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APPENDIX A

Sample Parameters

This appendix presents elliptic curve domain parameters D = (q, FR, S, a, b, P, n, h) that are suitable for cryptographic use; see §4.2 for a review of the notation. In §A.1, an algorithm for testing irreducibility of a polynomial is presented. This algorithm can be used to generate a reduction polynomial for representing elements of the ﬁnite ﬁeld F pm . Also included in §A.1 are tables of irreducible binary polynomials that are recommended by several standards including ANSI X9.62 and ANSI X9.63 as reduction polynomials for representing the elements of binary ﬁelds F2m . The 15 elliptic curves recommended by NIST in the FIPS 186-2 standard for U.S. federal government use are listed in §A.2.

# A.1 Irreducible polynomials

A polynomial f (z) = am zm + · · · + a1z + a0 F p[z] of degree m 1 is irreducible over F p if f (z) cannot be factored as a product of polynomials in F p[z] each of degree less than m. Since f (z) is irreducible if and only if am1 f (z) is irreducible, it sufﬁces to only consider monic polynomials (i.e., polynomials with leading coefﬁcient am = 1).

For any prime p and integer m 1, there exists at least one monic irreducible polynomial of degree m in F p[z]. In fact, the exact number of such polynomials is

Np (m) = 1 µ(d) pm/d ,

m d|m

258 A. Sample Parameters

where the summation index d ranges over all positive divisors of m, and the Mobius¨ function µ is deﬁned as follows:

 1, if d = 1, µ(d) = 0, if d is divisible by the square of a prime, ( 1)l , if d is the product of l distinct primes. It has been shown that − 1 Np (m) 1 ≤ ≈ . 2m pm m

Thus, if polynomials in F p[z] can be efﬁciently tested for irreducibility, then irreducible polynomials of degree m can be efﬁciently found by selecting random monic polynomials of degree m in F p[z] until an irreducible one is found—the expected number of trials is approximately m.

Algorithm A.1 is an efﬁcient test for deciding irreducibility. It is based on the fact

 that a polynomial f (z) of degree m is irreducible over F p if and only if gcd( f (z), z pi − m . z) = 1 for each i, 1 ≤ i ≤ 2 Algorithm A.1 Testing a polynomial for irreducibility INPUT: A prime p and a polynomial f (z) F p[z] of degree m ≥ 1. OUTPUT: Irreducibility of f (z).

1.u(z) z.

2.For i from 1 to m2 do:

2.1u(z) u(z) p mod f (z).

2.2d(z) gcd( f (z), u(z) z).

2.3If d(z) = 1 then return(“reducible”).

3.Return(“irreducible”).

For each m, 2 m 600, Tables A.1 and A.2 list an irreducible trinomial or pentanomial f (z) of degree m over F2. The entries in the column labeled “T ” are the degrees of the nonzero terms of the polynomial excluding the leading term zm and the constant term 1. For example, T = k represents the trinomial zm + zk + 1, and T = (k3 , k2, k1) represents the pentanomial zm + zk3 + zk2 + zk1 + 1. The following criteria from the ANSI X9.62 and ANSI X9.63 standards were used to select the reduction polynomials:

(i)If there exists an irreducible trinomial of degree m over F2, then f (z) is the irreducible trinomial zm + zk + 1 for the smallest possible k.

(ii)If there does not exist an irreducible trinomial of degree m over F2, then f (z) is the irreducible pentanomial zm + zk3 + zk2 + zk1 + 1 for which (a) k3 is the smallest possible; (b) for this particular value of k3, k2 is the smallest possible; and (c) for these particular values of k3 and k2, k1 is the smallest possible.

 A.1. Irreducible polynomials 259 m T m T m T m T m T m T

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

 − 51 6, 3, 1 101 7, 6, 1 151 3 201 14 251 7, 4, 2 1 52 3 102 29 152 6, 3, 2 202 55 252 15 1 53 6, 2, 1 103 9 153 1 203 8, 7, 1 253 46 1 54 9 104 4, 3, 1 154 15 204 27 254 7, 2, 1 2 55 7 105 4 155 62 205 9, 5, 2 255 52 1 56 7, 4, 2 106 15 156 9 206 10, 9, 5 256 10, 5, 2 1 57 4 107 9, 7, 4 157 6, 5, 2 207 43 257 12 4, 3, 1 58 19 108 17 158 8, 6, 5 208 9, 3, 1 258 71 1 59 7, 4, 2 109 5, 4, 2 159 31 209 6 259 10, 6, 2 3 60 1 110 33 160 5, 3, 2 210 7 260 15 2 61 5, 2, 1 111 10 161 18 211 11, 10, 8 261 7, 6, 4 3 62 29 112 5, 4, 3 162 27 212 105 262 9, 8, 4 4, 3, 1 63 1 113 9 163 7, 6, 3 213 6, 5, 2 263 93 5 64 4, 3, 1 114 5, 3, 2 164 10, 8, 7 214 73 264 9, 6, 2 1 65 18 115 8, 7, 5 165 9, 8, 3 215 23 265 42 5, 3, 1 66 3 116 4, 2, 1 166 37 216 7, 3, 1 266 47 3 67 5, 2, 1 117 5, 2, 1 167 6 217 45 267 8, 6, 3 3 68 9 118 33 168 15, 3, 2 218 11 268 25 5, 2, 1 69 6, 5, 2 119 8 169 34 219 8, 4, 1 269 7, 6, 1 3 70 5, 3, 1 120 4, 3, 1 170 11 220 7 270 53 2 71 6 121 18 171 6, 5, 2 221 8, 6, 2 271 58 1 72 10, 9, 3 122 6, 2, 1 172 1 222 5, 4, 2 272 9, 3, 2 5 73 25 123 2 173 8, 5, 2 223 33 273 23 4, 3, 1 74 35 124 19 174 13 224 9, 8, 3 274 67 3 75 6, 3, 1 125 7, 6, 5 175 6 225 32 275 11, 10, 9 4, 3, 1 76 21 126 21 176 11, 3, 2 226 10, 7, 3 276 63 5, 2, 1 77 6, 5, 2 127 1 177 8 227 10, 9, 4 277 12, 6, 3 1 78 6, 5, 3 128 7, 2, 1 178 31 228 113 278 5 2 79 9 129 5 179 4, 2, 1 229 10, 4, 1 279 5 1 80 9, 4, 2 130 3 180 3 230 8, 7, 6 280 9, 5, 2 3 81 4 131 8, 3, 2 181 7, 6, 1 231 26 281 93 7, 3, 2 82 8, 3, 1 132 17 182 81 232 9, 4, 2 282 35 10 83 7, 4, 2 133 9, 8, 2 183 56 233 74 283 12, 7, 5 7 84 5 134 57 184 9, 8, 7 234 31 284 53 2 85 8, 2, 1 135 11 185 24 235 9, 6, 1 285 10, 7, 5 9 86 21 136 5, 3, 2 186 11 236 5 286 69 6, 4, 1 87 13 137 21 187 7, 6, 5 237 7, 4, 1 287 71 6, 5, 1 88 7, 6, 2 138 8, 7, 1 188 6, 5, 2 238 73 288 11, 10, 1 4 89 38 139 8, 5, 3 189 6, 5, 2 239 36 289 21 5, 4, 3 90 27 140 15 190 8, 7, 6 240 8, 5, 3 290 5, 3, 2 3 91 8, 5, 1 141 10, 4, 1 191 9 241 70 291 12, 11, 5 7 92 21 142 21 192 7, 2, 1 242 95 292 37 6, 4, 3 93 2 143 5, 3, 2 193 15 243 8, 5, 1 293 11, 6, 1 5 94 21 144 7, 4, 2 194 87 244 111 294 33 4, 3, 1 95 11 145 52 195 8, 3, 2 245 6, 4, 1 295 48 1 96 10, 9, 6 146 71 196 3 246 11, 2, 1 296 7, 3, 2 5 97 6 147 14 197 9, 4, 2 247 82 297 5 5, 3, 2 98 11 148 27 198 9 248 15, 14, 10 298 11, 8, 4 9 99 6, 3, 1 149 10, 9, 7 199 34 249 35 299 11, 6, 4 4, 3, 2 100 15 150 53 200 5, 3, 2 250 103 300 5

Table A.1. Irreducible binary polynomials of degree m, 2 m 300.

 260 A. Sample Parameters m T m T m T m T m T m T 301 9, 5, 2 351 34 401 152 451 16, 10, 1 501 5, 4, 2 551 135 302 41 352 13, 11, 6 402 171 452 6, 5, 4 502 8, 5, 4 552 19, 16, 9 303 1 353 69 403 9, 8, 5 453 15, 6, 4 503 3 553 39 304 11, 2, 1 354 99 404 65 454 8, 6, 1 504 15, 14, 6 554 10, 8, 7 305 102 355 6, 5, 1 405 13, 8, 2 455 38 505 156 555 10, 9, 4 306 7, 3, 1 356 10, 9, 7 406 141 456 18, 9, 6 506 23 556 153 307 8, 4, 2 357 11, 10, 2 407 71 457 16 507 13, 6, 3 557 7, 6, 5 308 15 358 57 408 5, 3, 2 458 203 508 9 558 73 309 10, 6, 4 359 68 409 87 459 12, 5, 2 509 8, 7, 3 559 34 310 93 360 5, 3, 2 410 10, 4, 3 460 19 510 69 560 11, 9, 6 311 7, 5, 3 361 7, 4, 1 411 12, 10, 3 461 7, 6, 1 511 10 561 71 312 9, 7, 4 362 63 412 147 462 73 512 8, 5, 2 562 11, 4, 2 313 79 363 8, 5, 3 413 10, 7, 6 463 93 513 26 563 14, 7, 3 314 15 364 9 414 13 464 19, 18, 13 514 67 564 163 315 10, 9, 1 365 9, 6, 5 415 102 465 31 515 14, 7, 4 565 11, 6, 1 316 63 366 29 416 9, 5, 2 466 14, 11, 6 516 21 566 153 317 7, 4, 2 367 21 417 107 467 11, 6, 1 517 12, 10, 2 567 28 318 45 368 7, 3, 2 418 199 468 27 518 33 568 15, 7, 6 319 36 369 91 419 15, 5, 4 469 9, 5, 2 519 79 569 77 320 4, 3, 1 370 139 420 7 470 9 520 15, 11, 2 570 67 321 31 371 8, 3, 2 421 5, 4, 2 471 1 521 32 571 10, 5, 2 322 67 372 111 422 149 472 11, 3, 2 522 39 572 12, 8, 1 323 10, 3, 1 373 8, 7, 2 423 25 473 200 523 13, 6, 2 573 10, 6, 4 324 51 374 8, 6, 5 424 9, 7, 2 474 191 524 167 574 13 325 10, 5, 2 375 16 425 12 475 9, 8, 4 525 6, 4, 1 575 146 326 10, 3, 1 376 8, 7, 5 426 63 476 9 526 97 576 13, 4, 3 327 34 377 41 427 11, 6, 5 477 16, 15, 7 527 47 577 25 328 8, 3, 1 378 43 428 105 478 121 528 11, 6, 2 578 23, 22, 16 329 50 379 10, 8, 5 429 10, 8, 7 479 104 529 42 579 12, 9, 7 330 99 380 47 430 14, 6, 1 480 15, 9, 6 530 10, 7, 3 580 237 331 10, 6, 2 381 5, 2, 1 431 120 481 138 531 10, 5, 4 581 13, 7, 6 332 89 382 81 432 13, 4, 3 482 9, 6, 5 532 1 582 85 333 2 383 90 433 33 483 9, 6, 4 533 4, 3, 2 583 130 334 5, 2, 1 384 12, 3, 2 434 12, 11, 5 484 105 534 161 584 14, 13, 3 335 10, 7, 2 385 6 435 12, 9, 5 485 17, 16, 6 535 8, 6, 2 585 88 336 7, 4, 1 386 83 436 165 486 81 536 7, 5, 3 586 7, 5, 2 337 55 387 8, 7, 1 437 6, 2, 1 487 94 537 94 587 11, 6, 1 338 4, 3, 1 388 159 438 65 488 4, 3, 1 538 195 588 35 339 16, 10, 7 389 10, 9, 5 439 49 489 83 539 10, 5, 4 589 10, 4, 3 340 45 390 9 440 4, 3, 1 490 219 540 9 590 93 341 10, 8, 6 391 28 441 7 491 11, 6, 3 541 13, 10, 4 591 9, 6, 4 342 125 392 13, 10, 6 442 7, 5, 2 492 7 542 8, 6, 1 592 13, 6, 3 343 75 393 7 443 10, 6, 1 493 10, 5, 3 543 16 593 86 344 7, 2, 1 394 135 444 81 494 17 544 8, 3, 1 594 19 345 22 395 11, 6, 5 445 7, 6, 4 495 76 545 122 595 9, 2, 1 346 63 396 25 446 105 496 16, 5, 2 546 8, 2, 1 596 273 347 11, 10, 3 397 12, 7, 6 447 73 497 78 547 13, 7, 4 597 14, 12, 9 348 103 398 7, 6, 2 448 11, 6, 4 498 155 548 10, 5, 3 598 7, 6, 1 349 6, 5, 2 399 26 449 134 499 11, 6, 5 549 16, 4, 3 599 30 350 53 400 5, 3, 2 450 47 500 27 550 193 600 9, 5, 2

Table A.2. Irreducible binary polynomials of degree m, 301 m 600.

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