IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT OMRAN AHMADI AND ALFRED MENEZES

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, 2005. Key words and phrases. Fnte Felds, Irreducble Polynomals. 1

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 2. 2. 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 1 + + x + 1 = xn+1 + 1 x + 1 + 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

IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 3 polynomal for F 2 223. We have g(x) = x 224 + x 11 + x 10 + 1. 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 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 2 + + 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+1 2 1 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.

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 OMRAN AHMADI AND ALFRED MENEZES s p,p. Hence D u n+1 + v n+1 + u n vs 1 + uv n s 1 + 1 2 (u v 2 s 1, 1 + u 2 v s 1,1 ) + u n ws m 1 + v n ws m + 1 2 (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 1 + 1 2 u v 2 s 1, 1 We consder three cases. + 1 2 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}.

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, 100000] 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 3 + 1 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 OMRAN AHMADI AND ALFRED MENEZES n m n m n m 5 1 2 7 1 3 9 2 13 1 3 15 1 4 7 17 4 5 21 23 1 6 8 10 25 4 9 29 6 11 31 3 6 7 13 33 37 1 3 6 10 15 39 4 7 11 19 41 5 12 16 45 7 47 1 3 8 16 17 18 19 49 4 53 6 55 9 12 16 19 24 57 8 16 61 22 63 1 5 11 31 65 16 21 28 69 71 9 14 20 73 77 30 34 79 16 22 27 81 2 25 85 1 87 4 28 89 5 17 32 33 93 22 35 95 4 7 28 44 46 97 4 12 36 45 101 6 18 103 7 37 43 105 17 32 109 111 19 34 43 113 16 36 37 41 117 14 19 119 9 13 15 24 121 125 6 31 38 46 127 1 7 15 30 63 129 133 22 31 46 135 28 58 62 64 137 20 33 41 44 141 67 143 40 41 68 145 12 33 57 149 6 43 55 70 151 46 153 52 56 157 3 46 159 5 7 17 37 161 65 73 165 167 6 17 32 43 56 57 72 169 173 43 175 18 177 41 181 67 75 78 183 1 35 56 185 12 53 189 34 62 71 191 23 42 69 76 77 193 21 61 197 11 27 199 3 60 201 32 88 205 207 11 53 83 209 5 8 24 81 96 213 26 67 215 7 18 44 59 78 217 221 35 74 223 10 22 60 106 225 16 37 229 39 63 231 82 94 97 233 36 100 237 59 86 94 239 9 11 15 29 49 51 77 241 48 245 3 87 102 247 10 42 249 253 42 70 255 52 56 82 257 68 72 84 261 34 263 23 51 62 81 128 265 24 129 269 7 95 123 271 36 84 91 99 108 273 68 277 90 130 135 279 37 47 52 56 59 79 80 281 20 21 36 105 100 101 109 130 131 113 133 285 127 287 6 59 69 93 95 104 131 289 100 293 47 131 295 6 58 102 297 28 112 133 301 6 66 303 50 133 305 72 121 184 233 309 311 25 62 66 313 28 285 317 58 90 134 319 72 76 82 105 321 44 277 325 327 19 110 217 308 329 53 276 333 62 86 103 107 335 53 96 117 337 21 316 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).

IRREDUCIBLE POLYNOMIALS OF MAXIMUM WEIGHT 9 n 1 3 5 7 Total Cumulatve 3 200 92 10 92 182 376 376 201 400 96 0 112 220 428 804 401 600 94 2 106 226 428 1232 601 800 100 0 114 212 426 1658 801 1000 114 0 72 204 390 2048 1001 1200 86 2 120 202 410 2458 1201 1400 84 0 86 212 382 2840 1401 1600 114 0 90 206 410 3250 1601 1800 90 0 84 214 388 3638 1801 2000 116 0 94 192 402 4040 2001 2200 90 0 112 204 406 4446 2201 2400 116 0 112 194 422 4868 2401 2600 94 0 96 212 402 5270 2601 2800 96 2 88 200 386 5656 2801 3000 88 0 98 214 400 6056 3001 3200 84 0 112 202 398 6454 3201 3400 110 0 96 194 400 6854 3401 3600 112 0 116 176 404 7258 3601 3800 90 0 136 228 454 7712 3801 4000 108 0 130 204 442 8154 4001 4200 96 0 80 234 410 8564 4201 4400 104 0 102 210 416 8980 4401 4600 86 0 100 198 384 9364 4601 4800 96 0 112 214 422 9786 4801 5000 126 0 100 218 444 10230 5001 5200 114 0 140 156 410 10640 5201 5400 110 0 110 174 394 11034 5401 5600 94 0 94 216 404 11438 5601 5800 92 0 120 178 390 11828 5801 6000 104 0 100 222 426 12254 6001 6200 82 0 98 250 430 12684 6201 6400 104 0 110 178 392 13076 6401 6600 106 0 78 238 422 13498 6601 6800 78 0 120 216 414 13912 6801 7000 114 0 82 214 410 14322 7001 7200 102 0 64 168 334 14656 7201 7400 88 0 132 190 410 15066 7401 7600 92 0 142 188 422 15488 7601 7800 124 0 84 204 412 15900 7801 8000 114 0 102 180 396 16296 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 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 1 1 2 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 < 2 1 2 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 2 + + 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.

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, 2004. [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), 1-23. [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, 19-32. [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), 339-352. [7] A. Hales and D. Newhart, Irreducbles of tetranomal type, n Mathematcal Propertes of Sequences and Other Combnatoral Structures, Kluwer, 2003. [8] A. Hales and D. Newhart, Swan s theorem for bnary tetranomals, preprnt, 2004. [9] D. Hankerson, A. Menezes and S. Vanstone, Gude to Ellptc Curve Cryptography, Sprnger, 2003. [10] R. Ldl and H. Nederreter, Fnte Felds, Cambrdge Unversty Press, 1984. [11] G. Serouss, Table of low-weght bnary rreducble polynomals, Hewlett-Packard Techncal Report HPL-98-135, 1998. [12] I. Shparlnsk, On prmtve polynomals, Problemy Peredach Inform., 23, (1987), 100-103 (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), 122-134. [14] L. Stckelberger, Über ene neue Egenschaft der Dskrmnanten algebrascher Zahlkörper, Verh. 1 Internat. Math. Kongresses, Zurch 1897, 182-193. [15] R. Swan, Factorzaton of polynomals over fnte felds, Pacfc Journal of Mathematcs, 12 (1962), 1099-1106. [16] H. Wu, M. Anwar Hasan, I. Blake and S. Gao, Fnte feld multpler usng redundant representaton, IEEE Transactons on Computers, 51 (2002), 1306-1316. Dept. of Combnatorcs and Optmzaton, Unversty of Waterloo, Waterloo, Ontaro, Canada N2L 3G1 E-mal address: oahmadd@uwaterloo.ca ajmeneze@uwaterloo.ca