IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT OMRAN AHMADI AND ALFRED MENEZES

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1 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT OMRAN AHMADI AND ALFRED MENEZES Abstract. We establsh some necessary condtons for the exstence of rreducble polynomals of degree n and weght n over F 2. Such polynomals can be used to effcently mplement multplcaton n F 2 n. We also provde a smple proof of a result of Bluher concernng the reducblty of a certan famly of polynomals. 1 n 1. Introducton Let q be a prme power, and let I q (n) denote the number of monc rreducble polynomals of degree n over F q. It s well known that I q (n) = d n µ(d)qn/d where µ s the Möbus functon, and that I q (n) qn n. Many researchers have studed the dstrbuton of rreducble polynomals havng certan propertes. In partcular, much work has been done on the exstence and dstrbuton of rreducble trnomals over F 2 ; for example see [15, 3, 4] and the references theren. The followng theorem, due to Swan, s an mportant result about the non-exstence of rreducble trnomals over F 2. Theorem 1. [15] Let n > m > 0 and assume that exactly one of n, m s odd. Then x n + x m + 1 has an even number of rreducble factors over F 2 f and only f () n s even, m s odd, n 2m, and nm/2 0, 1 (mod 4). () n s odd, m s even, m 2n, and n ±3 (mod 8). () n s odd, m s even, m 2n, and n ±1 (mod 8). The case where n and m are both odd can be reduced to the case m even by consderng x n + x n m + 1. For example, f n 0 (mod 8) then Theorem 1() says that x n + x m + 1 has an even number of rreducble factors. Thus there does not exst an rreducble trnomal of degree n over F 2 when n 0 (mod 8). There s overwhelmng evdence n support of the conjecture that there exsts an rreducble pentanomal of degree n over F 2 for each n 4 [11]; however exstence has not yet been proven. More generally, one can ask about the exstence of an rreducble polynomal of degree n and weght t over F 2 for each odd t [3, n + 1]. (The weght of a polynomal s the number of ts coeffcents that are nonzero.) Date: January 12, Key words and phrases. Fnte Felds, Irreducble Polynomals. 1

2 2 OMRAN AHMADI AND ALFRED MENEZES Shparlnsk [12] and Ahmad [1] respectvely proved the exstence of rreducble degree-n polynomals of weght n 4 + o(n) and n 2 + o(n) over F 2. It s well known that there exsts an rreducble degree-n polynomal of weght n + 1 over F 2 f and only f n + 1 s prme (and hence n s even) and 2 s a generator of the multplcatve group of ntegers modulo n + 1. In ths paper, we consder the exstence of rreducble degree-n polynomals of weght n (where n s odd) over F 2. The remander of ths paper s organzed as follows. In Secton 2 we show that rreducble polynomals of weght n can be used to mplement fast multplcaton n the feld F 2 n. In Secton 3 we prove an analogue of Swan s theorem for weght-n polynomals over F 2. The results of a computer search for rreducble polynomals of weght n are summarzed n Secton 4. In Secton 5, we use the technques of Secton 3 to provde a smple proof of a theorem of Bluher about the reducblty of a certan famly of polynomals over F Fast multplcaton n F 2 n Let f(x) be an rreducble polynomal of degree n over F 2. Then F 2 n = F 2 [x]/(f) s a fnte feld of order 2 n, and f(x) s called the reducton polynomal. Elements of F 2 n are canoncally represented as polynomals n F 2 [x] of degree less than n. Multplcaton of a(x), b(x) F 2 n can be performed by frst computng the polynomal product c(x) of a(x) and b(x), and then reducng c(x) modulo f(x). The reducton operaton s consderably faster f f(x) has small weght and f ts mddle terms (the nonzero terms not ncludng the end terms x n and 1) are close to each other and preferably all have small degree (see [9, Secton 2.3.5]). Another strategy for fast reducton s to select f(x) so that t has a lowweght multple g(x) of degree slghtly greater than n. Multplcaton s then performed modulo g(x), followed by a reducton by f(x) whenever a representaton n canoncal form s desred. Ths strategy of usng a redundant representaton has been pursued by several authors; e.g., see [13, 6, 16]. For the case of weght-n polynomals, we have f(x) = F n,m (x) where (1) and we can take F n,m (x) = x n + x + + x m+1 + x m x + 1 = xn x x m g(x) = (x + 1)f(x) = x n+1 + x m+1 + x m + 1. The weght of g(x) s 4, and ts mddle terms are consecutve. If m s small, then the mddle terms also have small degree. Reducton usng g(x) nstead of F n,m (x) can be as effcent as f the reducton polynomal were a trnomal or a pentanomal. We llustrate the reducton operaton wth an example. The polynomal F 223,10 (x) s rreducble over F 2 and therefore can be used as the reducton

3 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 3 polynomal for F We have g(x) = x x 11 + x Let c(x) = 446 =0 c x be the product of two polynomals each of degree less than 224. On a 32-bt machne, c(x) may be stored n an array (C[13], C[12],..., C[0]) of 32-bt words, where the rghtmost bt of C[0] s c 0, the second leftmost bt of C[13] s c 446, and the leftmost bt of C[13] s unused (always set to 0). The hgh-order bts of c(x) can be reduced modulo g(x) one word at a tme startng wth C[13]. The pseudocode for the reducton operaton s short and smple: For from 13 downto 7 to: T C[]. C[ 7] C[ 7] T (T 10) (T 11). C[ 6] C[ 6] (T 22) (T 21). The result s (C[6], C[5],..., C[0]). Here, denotes btwse exclusve-or, U j s the rght shft of U by j postons, and U j s the left shft of U by j postons. 3. Non-exstence results Let K be a feld, and let F (x) K[x] be a polynomal of degree n wth leadng coeffcent a. The dscrmnant of F (x) s Dsc(F ) = a 2n 2 <j(x x j ) 2, where x 0, x 1,..., x are the roots of F (x) n some extenson of K. We have Dsc(F ) K. The followng result, whch s sometmes called the Stckelberger-Swan theorem, s our man tool for determnng reducblty of a polynomal n F 2 [x]. Theorem 2. [14, 15] Suppose that the degree-n polynomal f(x) F 2 [x] s the product of r parwse dstnct rreducble polynomals over F 2. Then r n (mod 2) f and only f Dsc(F ) 1 (mod 8) where F (x) Z[x] s any monc lft of f(x) to the ntegers. If n s odd and Dsc(F ) 1 (mod 8), then Theorem 2 asserts that f(x) has an even number of rreducble factors and therefore s reducble over F 2. Thus one can fnd necessary condtons for the rreducblty of f(x) by computng Dsc(F ) modulo 8. Let f(x), g(x) K[x]. Let f(x) = a s 1 =0 (x x ) and g(x) = b t 1 j=0 (x y j ), where a, b K and x 0, x 1,..., x s 1, y 0, y 1,..., y t 1 are n some extenson of K. The resultant of f(x) and g(x) s t 1 s 1 (2) Res(f, g) = ( 1) st b s f(y j ) = a t g(x ). j=0 =0 We wll use Lemma 3 to compute the dscrmnant of F.

4 4 OMRAN AHMADI AND ALFRED MENEZES Lemma 3. [7] Let K be a feld, and let F (x) K[x] have degree n. Suppose also that F s monc and F (0) = 1. Then Dsc(F ) = ( 1) n()/2 Res(F, nf xf ), where F denotes the dervatve of F wth respect to x. Let f(x) = x n + a 1 x + + a n K[x], and let x 0, x 1,..., x be the roots of f(x) n some extenson of K. Then t s well known that the coeffcents a k are the elementary symmetrc polynomals of x : a k = ( 1) k x 1 x 2 x k 0 1 < 2 < < k <n for 1 k n. Snce each a k K, t follows that S(x 0, x 1,..., x ) K for any symmetrc polynomal S K[X 0, X 1,..., X ]. Now for any ntegers k, p, q, let (3) s k = x k and s p,q = Then s 0 = n and =0 (4) s p,q = s p s q s p+q.,j=0 j x p xq j. Note also that f f(0) 0, then the power sum s p of f(x) s equal to the pth power sum of ts recprocal, x n f( ). Newton s dentty relates the coeffcents a k and power sums s k. Theorem 4. [10, Theorem 1.75] Let f(x) and x 0, x 1,..., x be as above. Then for 1 k n we have (5) s k + s k 1 a 1 + s k 2 a s 1 a k 1 + ka k = 0. A polynomal f(x) F 2 [x] havng the property that (x+1)f(x) has weght 4 s sad to be of tetranomal type. Note that polynomals of degree n and weght n are of tetranomal type. Hales and Newhart [7] obtaned a Swanlke theorem for a certan subset of polynomals of tetranomal type 1. Our man result s an analogue of Swan s theorem for all weght-n polynomals. Theorem 5. Let n > m > 0 and assume that n s odd. Then F n,m (x) = (x n+1 + 1)/(x + 1) + x m has an odd number of rreducble factors over F 2 f and only f one of the followng condtons hold: () n 1 (mod 8) and ether (a) m {2, n 2}; or (b) m 0, 1 (mod 4) and m {1, n 1, 2, n+1 2 }. () n 3 (mod 8) and m {2, n 2}. () n 5 (mod 8) and ether (a) m {1, }; or (b) m 2, 3 (mod 4) and m {2, n 2, } f n > 5. 2, n After completng ths paper, we were nformed that Hales and Newhart [8] have obtaned a Swan-lke theorem for all polynomals of tetranomal type. Theorem 2 of ther paper mples our Theorem 5.

5 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 5 (v) n 7 (mod 8) and m {2, n 2}. Proof. Snce F n,m (0) 0, we have gcd(f n,m, F n,m) = 1 and hence F n,m has no repeated factors. Let g(x) = (x + 1)F n,m (x). Then g(x) has degree n + 1 and G(x) = x n+1 + x m+1 + x m + 1 s a monc lft of g(x) to Z[x]. Suppose now that F n,m (x) s the product of r parwse dstnct rreducble polynomals over F 2. Then g(x) s the product of r + 1 parwse dstnct rreducble polynomals over F 2. Hence, by Theorem 2, n+1 r+1 (mod 2) or, equvalently, n r (mod 2), f and only f Dsc(G) 1 (mod 8). Thus the theorem can be proved by computng Dsc(G). Frst we apply Lemma 3 to G(x). We see that (n + 1)G(x) xg (x) = (n m)x m+1 + (n m + 1)x m + (n + 1). Now settng u = n m, v = n m + 1 and w = n + 1, we have (6) Dsc(G) = ( 1) n(n+1)/2 Res(G, ux m+1 + vx m + w). Let x 0, x 1,..., x n be the roots of G(x) n some extenson of the ratonal numbers. Usng (6) and (2) we have (7) Dsc(G) = ( 1) n(n+1)/2 n =0 (ux m+1 + vx m + w). Let D = ( 1) (n+1)n/2 Dsc(G). Upon expandng the rght hand sde of (7) and usng the fact that n =0 x = 1, we have D = u n+1 + v n+1 + u n v + u 2 v <j + u w 2 <j n =0 x x j + u n w ( + uv n n =0 ( n j ) m+1 + v w 2 <j x + u v 2 =0 <j n ) m+1 + v n w ( =0 ( j ) m j ) m (8) + u vw j x m 1 j + uv w x x m j + S(x 0, x 1,..., x n ), j where S(x 0, x 1,..., x n ) Z[x 0, x 1,..., x n ]. Snce Dsc(G) s a symmetrc polynomal n x 0, x 1,..., x n and all the terms gven explctly n the rght hand sde of equaton (8) are symmetrc polynomals, S(x 0, x 1,..., x n ) s also a symmetrc polynomal n x 0, x 1,..., x n. The coeffcents of the monomals of S have one of the followng forms: (a) u v n+1 wth 3 n 2; (b) u v n w wth 2 n 2; (c) u v j w 2 wth 1 and j 1; or (d) u v j w k wth k 3. Snce n s odd and u, v are consecutve ntegers, we have w uv 0 (mod 2) and so the coeffcents of all monomals n S(x 0, x 1,..., x n ) are dvsble by 8. Therefore S(x 0, x 1,..., x n ) s an nteger dvsble by 8. Also for any nteger p we have 2 <j xp xp j = j xp xp j =

6 6 OMRAN AHMADI AND ALFRED MENEZES s p,p. Hence D u n+1 + v n+1 + u n vs 1 + uv n s (u v 2 s 1, 1 + u 2 v s 1,1 ) + u n ws m 1 + v n ws m (u w 2 s m 1, m 1 + v w 2 s m, m ) + u vws 1, m 1 + uv ws 1, m (mod 8). Applyng Newton s dentty (5) to the polynomal G(x) and ts recprocal, x n+1 G( ), we can compute all the unknown terms n the above equaton and thus evaluate D mod 8 for all permssble values of m and n. For example, suppose that n 7 (mod 8). Then w 0 (mod 8) and D u n+1 + v n+1 + u n vs 1 + uv n s u v 2 s 1, 1 We consder three cases u2 v s 1,1 (mod 8). (a) If m {1, 2, n 2, }, then (5) mples that s 1 = s 2 = s 1 = s 2 = 0. Snce s 1,1 = s 2 1 s 2, we have s 1,1 = 0 and smlarly s 1, 1 = 0. Hence D u n+1 + v n+1 (mod 8). Now snce n + 1 s even and one of u, v s even and the other s odd, we have D 1 (mod 8). (b) If m = n 1, then s 1 = s 2 = 1 and s 1 = s 2 = 0, so s 1,1 = s 2 1 s 2 = 2 and s 1, 1 = s 2 1 s 2 = 0. Hence D u n+1 + v n+1 uv n + u 2 v (mod 8). Snce m = n 1, we have u = 1, v = 2 and D u n+1 1 (mod 8). Smlarly we have D 1 (mod 8) f m = 1. (c) If m = n 2, then s 1 = s 1 = s 2 = 0 and s 2 = 2 whence s 1,1 = 2, s 1, 1 = 0, and D u n+1 + v n+1 + u 2 v (mod 8). In ths case snce u = 2, v s odd, and n 1 s even, we have D 5 (mod 8). Smlarly we have D 5 (mod 8) f m = 2. Part (v) of the theorem now follows snce Dsc(G) = D when n 7 (mod 8). The cases n 1, 3, 5 (mod 8) are more tedous but can be handled n a smlar way. Corollary 6. Let n > m > 0 and assume that n s odd. F n,m (x) = (x n+1 + 1)/(x + 1) + x m s rreducble over F 2. Suppose that () If n 1 (mod 8) then ether m {2, n 2} or m 0, 1 (mod 4). Moreover, m {1, n 1, 2, n+1 2 }. () If n 3 (mod 8) then m {2, n 2}. () If n 5 (mod 8) then ether m {1, n 1} or m 2, 3 (mod 4). Moreover, f n > 5 then m {2, n 2, 2, n+1 2 }. (v) If n 7 (mod 8) then m {2, n 2}.

7 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 7 4. Exstence Corollary 6 states that f n 3 (mod 8) then F n,m (x) can only be rreducble f m = 2 or m = n 2. A computer search shows that the only ntegers n [3, ] congruent to 3 (mod 8) for whch F n,2 (x) s rreducble are n {3, 11, 35, 107, 195, 483, 1019, 2643}. One would expect there to be more rreducbles F n,m (x) for n 7 (mod 8) than for n 1, 5 (mod 8) snce Corollary 6 rules out only two values of m n the former case, and about half of all possble m n the latter case. Ths s reflected n Table 1 whch lsts all rreducble polynomals F n,m for n [5, 340] and n 1, 5, 7 (mod 8). Irreducbles F n,m (x) are more abundant than expected n the case n 7 (mod 8). A computer search shows that the only n [7, 5000] congruent to 7 (mod 8) for whch no rreducble polynomal F n,m (x) exsts are n {575, 823, 1543, 2063, 2103, 2335, 3439, 3607, 3847, 3895, 4167, 4375, 4567, 4911}. Blake, Gao and Lambert [4] observed expermentally that the number of rreducble trnomals of degree n s approxmately 3n. Smlarly, we have notced that the number of rreducble polynomals F n,m of degree n s approxmately 2n. Table 2 lsts the total number of such polynomals for n belongng to consecutve ntervals of length 200. There are approxmately 400 rreducble polynomals n each nterval, gvng an average of approxmately 2 rreducble weght-n polynomals for each degree n. An explanaton for ths phenomenon would be of nterest. 5. A famly of reducble polynomals over F 2 Expermental evdence was provded n [2] that f n ±3 (mod 8) and f(x) = x n + x m 1 + x m 2 + x m s an rreducble pentanomal over F 2, where m 1 > m 2 > m 3 > 0 and m 1, m 2, m 3 are odd, then m 1 n/3. (Such polynomals have the property that the correspondng polynomal bass has exactly one element of trace one.) Motvated by ths observaton Bluher [5] proved the followng. Theorem 7. [5] Let n ±3 (mod 8). Let I = { : even, 2n/3 < < n} and J = {j : j 0 (mod 4), 0 < j < n} \ I. Then the polynomal f(x) = x n + I a x n + j J a j x n j + 1 F 2 [x] s reducble over F 2. Bluher s proof nvolves computng Dsc(F ) mod 8 usng propertes of determnants. Here we use Newton s dentty to gve a smpler proof smlar to the one for Theorem 5.

8 8 OMRAN AHMADI AND ALFRED MENEZES n m n m n m Table 1. Irreducble F n,m (x) = (x n+1 + 1)/(x + 1) + x m wth m n/2, for 5 n 340 and n 1, 5, 7 (mod 8). The three tables lst n that are congruent to 5, 7, 1 (mod 8).

9 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 9 n Total Cumulatve Table 2. The total number of rreducble polynomals F n,m (x) = (x n+1 + 1)/(x + 1) + x m. The ranges for n are ndcated n the frst column. The second, thrd, fourth and ffth columns gve the total number for n 1, 3, 5, 7 (mod 8), respectvely.

10 10 OMRAN AHMADI AND ALFRED MENEZES Proof. Let F (x) Z[x] be any monc lft of f(x) wth F (0) = 1, and let x 0, x 1,..., x be the roots of F (x) n some extenson of the ratonal numbers. Then nf xf = a x n + ja j x n j + n. I j J Settng D = ( 1) n()/2 Dsc(F ) and usng (2) and Lemma 3 we obtan (9) D = k=0 I a x n k + j J Expandng the rght hand sde of (9) yelds D = n n + n I + n n 2 1, 2 I 1 < 2 + n n 2 I a x n k k=0 k 1,k 2 =0 k 1 k 2 k 1,k 2 =0 k 1 <k 2 ja j x n j k + n j J + n. ja j x n j k k=0 1 2 a a x n k 1 x n 2 k 2 2 a 2 x n k 1 x n k 2 + S(x 0, x 1,..., x ), where S(x 0, x 1,..., x ) Z[x 0, x 1,..., x ] s a symmetrc polynomal. It can easly be verfed that the coeffcents of each monomal n S s dvsble by 8, and hence S(x 0, x 1,..., x ) s an nteger dvsble by 8. Usng the notaton ntroduced n (3) for power sums of the x s, we have D n n + n I (10) + n n 2 a s n + n j J 1, 2 I 1 < a 1 a 2 s,n 2 + ja j s n j Now, f a k 0 for some 1 k 2n/3, then 4 k. dentty (5) smplfes to 1 2 nn 2 2 a 2 s n,n (mod 8). I s k + s k 1 a 1 + s k 2 a s 1 a k 1 0 (mod 4) Hence Newton s for 1 k 2n/3. It follows that s k 0 (mod 4) for 1 k 2n/3. Smlarly, snce 2 k for all k satsfyng a k 0 and 2n/3 < k n 1, one can conclude that s k 0 (mod 2) for 2n/3 < k n 1. Also, f p, q 1 and p + q 2n/3, then s p s q s p+q 0 (mod 4) and (4) mples that s p,q 0 (mod 4). Thus (10) smplfes to D n n (mod 8), and so Dsc(F ) 5 (mod 8) f n ±3 (mod 8). Snce Dsc(f) Dsc(F ) (mod 2), ths mples that f(x) has nonzero dscrmnant and hence no repeated factors. The reducblty of f(x) s now a consequence of Theorem 2.

11 IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 11 Acknowledgements We would lke to thank Alfred Hales for provdng us wth a copy of [8], and Antona Bluher for her comments on our proof of Theorem 7. References [1] O. Ahmad, The trace spectra of polynomal bases for F 2 n, preprnt, [2] O. Ahmad and A. Menezes, On the number of trace-one elements n polynomal bases for F 2 n, Desgns, Codes and Cryptography, to appear. [3] I. Blake, S. Gao and R. Lambert, Constructve problems for rreducble polynomals over fnte felds, Informaton Theory and Applcatons, Lecture Notes n Computer Scence 793 (1994), [4] I. Blake, S. Gao and R. Lambert, Constructon and dstrbuton problems for rreducble polynomals over fnte felds, Applcatons of Fnte Feld (D. Gollmann, Ed.), Clarendon Press, 1996, [5] A. Bluher, A Swan-lke theorem, Fnte Felds and Ther Applcatons, to appear. [6] W. Geselmann and H. Lukhaub, Redundant representaton of fnte felds Publc Key Cryptography PKC 2001, Lecture Notes n Computer Scence 1992 (2001), [7] A. Hales and D. Newhart, Irreducbles of tetranomal type, n Mathematcal Propertes of Sequences and Other Combnatoral Structures, Kluwer, [8] A. Hales and D. Newhart, Swan s theorem for bnary tetranomals, preprnt, [9] D. Hankerson, A. Menezes and S. Vanstone, Gude to Ellptc Curve Cryptography, Sprnger, [10] R. Ldl and H. Nederreter, Fnte Felds, Cambrdge Unversty Press, [11] G. Serouss, Table of low-weght bnary rreducble polynomals, Hewlett-Packard Techncal Report HPL , [12] I. Shparlnsk, On prmtve polynomals, Problemy Peredach Inform., 23, (1987), (n Russan). [13] J. Slverman, Fast multplcaton n fnte felds GF (2 N ), Cryptographc Hardware and Embedded Systems CHES 99, Lecture Notes n Computer Scence 1717 (1999), [14] L. Stckelberger, Über ene neue Egenschaft der Dskrmnanten algebrascher Zahlkörper, Verh. 1 Internat. Math. Kongresses, Zurch 1897, [15] R. Swan, Factorzaton of polynomals over fnte felds, Pacfc Journal of Mathematcs, 12 (1962), [16] H. Wu, M. Anwar Hasan, I. Blake and S. Gao, Fnte feld multpler usng redundant representaton, IEEE Transactons on Computers, 51 (2002), Dept. of Combnatorcs and Optmzaton, Unversty of Waterloo, Waterloo, Ontaro, Canada N2L 3G1 E-mal address: oahmadd@uwaterloo.ca ajmeneze@uwaterloo.ca