Calculating cyclotomic polynomials

Transcription

1 Calculatig cyclotomic polyomials Adrew Arold Michael Moaga July 17, 2008 Abstract We preset two algorithms to calculate Φ (z), the th cyclotomic polyomial. The first algorithm calculates Φ (z) by a series of polyomial divisios, optimized usig the discrete Fourier trasform. The secod algorithm calculates Φ (z) as a quotiet of sparse power series. These algorithms were used to calculate cyclotomic polyomials of large height ad legth. I particular we fid cyclotomic polyomials Φ (z) of miimal order whose height is greater tha, 2, 3, ad 4, respectively. 1 Itroductio The th cyclotomic polyomial, Φ (z), is the moic polyomial whose φ() distict roots are exactly the th primitive roots of uity. Φ (z) = k=1 gcd(k,)=1 It is a irreducible polyomial i Z. We write φ() ( ) z e 2πi k. Φ (z) = a (k)z k, k=1 For < 105, the coefficiets of Φ (z) are strictly 1, 0, or 1. Deote by A() ad S() the height ad legth, respectively, of the th cyclotomic polyomial. That is, φ() A() = max a (k), S() = a (k). 0 k φ() Paul Erdos [5] proved that A() is ot bouded above by ay polyomial of. There is a wealth of material o the behaviour of A() ad the size of cyclotomic polyomial coefficiets [3], [2], [10], [11]; however, comparatively little work has goe ito actually calculatig these values. Koshiba calculated k=0 1

2 A( ) = [8] ad foud the coefficiets of Φ (z) with degree less tha φ()/10 for = = [9]. Bosma foud the least value of k for which a occurs as a (k) for some, for a < 50 [4]. We preset ew results o A(), S(), ad o the bouds of a (k) for fixed k. 1.1 Orgaizatio of paper. We preset two algorithms to calculate cyclotomic polyomials. Our first algorithm calculates Φ (z) for squarefree via a series of polyomial divisios. We use the discrete Fourier trasform to do these divisios quickly. The secod algorithm calculates Φ (z) as a quotiet of sparse power series. Usig these two algorithms, we have produced a wealth of data o the heights ad leghts of cyclotomic polyomials. Amogst our results, we have foud: A() ad S() for all < the smallest values of for which A() >, 2 ad 3 respectively a order for which A() > 4 the smallest such that A() > 2 64 (machie precisio) A() ad S() for squarefree < with six or more prime factors A() ad S() for squarefree < 10 9 with seve or more prime factors the maximum ad miimum values of a (k) for fixed k, for k < 138 the smallest istace of k for which a (k) = a for some, for a < Prelimiaries We are iterested i fidig cyclotomic polyomials with large coefficiets. The followig two idetities are useful: Lemma 1. Let > 1 be odd. The Φ 2 (z) = Φ ( z). Lemma 2. Let p be a prime that divides. The Φ p (z) = Φ (z p ) Lemma 1 tells us A(2) = A() ad S(2) = S() for odd. Lemma 2 says that A(p) = A() ad S(p) = S() for p dividig. For the remaider of the paper, we will be strictly cocered with the calculatio of cyclotomic polyomials of squarefree, odd order. Lemmas 1 ad 2 provide a easy meas of calculatig Φ (z) for eve or squarefree, provided we ca calculate Φ m (z), where m is the product of the odd prime factors of. 2

3 2 Usig the discrete Fourier trasform to calculate Φ (z) Our first algorithm calculates Φ (z) by a series of polyomial divisios. For primes p m, Φ mp (z) = Φ m(z p ) Φ m (z). (1) We thus are able to calculate Φ (z) by repeated polyomial divisio, as detailed i algorithm 1. Iput: = p 1 p 2 p k, a product of k distict primes. Output: Φ (z), the th cyclotomic polyomial. m 1 Φ m (z) z 1 for j = 1 to k do m mp j Φ m (z) Φ m(z pj ) Φ m (z) m m Algorithm 1: solvig Φ (z) by repeated divisio Our key optimizatio was to perform the polyomial divisio i algorithm 1 by way of the discrete Fourier trasform (DFT)[6]. Algorithm 2 gives a highlevel descriptio of the divisio calculatio. Usig the discrete Fourier trasform, we ca calculate Φm(zp ) Φ m(z) i O(N log(n)) arithmetic operatios, where N is the smallest power of 2 greater tha φ(m) p, the degree of the umerator. Observe that algorithm 2 requires that B i is ozero for ω i, 0 i < N. This is ot a problem for odd orders m, however, so log as our modulus q does ot divide the order m. Every power of ω is a 2 k th root of uity modulo q for some k < s. For m > 1 it holds that Φ m (z) divides z m 1 z 1 = zm 1 + z m z + 1. (2) Thus if q does ot divide m, ay root of Φ m (z) mod q is ecessarily a m th root of uity ot equal to 1. Give m is always odd for our purposes, the oly m th root of uity that is also a 2 k th root of uity modulo q for some k is exactly 1, ad so Φ m (ω i ) 0 mod q for 0 i < N. 2.1 Implemetatio Details Our implemetatio of the DFT modulo a 42-bit prime q requires 12N bytes of storage: 8N to store the iput polyomial ad the result of the DFT, ad aother 4N bytes i which we use for the DFT computatio. We require that N is a power of two greater tha φ(/p k )p k, the degree of the umerator Φ /pk (z p k ). 3

4 Iput: Φ m (z) p, a prime ot dividig N = 2 s, a power of 2 greater tha φ(m) p + 1, the umber of coefficiets of Φ m (z p ) q, a prime of the form q = r N + 1 ω, a primitive N th root of uity modulo q Output: Φ mp (z) Calculate A i ad B i by the DFT: A i Φ (ω ip ) mod q for i = 0, 1,..., N 1 B i Φ (ω i ) mod q for i = 0, 1,..., N 1 C i A i B i for i = 0, 1,..., N 1 /* C i = Φ mp (ω i ) */ Iterpolate C by the iverse discrete fast Fourier trasform to get the coefficiets of Φ mp (z). Algorithm 2: A brief descriptio of polyomial divisio via the discrete fast Fourier trasform For 32-bit primes q, N is o greater tha For cyclotomic polyomials of larger degree, we require larger primes. Choosig uecessarily large primes for the DFT, however, would require multiprecisio arithmetic. Usig 64-bit arithmetic, we are able to do multiplicatio modulo prime umbers as large as 42-bits (see algorithm 3); however, the multiplicatio is roughly four times slower tha multiplyig two itegers. We typically use a 32-bit prime uless a 42-bit prime is ecessary. Our implemetatio of algorithm 2 requires 20N bytes of storage: 16N bytes to store the itermediate results A i ad B i for 0 i < N, ad additioal 4N bytes of memory for the DFT computatio. Iput: a = a 41 a 40 a 1 a 0, b = b 41 b 40 b 1 b 0, two 42-bit primes modulo a 42-bit prime q Output: c = ab mod q A 0 a 20 a 19 a 0 /* A 0 = a mod 2 21 */ A 1 a 41 a 40 a 21 /* A 1 = (a A 0 )/2 21 */ B 0 b 20 b 19 b 0 /* B 0 = b mod 2 21 */ B 1 b 41 b 40 b 21 /* B 1 = (b B 0 )/2 21 */ c (A 1 )(B 1 )(2 21 ) + (A 1 )(B 0 ) + (A 0 )(B 1 ) c c mod q c (c)(2 21 ) + (A 0 )(B 0 ) mod q Algorithm 3: multiplicatio modulo a 42-bit prime The DFT oly gives us the coefficiets of Φ (z) modulo a prime q 1. Our resultig polyomial, call it H (z), will ot equal Φ (z) if A() > q1 2. If the height of H (z) is close to q1 2, we calculate Φ (z) modulo aother prime q 2. We 4

5 the recostruct H (z) Φ (z) mod q 1 q 2 by Chiese remaiderig. We do this process with primes q 1, q 2... q l util the height of H (z) is less tha q1q2...q l 2 by a factor of 2 20 or more. We the take our solutio H (z) ad use the DFT to test that H (ω j ) Φ /pk (ω j ) Φ /pk (ω jp k ) 0 mod q l+1, (3) for some ew prime q l+1 with N th primitive root ω. If equatio (3) holds for all j, H (z) Φ (z) (mod Q = q 1 q 2 q l q l+1 ). For q l+1 > 2 40, it follows that all of the coefficiets of Φ (z), modulo Q, lie i the iterval ( Q 2, Q 60 2 ). We cosider redudat bits sufficiet. All our results obtaied by this method have bee cosistet with results we have obtaied by o-modular algorithms. It is very ulikely that Φ (z) would have coefficiets strictly i such a small rage mod Q provided its height A() were greater tha Q 2. Table 1 lists the primes ad the primitive roots we used i our computatios. q = r N + 1 size of q ω q = bits 243 q = bits q = bits q = bits q = bits Table 1: primes ad the primitive roots used i our DFT calculatios The brut of the computatio takes place i the last divisio step, as each successive divisio step ivolves polyomials of greater degree, which requires us to use larger N for the DFT. For squarefree with largest prime divisor p, we ca compute Φ (z) i O(φ( p )p log( p p)) arithmetic operatios. Our implemetatio of algorithm 2 modulo oe 42-bit prime q requires 20N bytes of storage: 8N bytes are used to store the umerator ad the fial result; a additioal 4N bytes are used i the DFT computatio. 3 Calculatig Φ (z) as a quotiet of sparse polyomials It is well kow that Φ (z) = d ( 1 z d ) µ( d ), (4) where µ is the Mőbius fuctio. For istace, for = 105 = 3 5 7, Φ 105 (z) = (1 z3 )(1 z 5 )(1 z 7 )(1 z 105 ) (1 z)(1 z 15 )(1 z 21 )(1 z 35 ) The sparseess of each term i this quotiet leds itself to fast polyomial arithmetic. For the purposes of our algorithm, we treat Φ (z) as a power series. 5

6 Multiplyig a power series C(z) = i=0 c z by 1 z d is easy: ( )( ) d 1 c z 1 z d = c z i + (c i c i d )z i (5) i=0 To divide by 1 z d we merely multiply by the power series for 1 i=0 i=0 i=d 1 z d : ( ) c z )(1 + z d + z 2d + = (c i + c i d + c i 2d + )z i (6) Observe that the coefficiets of C(z)(1+z d ) ad C()(1+z d ) 1 deped strictly o coefficiets of lesser degree i C(z). I additio, we kow that the φ() + 1 coefficiets of Φ (z), {a (0), a (1),... a (φ())}, are palidromic, that is, a (k) = a (φ() k). So, to calculate the Φ (z) as a power series, we oly eed compute the first φ() terms. We may select the divisors d of i Iput: = p 1 p 2 p k, a product of k distict primes. Output: a (0), a (1),, a ( φ() 2 + 1), the first half of the coefficiets of Φ (z) M φ() a (0) 1 for 1 i M do a (i) 0 i=0 for d, d > 0 do if d has a eve umber of prime factors the i M while i d do c i a (i) a (i d) i i 1 else i d while i M do a (i) a (i) + a (i d) i i + 1 Algorithm 4: solvig Φ (z) as a quotiet of sparse power series ay order i algorithm 4. We could select all the values of d correspodig to terms i the deomiator first, for istace. That method, however, appears to result i some itermediate terms to become very large, typically larger tha A(). We select d i the order give by algorithm 5, usig the bits of iterator i to determie which primes divisors p of to iclude i our divisor d. 6

7 Iput: = p 1 p 2 p k, a product of k distict primes. Output: d 0, d 1,, d 2 1, a orderig of the positive divisors of k for i 0 to 2 k 1 do j i d i 1 for k 1to k do if j = 1 mod 2 the d i d i p k j j = 2 Algorithm 5: orderig the divisors of 3.1 A compariso of the two algorithms. Calculatig Φ (z) for, a product of k distict primes takes O(2 k φ() arithmetic operatios ad 4φ() bytes of memory to store the terms with 64-bit precisio. We expect that our secod approach is slower for a product of may distict small primes; however, we curretly caot calculate Φ (z) for odd with more tha 9 distict prime factors. Calculatig Φ (z) where = , the product of the smallest te odd primes, would require φ() 2 = bytes (approximately 114 GB) of memory merely to store the polyomial coefficiets up to 64-bit precisio. I practise, the power series method is appreciably faster tha the DFT approach. Our implemetatio of algorithm 4 has several advatages. First, we ca perform the calculatios i memory; aside from a small overhead, all the memory used i the power series algorithm is to store the coefficiets. The power series algorithm makes better use of the memory used to store terms. Usig 64-bits of storage for oe term gives us exactly 64-bit precisio usig algorithm 4, whereas algorithm 1 uses 64-bits to store a 42-bit terms. I additio, the arithmetic operatios used i algorithm 4 are strictly additios ad substractios, which take fewer CPU cycles tha multiplicatio ad divisio operatios. I practise, algorithm 4 is approximately 30 times faster tha algorithm 1. 4 Results 4.1 Heights ad Legths of cyclotomic polyomials To fid cyclotomic polyomials with large heights, we strictly looked at cyclotomic polyomials of squarefree, odd order. For odd > 1, it holds that Φ 2 (z) = Φ ( z), (7) ad so A(2) = A(); moreoever, for ay prime p dividig some iteger, Φ p (z) = Φ (z p ), (8) 7

8 thus A(p) = A(). We also cosidered the umber of prime factors of. Bag showed that for = pqr, a product of three primes with p < q < r, that A() < p. For, a product of two primes, A() = 1 [1]. Bloom later proved for = pqrs, a product of four primes with p < q < r < s, that A() < p(p 1)(pq 1) [3]. Batema proved a more geeral albeit slightly weaker result [2]: for = p 1 p 2 p k, a product of k distict primes with p 1 < p 2 < p k, A() j 2 k=1 p 2j k 1 1 k. (9) For istace, for = p 1 p 2 p 3 p 4 p 5, A() < p 7 1 p 3 2 p 1 3 < 2. Usig the two methods detailed i this paper, we have created a library of data o A() ad S(). We iclude here our more oteworthy results. Table 2 shows those cyclotomic polyomials we have foud whose height is greater tha all those of smaller order. Table 2 also shows the growth of log A(), which we foud of iterest. Our results iclude the first istaces of such that A() >, A() > 2, A() > 3, ad A() > 4. Tables 3 through 8 have large A() for, a product of three distict primes, up to, a product of eight primes. Table 9 shows A() for, a product of the s smallest odd primes, for 1 s 9. Table 10 shows A() for various multiples of A( ) was is first istace of A() such that A() > 2. We expected that if order produced a large height, that multiplyig by aother prime would result i large heights as well. Table 10 supports this. Ideed, give for primes p such that p Φ p(z) = Φ (z p ) Φ (z), (10) it holds that A(p) A()S(). This follows by ispectio of log divisio. Table 10 otably has the first we have foud for which A() > 4. We iclude every istace of A() which required double-precisio (128 bit) arithmetic, for < , ad every istace of A() which required tripleprecisio (192 bit) arithmetic i tables. Table 11 presets cyclotomic polyomials of large legth. I seems, usurprisigly, that the cyclotomic polyomials with the greatest heights had the greatest legths as well. 4.2 Extreme values for the k th cyclotomic polyomial coefficiet a (k) Tables 12 ad 13 give the the maximum ad miimum values of the k th cyclotomic polyomial coefficiet a (k), ad the smallest order for which we obtai those extrema. Algorithm 4 tells us that a (k) depeds strictly o the divisors of that are less tha or equal to k. To fid the extreme values of a (k) for fixed k, we calculate a (k) for all squarefree whose prime divisors are ot greater tha k. It is easy to check by ispectio whether a o-squarefree order will produce a larger coefficiet. By lemma 2, if p divides, the a p (k) = a ( k p ) 8

9 if p k, ad 0 otherwise. Thus to prove that some o-squarefree order will ot produce a larger coefficiet a (k), it suffices to show that for every positive divisor d of k, that max a (d) < max a (k), (11) ad similarly for the miimum values. To fid the miimum ad maximum values of a (k) for 0 k K, where there are P primes less tha or equal to K, we eed to calculate the the first K + 1 terms of some 2 P cyclotomic polyomials. Teatively we have results for K = 138. We fid the k = 119 is the smallest k > 0 such that a (k) > k for some, as mi a (119) = 136. Tables 14 ad 15 show, for 248 a 248, the smallest k such that a (k) = a, ad for that a ad k, the smallest order. We exted results from [4] ad [7]. Refereces [1] A. S. Bag. Om ligige φ (x) = 0. Nyt Tidsskrift for Mathematik, (6):6 12, [2] P. T. Batema, C. Pomerace, ad R. C. Vaugha. O the size of the coefficiets of the cyclotomic polyomial. I Topics i classical umber theory, Vol. I, II (Budapest, 1981), volume 34 of Colloq. Math. Soc. Jáos Bolyai, pages North-Hollad, Amsterdam, [3] D. M. Bloom. O the coefficiets of the cyclotomic polyomials. Amer. Math. Mothly, 75: , [4] Wieb Bosma. Computatio of cyclotomic polyomials with Magma. I Computatioal algebra ad umber theory (Sydey, 1992), volume 325 of Math. Appl., pages Kluwer Acad. Publ., Dordrecht, [5] Paul Erdős ad R.C. Vaugh. O the coefficiets of the cyclotomic polyomial. Bull. Amer. Math. Soc., 52: , [6] Keith O. Geddes, Stephe R. Czapor, ad George Labah. Algorithms for Computer Algebra. Kluwer Academic Publishers, 101 Philip Drive, Assiippi Park, Norwell, Massachusetts USA, [7] A. Grytczuk ad B. Tropak. A umerical method for the determiatio of the cyclotomic polyomial coefficiets. I Computatioal umber theory (Debrece, 1989), pages de Gruyter, Berli, [8] Yôichi Koshiba. O the calculatios of the coefficiets of the cyclotomic polyomials. Rep. Fac. Sci. Kagoshima Uiv., (31):31 44, [9] Yôichi Koshiba. O the calculatios of the coefficiets of the cyclotomic polyomials. II. Rep. Fac. Sci. Kagoshima Uiv., (33):55 59,

10 [10] Helmut Maier. The size of the coefficiets of cyclotomic polyomials. I Aalytic umber theory, Vol. 2 (Allerto Park, IL, 1995), volume 139 of Progr. Math., pages Birkhäuser Bosto, Bosto, MA, [11] R. Thagadurai. O the coefficiets of cyclotomic polyomials. I Cyclotomic fields ad related topics (Pue, 1999), pages Bhaskaracharya Pratishthaa, Pue,

11 5 Appedix Table 2: such that A() > A(m) for m < A() log A()

12 Table 3: A() for a product of three distict odd primes, < factorizatio of A()

13 Table 4: A() for a product of four distict odd primes, < factorizatio of A()

14 Table 5: A() for a product of five distict odd primes, < 10 8 factorizatio of A()

15 Table 6: A() for a product of six distict primes, < factorizatio of A() Table 7: Large A() for a product of seve distict primes, < factorizatio of A()

16 Table 8: Large A() for a product of eight distict primes, < factorizatio of A() Table 9: A() for a product of the smallest odd primes factorizatio of A() * *(Koshiba,2002). Table 10: A() for a multiple of m = A() 7m = m = m = m = m = m = m = m = m = m =

17 Table 11: S() such that S() > S(m) for > m, for < 10 9 S() Cotiued o Next Page... 17

18 Table 11 Cotiued S()

19 Table 12: The maximum values of a (k) for fixed k, 0 k 138 k max (k) Cotiued o Next Page... 19

20 Table 12 Cotiued k max (k) Cotiued o Next Page... 20

21 Table 12 Cotiued k max (k) Cotiued o Next Page... 21

22 Table 12 Cotiued k max (k)

23 Table 13: The miimum values of a (k) for fixed k, 0 k 138 k mi a (k) Cotiued o Next Page... 23

24 Table 13 Cotiued k mi a (k) Cotiued o Next Page... 24

25 Table 13 Cotiued k mi a (k) Cotiued o Next Page... 25

26 Table 13 Cotiued k mi a (k)

27 Table 14: Least k for which a occurs as a (k), 0 < a 248 a k Cotiued o Next Page... 27

28 Table 14 Cotiued a k Cotiued o Next Page... 28

29 Table 14 Cotiued a k Cotiued o Next Page... 29

30 Table 14 Cotiued a k Cotiued o Next Page... 30

31 Table 14 Cotiued a k Cotiued o Next Page... 31

32 Table 14 Cotiued a k

33 Table 15: Least k for which a occurs as a (k), 248 a < 0 a k Cotiued o Next Page... 33

34 Table 15 Cotiued a k Cotiued o Next Page... 34

35 Table 15 Cotiued a k Cotiued o Next Page... 35

36 Table 15 Cotiued a k Cotiued o Next Page... 36

37 Table 15 Cotiued a k Cotiued o Next Page... 37