/usr/share/doc/libntl-dev/NTL/lzz_pEX.txt is in libntl-dev 9.9.1-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 | /**************************************************************************\
MODULE: zz_pEX
SUMMARY:
The class zz_pEX represents polynomials over zz_pE,
and so can be used, for example, for arithmentic in GF(p^n)[X].
However, except where mathematically necessary (e.g., GCD computations),
zz_pE need not be a field.
\**************************************************************************/
#include <NTL/lzz_pE.h>
#include <NTL/vec_lzz_pE.h>
class zz_pEX {
public:
zz_pEX(); // initial value 0
zz_pEX(const zz_pEX& a); // copy
zz_pEX(const zz_pE& a); // promotion
zz_pEX(const zz_p& a);
zz_pEX(long a);
zz_pEX& operator=(const zz_pEX& a); // assignment
zz_pEX& operator=(const zz_pE& a);
zz_pEX& operator=(const zz_p& a);
zz_pEX& operator=(long a);
~zz_pEX(); // destructor
zz_pEX(INIT_MONO_TYPE, long i, const zz_pE& c);
zz_pEX(INIT_MONO_TYPE, long i, const zz_p& c);
zz_pEX(INIT_MONO_TYPE, long i, long c);
// initilaize to c*X^i; invoke as zz_pEX(INIT_MONO, i, c)
zz_pEX(INIT_MONO_TYPE, long i);
// initilaize to X^i; invoke as zz_pEX(INIT_MONO, i)
// typedefs to aid in generic programming
typedef zz_pE coeff_type;
typedef zz_pEXModulus modulus_type;
// ...
};
/**************************************************************************\
Accessing coefficients
The degree of a polynomial f is obtained as deg(f),
where the zero polynomial, by definition, has degree -1.
A polynomial f is represented as a coefficient vector.
Coefficients may be accesses in one of two ways.
The safe, high-level method is to call the function
coeff(f, i) to get the coefficient of X^i in the polynomial f,
and to call the function SetCoeff(f, i, a) to set the coefficient
of X^i in f to the scalar a.
One can also access the coefficients more directly via a lower level
interface. The coefficient of X^i in f may be accessed using
subscript notation f[i]. In addition, one may write f.SetLength(n)
to set the length of the underlying coefficient vector to n,
and f.SetMaxLength(n) to allocate space for n coefficients,
without changing the coefficient vector itself.
After setting coefficients using this low-level interface,
one must ensure that leading zeros in the coefficient vector
are stripped afterwards by calling the function f.normalize().
NOTE: the coefficient vector of f may also be accessed directly
as f.rep; however, this is not recommended. Also, for a properly
normalized polynomial f, we have f.rep.length() == deg(f)+1,
and deg(f) >= 0 => f.rep[deg(f)] != 0.
\**************************************************************************/
long deg(const zz_pEX& a); // return deg(a); deg(0) == -1.
const zz_pE& coeff(const zz_pEX& a, long i);
// returns the coefficient of X^i, or zero if i not in range
const zz_pE& LeadCoeff(const zz_pEX& a);
// returns leading term of a, or zero if a == 0
const zz_pE& ConstTerm(const zz_pEX& a);
// returns constant term of a, or zero if a == 0
void SetCoeff(zz_pEX& x, long i, const zz_pE& a);
void SetCoeff(zz_pEX& x, long i, const zz_p& a);
void SetCoeff(zz_pEX& x, long i, long a);
// makes coefficient of X^i equal to a; error is raised if i < 0
void SetCoeff(zz_pEX& x, long i);
// makes coefficient of X^i equal to 1; error is raised if i < 0
void SetX(zz_pEX& x); // x is set to the monomial X
long IsX(const zz_pEX& a); // test if x = X
zz_pE& zz_pEX::operator[](long i);
const zz_pE& zz_pEX::operator[](long i) const;
// indexing operators: f[i] is the coefficient of X^i ---
// i should satsify i >= 0 and i <= deg(f).
// No range checking (unless NTL_RANGE_CHECK is defined).
void zz_pEX::SetLength(long n);
// f.SetLength(n) sets the length of the inderlying coefficient
// vector to n --- after this call, indexing f[i] for i = 0..n-1
// is valid.
void zz_pEX::normalize();
// f.normalize() strips leading zeros from coefficient vector of f
void zz_pEX::SetMaxLength(long n);
// f.SetMaxLength(n) pre-allocate spaces for n coefficients. The
// polynomial that f represents is unchanged.
/**************************************************************************\
Comparison
\**************************************************************************/
long operator==(const zz_pEX& a, const zz_pEX& b);
long operator!=(const zz_pEX& a, const zz_pEX& b);
long IsZero(const zz_pEX& a); // test for 0
long IsOne(const zz_pEX& a); // test for 1
// PROMOTIONS: ==, != promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
/**************************************************************************\
Addition
\**************************************************************************/
// operator notation:
zz_pEX operator+(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator-(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator-(const zz_pEX& a);
zz_pEX& operator+=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator+=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator+=(zz_pEX& x, const zz_p& a);
zz_pEX& operator+=(zz_pEX& x, long a);
zz_pEX& operator++(zz_pEX& x); // prefix
void operator++(zz_pEX& x, int); // postfix
zz_pEX& operator-=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator-=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator-=(zz_pEX& x, const zz_p& a);
zz_pEX& operator-=(zz_pEX& x, long a);
zz_pEX& operator--(zz_pEX& x); // prefix
void operator--(zz_pEX& x, int); // postfix
// procedural versions:
void add(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a + b
void sub(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a - b
void negate(zz_pEX& x, const zz_pEX& a); // x = - a
// PROMOTIONS: +, -, add, sub promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
/**************************************************************************\
Multiplication
\**************************************************************************/
// operator notation:
zz_pEX operator*(const zz_pEX& a, const zz_pEX& b);
zz_pEX& operator*=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator*=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator*=(zz_pEX& x, const zz_p& a);
zz_pEX& operator*=(zz_pEX& x, long a);
// procedural versions:
void mul(zz_pEX& x, const zz_pEX& a, const zz_pEX& b); // x = a * b
void sqr(zz_pEX& x, const zz_pEX& a); // x = a^2
zz_pEX sqr(const zz_pEX& a);
// PROMOTIONS: *, mul promote {long,zz_p,zz_pE} to zz_pEX on (a, b).
void power(zz_pEX& x, const zz_pEX& a, long e); // x = a^e (e >= 0)
zz_pEX power(const zz_pEX& a, long e);
/**************************************************************************\
Shift Operations
LeftShift by n means multiplication by X^n
RightShift by n means division by X^n
A negative shift amount reverses the direction of the shift.
\**************************************************************************/
// operator notation:
zz_pEX operator<<(const zz_pEX& a, long n);
zz_pEX operator>>(const zz_pEX& a, long n);
zz_pEX& operator<<=(zz_pEX& x, long n);
zz_pEX& operator>>=(zz_pEX& x, long n);
// procedural versions:
void LeftShift(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX LeftShift(const zz_pEX& a, long n);
void RightShift(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX RightShift(const zz_pEX& a, long n);
/**************************************************************************\
Division
\**************************************************************************/
// operator notation:
zz_pEX operator/(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator/(const zz_pEX& a, const zz_pE& b);
zz_pEX operator/(const zz_pEX& a, const zz_p& b);
zz_pEX operator/(const zz_pEX& a, long b);
zz_pEX operator%(const zz_pEX& a, const zz_pEX& b);
zz_pEX& operator/=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator/=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator/=(zz_pEX& x, const zz_p& a);
zz_pEX& operator/=(zz_pEX& x, long a);
zz_pEX& operator%=(zz_pEX& x, const zz_pEX& a);
// procedural versions:
void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEX& b);
// q = a/b, r = a%b
void div(zz_pEX& q, const zz_pEX& a, const zz_pEX& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_pE& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_p& b);
void div(zz_pEX& q, const zz_pEX& a, long b);
// q = a/b
void rem(zz_pEX& r, const zz_pEX& a, const zz_pEX& b);
// r = a%b
long divide(zz_pEX& q, const zz_pEX& a, const zz_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
long divide(const zz_pEX& a, const zz_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0
/**************************************************************************\
GCD's
These routines are intended for use when zz_pE is a field.
\**************************************************************************/
void GCD(zz_pEX& x, const zz_pEX& a, const zz_pEX& b);
zz_pEX GCD(const zz_pEX& a, const zz_pEX& b);
// x = GCD(a, b), x is always monic (or zero if a==b==0).
void XGCD(zz_pEX& d, zz_pEX& s, zz_pEX& t, const zz_pEX& a, const zz_pEX& b);
// d = gcd(a,b), a s + b t = d
/**************************************************************************\
Input/Output
I/O format:
[a_0 a_1 ... a_n],
represents the polynomial a_0 + a_1*X + ... + a_n*X^n.
On output, all coefficients will be polynomials of degree < zz_pE::degree() and
a_n not zero (the zero polynomial is [ ]). On input, the coefficients
are arbitrary polynomials which are reduced modulo zz_pE::modulus(),
and leading zeros stripped.
\**************************************************************************/
istream& operator>>(istream& s, zz_pEX& x);
ostream& operator<<(ostream& s, const zz_pEX& a);
/**************************************************************************\
Some utility routines
\**************************************************************************/
void diff(zz_pEX& x, const zz_pEX& a); // x = derivative of a
zz_pEX diff(const zz_pEX& a);
void MakeMonic(zz_pEX& x);
// if x != 0 makes x into its monic associate; LeadCoeff(x) must be
// invertible in this case
void reverse(zz_pEX& x, const zz_pEX& a, long hi);
zz_pEX reverse(const zz_pEX& a, long hi);
void reverse(zz_pEX& x, const zz_pEX& a);
zz_pEX reverse(const zz_pEX& a);
// x = reverse of a[0]..a[hi] (hi >= -1);
// hi defaults to deg(a) in second version
void VectorCopy(vec_zz_pE& x, const zz_pEX& a, long n);
vec_zz_pE VectorCopy(const zz_pEX& a, long n);
// x = copy of coefficient vector of a of length exactly n.
// input is truncated or padded with zeroes as appropriate.
/**************************************************************************\
Random Polynomials
\**************************************************************************/
void random(zz_pEX& x, long n);
zz_pEX random_zz_pEX(long n);
// x = random polynomial of degree < n
/**************************************************************************\
Polynomial Evaluation and related problems
\**************************************************************************/
void BuildFromRoots(zz_pEX& x, const vec_zz_pE& a);
zz_pEX BuildFromRoots(const vec_zz_pE& a);
// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()
void eval(zz_pE& b, const zz_pEX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a)
void eval(zz_pE& b, const zz_pX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a); uses ModComp algorithm for zz_pX
void eval(vec_zz_pE& b, const zz_pEX& f, const vec_zz_pE& a);
vec_zz_pE eval(const zz_pEX& f, const vec_zz_pE& a);
// b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()
void interpolate(zz_pEX& f, const vec_zz_pE& a, const vec_zz_pE& b);
zz_pEX interpolate(const vec_zz_pE& a, const vec_zz_pE& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i].
/**************************************************************************\
Arithmetic mod X^n
Required: n >= 0; otherwise, an error is raised.
\**************************************************************************/
void trunc(zz_pEX& x, const zz_pEX& a, long n); // x = a % X^n
zz_pEX trunc(const zz_pEX& a, long n);
void MulTrunc(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, long n);
zz_pEX MulTrunc(const zz_pEX& a, const zz_pEX& b, long n);
// x = a * b % X^n
void SqrTrunc(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX SqrTrunc(const zz_pEX& a, long n);
// x = a^2 % X^n
void InvTrunc(zz_pEX& x, const zz_pEX& a, long n);
zz_pEX InvTrunc(zz_pEX& x, const zz_pEX& a, long n);
// computes x = a^{-1} % X^m. Must have ConstTerm(a) invertible.
/**************************************************************************\
Modular Arithmetic (without pre-conditioning)
Arithmetic mod f.
All inputs and outputs are polynomials of degree less than deg(f), and
deg(f) > 0.
NOTE: if you want to do many computations with a fixed f, use the
zz_pEXModulus data structure and associated routines below for better
performance.
\**************************************************************************/
void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b, const zz_pEX& f);
zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEX& f);
// x = (a * b) % f
void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX SqrMod(const zz_pEX& a, const zz_pEX& f);
// x = a^2 % f
void MulByXMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX MulByXMod(const zz_pEX& a, const zz_pEX& f);
// x = (a * X) mod f
void InvMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
zz_pEX InvMod(const zz_pEX& a, const zz_pEX& f);
// x = a^{-1} % f, error is a is not invertible
long InvModStatus(zz_pEX& x, const zz_pEX& a, const zz_pEX& f);
// if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise,
// returns 1 and sets x = (a, f)
/**************************************************************************\
Modular Arithmetic with Pre-Conditioning
If you need to do a lot of arithmetic modulo a fixed f, build
zz_pEXModulus F for f. This pre-computes information about f that
speeds up subsequent computations.
As an example, the following routine the product modulo f of a vector
of polynomials.
#include <NTL/lzz_pEX.h>
void product(zz_pEX& x, const vec_zz_pEX& v, const zz_pEX& f)
{
zz_pEXModulus F(f);
zz_pEX res;
res = 1;
long i;
for (i = 0; i < v.length(); i++)
MulMod(res, res, v[i], F);
x = res;
}
NOTE: A zz_pEX may be used wherever a zz_pEXModulus is required,
and a zz_pEXModulus may be used wherever a zz_pEX is required.
\**************************************************************************/
class zz_pEXModulus {
public:
zz_pEXModulus(); // initially in an unusable state
zz_pEXModulus(const zz_pEX& f); // initialize with f, deg(f) > 0
zz_pEXModulus(const zz_pEXModulus&); // copy
zz_pEXModulus& operator=(const zz_pEXModulus&); // assignment
~zz_pEXModulus(); // destructor
operator const zz_pEX& () const; // implicit read-only access to f
const zz_pEX& val() const; // explicit read-only access to f
};
void build(zz_pEXModulus& F, const zz_pEX& f);
// pre-computes information about f and stores it in F. Must have
// deg(f) > 0. Note that the declaration zz_pEXModulus F(f) is
// equivalent to zz_pEXModulus F; build(F, f).
// In the following, f refers to the polynomial f supplied to the
// build routine, and n = deg(f).
long deg(const zz_pEXModulus& F); // return n=deg(f)
void MulMod(zz_pEX& x, const zz_pEX& a, const zz_pEX& b,
const zz_pEXModulus& F);
zz_pEX MulMod(const zz_pEX& a, const zz_pEX& b, const zz_pEXModulus& F);
// x = (a * b) % f; deg(a), deg(b) < n
void SqrMod(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX SqrMod(const zz_pEX& a, const zz_pEXModulus& F);
// x = a^2 % f; deg(a) < n
void PowerMod(zz_pEX& x, const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F);
zz_pEX PowerMod(const zz_pEX& a, const ZZ& e, const zz_pEXModulus& F);
void PowerMod(zz_pEX& x, const zz_pEX& a, long e, const zz_pEXModulus& F);
zz_pEX PowerMod(const zz_pEX& a, long e, const zz_pEXModulus& F);
// x = a^e % f; e >= 0, deg(a) < n. Uses a sliding window algorithm.
// (e may be negative)
void PowerXMod(zz_pEX& x, const ZZ& e, const zz_pEXModulus& F);
zz_pEX PowerXMod(const ZZ& e, const zz_pEXModulus& F);
void PowerXMod(zz_pEX& x, long e, const zz_pEXModulus& F);
zz_pEX PowerXMod(long e, const zz_pEXModulus& F);
// x = X^e % f (e may be negative)
void rem(zz_pEX& x, const zz_pEX& a, const zz_pEXModulus& F);
// x = a % f
void DivRem(zz_pEX& q, zz_pEX& r, const zz_pEX& a, const zz_pEXModulus& F);
// q = a/f, r = a%f
void div(zz_pEX& q, const zz_pEX& a, const zz_pEXModulus& F);
// q = a/f
// operator notation:
zz_pEX operator/(const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX operator%(const zz_pEX& a, const zz_pEXModulus& F);
zz_pEX& operator/=(zz_pEX& x, const zz_pEXModulus& F);
zz_pEX& operator%=(zz_pEX& x, const zz_pEXModulus& F);
/**************************************************************************\
vectors of zz_pEX's
\**************************************************************************/
typedef Vec<zz_pEX> vec_zz_pEX; // backward compatibility
/**************************************************************************\
Modular Composition
Modular composition is the problem of computing g(h) mod f for
polynomials f, g, and h.
The algorithm employed is that of Brent & Kung (Fast algorithms for
manipulating formal power series, JACM 25:581-595, 1978), which uses
O(n^{1/2}) modular polynomial multiplications, and O(n^2) scalar
operations.
\**************************************************************************/
void CompMod(zz_pEX& x, const zz_pEX& g, const zz_pEX& h,
const zz_pEXModulus& F);
zz_pEX CompMod(const zz_pEX& g, const zz_pEX& h,
const zz_pEXModulus& F);
// x = g(h) mod f; deg(h) < n
void Comp2Mod(zz_pEX& x1, zz_pEX& x2, const zz_pEX& g1, const zz_pEX& g2,
const zz_pEX& h, const zz_pEXModulus& F);
// xi = gi(h) mod f (i=1,2); deg(h) < n.
void Comp3Mod(zz_pEX& x1, zz_pEX& x2, zz_pEX& x3,
const zz_pEX& g1, const zz_pEX& g2, const zz_pEX& g3,
const zz_pEX& h, const zz_pEXModulus& F);
// xi = gi(h) mod f (i=1..3); deg(h) < n.
/**************************************************************************\
Composition with Pre-Conditioning
If a single h is going to be used with many g's then you should build
a zz_pEXArgument for h, and then use the compose routine below. The
routine build computes and stores h, h^2, ..., h^m mod f. After this
pre-computation, composing a polynomial of degree roughly n with h
takes n/m multiplies mod f, plus n^2 scalar multiplies. Thus,
increasing m increases the space requirement and the pre-computation
time, but reduces the composition time.
\**************************************************************************/
struct zz_pEXArgument {
vec_zz_pEX H;
};
void build(zz_pEXArgument& H, const zz_pEX& h, const zz_pEXModulus& F, long m);
// Pre-Computes information about h. m > 0, deg(h) < n.
void CompMod(zz_pEX& x, const zz_pEX& g, const zz_pEXArgument& H,
const zz_pEXModulus& F);
zz_pEX CompMod(const zz_pEX& g, const zz_pEXArgument& H,
const zz_pEXModulus& F);
extern long zz_pEXArgBound;
// Initially 0. If this is set to a value greater than zero, then
// composition routines will allocate a table of no than about
// zz_pEXArgBound KB. Setting this value affects all compose routines
// and the power projection and minimal polynomial routines below,
// and indirectly affects many routines in zz_pEXFactoring.
/**************************************************************************\
power projection routines
\**************************************************************************/
void project(zz_pE& x, const zz_pEVector& a, const zz_pEX& b);
zz_pE project(const zz_pEVector& a, const zz_pEX& b);
// x = inner product of a with coefficient vector of b
void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k,
const zz_pEX& h, const zz_pEXModulus& F);
vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k,
const zz_pEX& h, const zz_pEXModulus& F);
// Computes the vector
// project(a, 1), project(a, h), ..., project(a, h^{k-1} % f).
// This operation is the "transpose" of the modular composition operation.
void ProjectPowers(vec_zz_pE& x, const vec_zz_pE& a, long k,
const zz_pEXArgument& H, const zz_pEXModulus& F);
vec_zz_pE ProjectPowers(const vec_zz_pE& a, long k,
const zz_pEXArgument& H, const zz_pEXModulus& F);
// same as above, but uses a pre-computed zz_pEXArgument
class zz_pEXTransMultiplier { /* ... */ };
void build(zz_pEXTransMultiplier& B, const zz_pEX& b, const zz_pEXModulus& F);
void UpdateMap(vec_zz_pE& x, const vec_zz_pE& a,
const zz_pEXMultiplier& B, const zz_pEXModulus& F);
vec_zz_pE UpdateMap(const vec_zz_pE& a,
const zz_pEXMultiplier& B, const zz_pEXModulus& F);
// Computes the vector
// project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f)
// Required: a.length() <= deg(F), deg(b) < deg(F).
// This is "transposed" MulMod by B.
// Input may have "high order" zeroes stripped.
// Output always has high order zeroes stripped.
/**************************************************************************\
Minimum Polynomials
These routines should be used only when zz_pE is a field.
All of these routines implement the algorithm from [Shoup, J. Symbolic
Comp. 17:371-391, 1994] and [Shoup, J. Symbolic Comp. 20:363-397,
1995], based on transposed modular composition and the
Berlekamp/Massey algorithm.
\**************************************************************************/
void MinPolySeq(zz_pEX& h, const vec_zz_pE& a, long m);
zz_pEX MinPolySeq(const vec_zz_pE& a, long m);
// computes the minimum polynomial of a linealy generated sequence; m
// is a bound on the degree of the polynomial; required: a.length() >=
// 2*m
void ProbMinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void ProbMinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX ProbMinPolyMod(const zz_pEX& g, const zz_pEXModulus& F);
// computes the monic minimal polynomial if (g mod f). m = a bound on
// the degree of the minimal polynomial; in the second version, this
// argument defaults to n. The algorithm is probabilistic, always
// returns a divisor of the minimal polynomial, and returns a proper
// divisor with probability at most m/2^{zz_pE::degree()}.
void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void MinPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX MinPolyMod(const zz_pEX& g, const zz_pEXModulus& F);
// same as above, but guarantees that result is correct
void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& F, long m);
void IrredPolyMod(zz_pEX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pEX IrredPolyMod(const zz_pEX& g, const zz_pEXModulus& F);
// same as above, but assumes that f is irreducible, or at least that
// the minimal poly of g is itself irreducible. The algorithm is
// deterministic (and is always correct).
/**************************************************************************\
Composition and Minimal Polynomials in towers
These are implementations of algorithms that will be described
and analyzed in a forthcoming paper.
The routines require that p is prime, but zz_pE need not be a field.
\**************************************************************************/
void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEXArgument& h,
const zz_pEXModulus& F);
zz_pEX CompTower(const zz_pX& g, const zz_pEXArgument& h,
const zz_pEXModulus& F);
void CompTower(zz_pEX& x, const zz_pX& g, const zz_pEX& h,
const zz_pEXModulus& F);
zz_pEX CompTower(const zz_pX& g, const zz_pEX& h,
const zz_pEXModulus& F);
// x = g(h) mod f
void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F,
long m);
zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void ProbMinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX ProbMinPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Uses a probabilistic algorithm to compute the minimal
// polynomial of (g mod f) over zz_p.
// The parameter m is a bound on the degree of the minimal polynomial
// (default = deg(f)*zz_pE::degree()).
// In general, the result will be a divisor of the true minimimal
// polynomial. For correct results, use the MinPoly routines below.
void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void MinPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX MinPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Same as above, but result is always correct.
void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F, long m);
zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F, long m);
void IrredPolyTower(zz_pX& h, const zz_pEX& g, const zz_pEXModulus& F);
zz_pX IrredPolyTower(const zz_pEX& g, const zz_pEXModulus& F);
// Same as above, but assumes the minimal polynomial is
// irreducible, and uses a slightly faster, deterministic algorithm.
/**************************************************************************\
Traces, norms, resultants
\**************************************************************************/
void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEXModulus& F);
zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& F);
void TraceMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f);
zz_pE TraceMod(const zz_pEX& a, const zz_pEXModulus& f);
// x = Trace(a mod f); deg(a) < deg(f)
void TraceVec(vec_zz_pE& S, const zz_pEX& f);
vec_zz_pE TraceVec(const zz_pEX& f);
// S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f)
// The above trace routines implement the asymptotically fast trace
// algorithm from [von zur Gathen and Shoup, Computational Complexity,
// 1992].
void NormMod(zz_pE& x, const zz_pEX& a, const zz_pEX& f);
zz_pE NormMod(const zz_pEX& a, const zz_pEX& f);
// x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)
void resultant(zz_pE& x, const zz_pEX& a, const zz_pEX& b);
zz_pE resultant(const zz_pEX& a, const zz_pEX& b);
// x = resultant(a, b)
// NormMod and resultant require that zz_pE is a field.
/**************************************************************************\
Miscellany
\**************************************************************************/
void clear(zz_pEX& x) // x = 0
void set(zz_pEX& x); // x = 1
void zz_pEX::kill();
// f.kill() sets f to 0 and frees all memory held by f. Equivalent to
// f.rep.kill().
zz_pEX::zz_pEX(INIT_SIZE_TYPE, long n);
// zz_pEX(INIT_SIZE, n) initializes to zero, but space is pre-allocated
// for n coefficients
static const zz_pEX& zero();
// zz_pEX::zero() is a read-only reference to 0
void zz_pEX::swap(zz_pEX& x);
void swap(zz_pEX& x, zz_pEX& y);
// swap (via "pointer swapping")
zz_pEX::zz_pEX(long i, const zz_pE& c);
zz_pEX::zz_pEX(long i, const zz_p& c);
zz_pEX::zz_pEX(long i, long c);
// initilaize to c*X^i; provided for backward compatibility
|