, then the result s trval. Therefore, n the followng we assue that t. Note that a pt = e u + p t p t p = + p t+e t u + p e u. 0 On one hand, for, t can be deduced fro = that t = pt pt pt pt + t = t.

7 80 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons If p, e,, then t can be seen that and so 0, and ply that e + t < e + t, for, a p t = e + t = a + t. If p, e =,, thensnce t a t = u + t = u u + u + +, = t follows that v a t = v u + + t +. Ths copletes the proof. Lea 0. Let p be a pre nuber and u be a postve nteger copre wth p. Suppose a = p e u + where e. For any postve nteger n, f ether p, e, or n = 0,and a n = a + n f p, e =, and n>0. a n = u n Proof. Assue n = p l n where p,n = and l 0. It s clear that n = l. Sncea odp, we have a pl n + a pl n + +a p l + n 0 od p. Hence, t follows fro the factorzaton a n = a pl a pl n + a pl n + +a p l + and Lea 9 that a n = a p l = a + l = a + n f ether p, e, or n = 0, and a n = a p l = u + + l + = u n f p, e =, and n>0. Ths copletes the proof.

8 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons Lea. Let p be an odd pre nuber and b be a postve nteger wth b, p =. Then for any postve nteger n, b n /b n f and only f b n s not dvsble by p. Proof. If b n s not dvsble by p, then t s obvous that b n /b = 0, whch s not greater than n. Conversely, f b n /b n and b n s dvsble by p, thensnceb snot dvsble by p, we have b n = b n. b It s clear that b s copre wth p, for otherwse, b n s copre wth p. Let ord p b be the ultplcatve order of b odulo p. Snceb n odp, t follows that n s dvsble by ord p b. By takng a = p ordpb n Lea 0 we obtan b n = n ord p b + b ord p b. Note that ord p b s a factor of p whch s copre wth p, and so b n = n + b ord p b n + b a contradcton to the assupton b n /b n. Therefore, f b n /b n then b n s not dvsble by p. Now we contnue our dscusson on the exstence of typcal prtve polynoals over Z/pq where p and q are two dstnct pre nubers. In the followng two theores, necessary and suffcent condtons are proved for pq od and pq 0od,respectvely. Theore. Let p and p be two dstnct odd pre nubers and be a postve nteger. For =,, suppose d = r r v rp, where n the product r runs through all the pre dvsors of p, p. Then a prtve polynoal f x over Z/p p of degree s a typcal prtve polynoal f and only f and v p + = v p + f s even. p, p d p =, p d =, Proof. By Lea 7, f x s a typcal prtve polynoal over Z/p p f and only f θ p p, p = p, =,. 3 p

9 804 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons Note that θ p p = lc p, p p = p p p p, p, p p whch s dvsble by p /p for =,, and so 3 holds f and only f p +od p +od p, p p, p =, =,, that s, p +od p +od p, p p, r = or equvalently +od p v r v r p +od p 4 for {, } and every pre dvsor r of p. If r s a pre dvsor of p but not p, p, then t s clear that r s copre wth p. Note that p, p s dvsble by, and so r >. It follows fro Lea 0 that v r = v r p p v r p = v r. Then by Lea, we obtan that v r v r 5 p f and only f p s not dvsble by r. Syetrcally, f r s a pre dvsor of p but not p, p, then v r v r p p f and only f p s not dvsble by r. If r s a pre dvsor of p, p, then t follows fro Lea 0 that v r = v r p v r p = v r p for =,. Ths shows that 4 holds for pre nuber r. Fnally, t reans to dscuss the pre nuber, a pre dvsor of p, p. Wrte p = e u + and p = e u + where u and u are odd nubers and e and e.

10 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons If ether s odd or e > and e >, then t follows fro Lea 0 that v = v p v p = v p for =,, and so 4 holds for pre nuber. If s even and at least one of e and e s equal to, then accordng to the dfferent values of e and e, we have the followng three cases. Case. If e > and e =, then t follows fro Lea 0 that v = v p p v p = v and v = v p p v p = v + v u + >v. Hence, 4 does not hold for r =, and n ths case we have p p, p p, = >. Case. If e = and e >, then slar to Case, 4 does not hold for r =, and we have p p, p p, Case 3. If e = e =, then t follows fro Lea 0 that for =,. Ths ples that = >. v = v p v p = v + v u + p p p, p p, = p p, p p, = f and only f p + p + v = v u + = v u + = v. We note that f e = and e > ore > and e =, then v p + v p +,

11 806 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons and otherwse v p + = v p + =. Therefore, f s even, we arrve at the fnal concluson that p p, p p, = p p, p p, = f and only f v p + = v p +. Cobnng all the above analyss shows that the theore holds. Reark 3. It can be seen that Theore s a copleton of the dscusson about the exstence of nteger S ade by [, Secton 3.]. Thus, actually the authors of [] proved the dstnctness of axal length sequences generated by typcal prtve polynoals over Z/pq odulo whch ght be thought as the true prtve polynoals over Z/pq, referrng to the behavor of prtve polynoals over fnte felds. Theore 4. Let p be an odd pre nuber. Then a prtve polynoal f x over Z/p of degree s a typcal prtve polynoal f and only f s copre wth p. Proof. Note that θ p = lc, p p. By Lea 7, f x s a typcal prtve polynoal over Z/p f and only f θ p, p = p p, that s,, p p, p =. 6 Furtherore, t can be seen that 6 holds f and only f v r v r p 7 for every pre dvsor of r of p. Snce s copre wth, we only need to consder odd pre dvsors of p. Assue r s an odd pre dvsor of p. By Lea 0 we get v r = v r p v r p = v r, p and so t follows fro Lea that 7 holds for r f and only f s not dvsble by r.

12 T. Tan, W.-F. Q / Fnte Felds and Ther Applcatons Conclusons Let N be a postve nteger whch s a product of dstnct pre nubers and Z/N be the nteger resdue rng odulo N. Ths paper copletely solves the exstence of typcal prtve polynoals over Z/N. The result and the ethod can be generalzed to general nteger resdue rngs, but t should be noted that for a pre nuber p and an nteger e, the subexponents of prtve polynoals over Z/p e of degree have any possble values, say p p /p, e. Thus, unlke the case of Z/N, the exstence of typcal prtve polynoals of degree over Z/p e M depends on ndvdual prtve polynoals besdes and p e M, where M s an arbtrary postve nteger. References [] H.J. Chen, W.F. Q, On the dstnctness of axal length sequences over Z/pq odulo, Fnte Felds Appl [] R. Ldl, H. Nederreter, Fnte Felds, Addson Wesley, Readng, MA, 983. [3] J.W.P. Hrschfeld, Cyclc projectvtes n PGn, q, Teore Cobnatore, Accad. Naz. Lnce Roa [4] J.W.P. Hrschfeld, Projectve Geoetry Over Fnte Felds, Clarendon Press, Oxford, 979.