A Note on Irreducible Polynomials and Identity Testing

Transcription

1 A Note on Irrecible Polynomials an Ientity Testing Chanan Saha Department of Compter Science an Engineering Inian Institte of Technology Kanpr Abstract We show that, given a finite fiel F q an an integer > 0, there is a eterministic algorithm that fins an irrecible polynomial g over F q in time polynomial in an log q sch that, log q < egg) < where c is a constant. This reslt follows easily from Aleman an Lenstra s reslt [AJ86] on irrecible polynomials over prime fiels an Lenstra s reslt [Jr.91] on isomorphisms between finite fiels. 1 As an application, we show that sch constrction of irrecible polynomials can be se to bil a sample space of coprime polynomials for the Agrawal- Biswas [AB03] polynomial ientity testing algorithm. 1 Introction The problem of fining irrecible polynomials over finite fiels is an important problem in algorithmic algebra with many applications in coing theory, cryptography an complexity theory. In many sch applications the primary se of irrecible polynomials is in the constrction of larger finite fiels. Althogh a ranom polynomial is irrecible with reasonable probability, there is no known eterministic polynomial time algorithm for contrcting irrecible polynomial of a given egree. An efficient algorithm was esigne by Aleman an Lenstra [AJ86] in case of prime fiels ner the assmption of the Extene Riemann Hypothesis. In the same paper they gave another reslt as state by the following theorem. Theorem 1.1 There is a eterministic algorithm that on inpt a prime p an an integer > 0, otpts an irrecible polynomial g F p [x] sch that c log p < egg) < where c is a constant. The algorithm takes log p) O1) time. We show that this reslt can be extene to any finite fiel as state by the following theorem. Theorem 1.2 Let F q be a finite fiel, with q = p an p prime, that is explicitly given by an irrecible polynomial f F p [x] of egree i.e. F q = Fp[x]. Given an integer > 0, there is a eterministic algorithm that otpts an irrecible polynomial g F q [x] sch that, log q < egg) < where c is a constant. The algorithm takes log q) O1) time. 1 Henrik W. Lenstra Jr. pointe ot that a stronger reslt follows from [AJ86] an [Jr.91] i.e. it is possible to fin a egree irrecible polynomial over F q from a egree irrecible polynomial over F p. 1

2 Or proof is fairly straightforwar an it makes se of Theorem 1.1 an a reslt by Lensta [Jr.91] on isomorphism between finite fiels, Theorem 1.3 There is an algorithm that, given a finite fiel F, a positive integer, an two fiel extensions F 1, F 2 of F of egree, constrcts an F -isomophism F 1 F 2 in time log F 1 ) O1). In the absence of an efficient eterministic algorithm for fining irrecible polynomial of a given egree, partial reslt like Theorem 1.2 col be sefl as there are applications where an irrecible polynomial of roghly the esire egree is jst as goo. We have chosen one sch application from complexity theory - the problem of testing whether a polynomial, given as a circit, is ientically zero. We show that Theorem 1.2 is sefl in the constrction of a sample space of polynomials se by Agrawal an Biswas s algorithm [AB03] to check if an inpt polynomial is ientically zero. In their paper [AB03], Agrawal an Biswas showe an elegant way of constrcting a small space of almost coprime low egree polynomials. Their algorithm works by ranomly selecting a polynomial from the sample space an compting the otpt of the circit molo the chosen polynomial. The inpt polynomial is eclare as ientically zero if an only if the otpt of the circit is evalate to zero. An important featre of their algorithm is that it achieves the time-error traeoff with a rnning time that is only polynomial in the size of the circit an the error parameter. For the sake of theoretical interest, it is natral to ask if one can get a similar reslt sing a sample space of mtally coprime polynomials instea of almost coprime polynomials. We answer this qestion affirmatively. The time-error traeoff property of the algorithm is also preserve. A slight avantage of sing coprime polynomials over almost coprime polynomials is that the egree of each polynomial in the sample space can be chosen to be smaller than the polynomials in the sample space of almost coprime polynomials. This makes molo operations less costly which gives a slightly better rnning time of the algorithm for large enogh circits. 2 Fining Irrecible Polynomials In this section we prove Theorem 1.2. Theorem 2.1 Let F q be a finite fiel, with q = p an p prime, that is explicitly given by an irrecible polynomial f F p [z] of egree i.e. F q = Fp[z]. Given an integer > 0, there is a eterministic algorithm that otpts an irrecible polynomial g F q [y] sch that, log q < egg) < where c is a constant. The algorithm takes log q) O1) time. Proof: The fiel F q enotes the explicitly given fiel Fp[z]. The otline of the proof is as follows. 1. Constrct a sfficiently large fiel E that contains F q, an isomorphic copy of F q. The extension egree [E : F q] shol be sfficiently large so that step 2 is feasible. 2. Constrct an irrecible polynomial g F q[y] sch that c log p < egg ) < log q log p. 2

3 3. Use an isomorphism to fin an irrecible polynomial g F q [y] sch that egg) = egg ). The rest of the proof shows how to perform each of these steps. Step 1: Constrct a sfficiently large fiel. Fining a sfficiently large extension of F p is mae easy by Theorem 1.1, bt the only complication being that this extene fiel might not contain an isomorphic copy of F q. This is becase Theorem 1.1 oes not provie a hanle on the exact vale of the extension egree. The following iscssion shows how to circmvent this problem. Using Theorem 1.1 first fin an irrecible polynomial h F p [x] of egree m sch that, c log p < m <. The fiel F p m = Fp[x] h) may not contain an isomorphic copy F q as may not ivie m). Therefore, we inten to constrct the fiel F p m where m = lcmm, ). To o this first fin the prime factorizations of m an an fin m = lcmm, ). Sppose v be a prime sch that v k m v k exactly ivies m ). Then v k exactly ivies m or. Assme withot loss of generality that v k m. Let F = F p m an K = F p v k. The iea is to fin an element that generates K over F p. For this we make se of the trace fnction. Recall that, if M = F q k efine as the sm, is an extension fiel of N = F q then the fnction T r M N : M N is T r M N α) = α qm 1 + α qm α q + α, for every α M. Fnction T r M N is a linear map from M onto N. Consier taking trace of the elements, X,..., X m 1, of F over K, T r F K X i ) for 0 < i < m, where X = x mo h. Since the elements 1, X,..., X m 1 form the basis of F over F p an v is prime, there mst exist an i, 0 < i < m, sch that, K = F p v k = F pt r F K X i ) ). Otherwise the fact that T r F K maps F onto K is contraicte. If α v = T r F K X i ) is the generator of K over F p, then the minimal polynomial of α v over F p is an irrecible polynomial h v of egree v k. The task of fining α v an h v is efficient becase fining minimal polynomial an testing for irrecibility can be one in eterministic polynomial time. By repeating the above process for other prime factors of m, we can fin all irrecible polynomials of egree w l sch that w l m an w is a prime. Sppose h v x) an h w y) be two irrecible polynomials of relatively coprime egrees v k an w l respectively. Let F p X) = Fp[x] h v) an F p Y ) = Fp[y] h where X = x mo h w) v an Y = y mo h w. Then it is not ifficlt to verify that F p X, Y ) = F p X + Y ) = F p v k w l. This is becase any maximal proper sbfiel of F p vk w l mst contain either X or Y an hence cannot contain X + Y. Therefore, the minimal polynomial of X + Y over F p yiels an irrecible polynomial of egree v k w l. Repeat this process 3

4 till we get an irrecible polynomial of egree m over F p. Let this polynomial be h. The fiel E = Fp[x] h ) = F p m ths contains an isomorphic copy of F q which we enote by F q. Step 2: Constrcting irrecible polynomial over F q. As before, assme that E = Fp[x] h ) an X = x mo h. The minimal polynomial of X E over F q is an irrecible polynomial of egree m over F q. The process of fining a monic polynomial of egree m with coefficients taken from F q reces to solving a bnch of linear eqations over F p. To see this rection assme that X satisfies the monic polynomial g y) F q[y], Take each β i E to be, g y) = m / i=0 β i y i where β i F q an β m = 1. β i = m 1 j=0 b ij x j where b ij s are nknowns in F p. We can ensre that each β i belongs to F q sing the eqation β q i = β i, which in trn reces to linear eqations in b ij s. Frther, since X is a root of g y), g X) = m / i=0 β i x i mo h = 0 which yiels more linear eqations in b ij s. By solving the eqations for b ij s we get the niqe irrecible polynomial g y) F q[y] of egree m. Step 3: Fining an irrecible polynomial over F q throgh an isomorphism. Sppose that v is a prime factor of sch that v k an let K = F p vk. As arge before, there exists an i 0 i < m ) sch that for α = T r E K X i ), K = F p α). Moreover, if F p v k = F pα) an F p w l = F pβ) where v an w are istinct prime ivisors of then F p v k w l = F pα + β). This way we can fin an element γ E sch that F q = F p γ). Let f y) F p [y] be the minimal polynomial of γ over F p. Using Theorem 1.3 fin an isomorphism σz) from Fp[z] f ) to F q = Fp[z]. This means that the element z in Fp[z] f ) maps to the element σz) in F q = Fp[z]. Since g y) belongs to F q[y], we can express each coefficient of g y) as an F p -linear combinations of the basis elements {γ j } 1 j<. This is one by solving linear eqations over F p. Ths if β = 1 i=0 b iγ i is an element in F q then 1 i=0 b iσz) i mo fz) is the image of β in F q by ientifying γ with the element z in Fp[z] f ) ). This isomorphism when applie to the coefficients of g y) yiels an irrecible polynomial gy) F q [y] of egree m. Since, hence, c log p < m < an m m m, log q < egg) <. 4

5 3 Constrcting the sample space for Ientity Testing Let C be a circit of size s that comptes the polynomial P x 1,..., x n ) F q [x 1,..., x n ]. The task is to check whether the polynomial is ientically zero. Assme that the egree in each variable is bone by 1. As in [AB03], first convert the mltivariate polynomial to an nivariate polynomial P x) by sbstitting x i by x i 1 this is also known as Kronecker sbstittion). This sbstittion has the property that P x 1,..., x n ) is a zero polynomial if an only if P x) is a zero polynomial. Note that the egree of polynomial P x) col be as high as n 1. The nmber of coprime polynomials of egree t that ivies P x) is at most n t. Sppose that we have a sample space of 2 r coprime polynomials of egree t, then the probability that a ranom polynomial from the sample space ivies P x) is at most n t 2. Ths if or sample r space is large enogh i.e. 2 r > n then the error probability is bone by 1 t. Constrcting sch a sample space is mae easy by Theorem 1.2. First exten the fiel F q to another fiel F q sing an appropriate irrecible polynomial sch that q > n. Since the fiel F q is explicitly given by an irrecible polynomial over F q, we can efine a natral orering among the elements in F q sing the orering of elements in F p. Ths with every a F q we can associate a non-negative integer inexa) < q. Also note that given an integer i < n we can niqely compte the element a F q sch that inexa) = i. Now efine the sample space of polynomials as, S = {X t + a : a F q an inexa) < n } Sch a sample space can be efine sing only r = n log ranom bits. The time taken for evalating Kronecker sbstittions molo a polynomial X t +a) is Õn2 log log q ɛ ), where t = 1 ɛ an ɛ is the error parameter. The total time taken for molar operations in the circit is Õs 1 ɛ n log log q), where s is the size of the circit. In aition the time taken for extening the fiel is n log ) c for a constant c. Ths, when s is greater than n log ) c, the total rnning time is Õs 1 ɛ n log log q). This is slightly better than the rnning time of Õs + n 2 log ) n log ) 2 n log + 1 ɛ ) log q)2 ) given in [AB03]. 4 Remarks The argments se to prove Theorem 1.2 can also be se to exten Alleman an Lensta s [AJ86] other reslt on fining irrecible polynomial of a given egree over a prime fiel to any finite fiel, ner the assmption of the Extene Riemann Hypothesis. It wol be nice to fin other applications of Theorem 1.2. References [AB03] Maninra Agrawal an Somenath Biswas. Primality an Ientity Testing via Chinese Remainering. Jornal of the ACM, 504): , [AJ86] Leonar M. Aleman an Henrik W. Lenstra Jr. Fining Irrecible Polynomials over Finite Fiels. STOC, [Jr.91] Henrik W. Lenstra Jr. Fining Isomorphisms Between Finite Fiels. Mathematics of Comptation, 56193): ,