/* mpn_gcdext -- Extended Greatest Common Divisor. Copyright 1996, 1998, 2000, 2001 Free Software Foundation, Inc. This file is part of the GNU MP Library. The GNU MP Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The GNU MP Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MP Library; see the file COPYING.LIB. If not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */ #include "gmp.h" #include "gmp-impl.h" #include "longlong.h" #ifndef GCDEXT_THRESHOLD #define GCDEXT_THRESHOLD 17 #endif #ifndef EXTEND #define EXTEND 1 #endif #if STAT int arr[BITS_PER_MP_LIMB]; #endif /* mpn_gcdext (GP, SP, SSIZE, UP, USIZE, VP, VSIZE) Compute the extended GCD of {UP,USIZE} and {VP,VSIZE} and store the greatest common divisor at GP (unless it is 0), and the first cofactor at SP. Write the size of the cofactor through the pointer SSIZE. Return the size of the value at GP. Note that SP might be a negative number; this is denoted by storing the negative of the size through SSIZE. {UP,USIZE} and {VP,VSIZE} are both clobbered. The space allocation for all four areas needs to be USIZE+1. Preconditions: 1) U >= V. 2) V > 0. */ /* We use Lehmer's algorithm. The idea is to extract the most significant bits of the operands, and compute the continued fraction for them. We then apply the gathered cofactors to the full operands. Idea 1: After we have performed a full division, don't shift operands back, but instead account for the extra factors-of-2 thus introduced. Idea 2: Simple generalization to use divide-and-conquer would give us an algorithm that runs faster than O(n^2). Idea 3: The input numbers need less space as the computation progresses, while the s0 and s1 variables need more space. To save memory, we could make them share space, and have the latter variables grow into the former. Idea 4: We should not do double-limb arithmetic from the start. Instead, do things in single-limb arithmetic until the quotients differ, and then switch to double-limb arithmetic. */ /* Division optimized for small quotients. If the quotient is more than one limb, store 1 in *qh and return 0. */ static mp_limb_t div2 (mp_limb_t *qh, mp_limb_t n1, mp_limb_t n0, mp_limb_t d1, mp_limb_t d0) { if (d1 == 0) { *qh = 1; return 0; } if ((mp_limb_signed_t) n1 < 0) { mp_limb_t q; int cnt; for (cnt = 1; (mp_limb_signed_t) d1 >= 0; cnt++) { d1 = (d1 << 1) | (d0 >> (BITS_PER_MP_LIMB - 1)); d0 = d0 << 1; } q = 0; while (cnt) { q <<= 1; if (n1 > d1 || (n1 == d1 && n0 >= d0)) { sub_ddmmss (n1, n0, n1, n0, d1, d0); q |= 1; } d0 = (d1 << (BITS_PER_MP_LIMB - 1)) | (d0 >> 1); d1 = d1 >> 1; cnt--; } *qh = 0; return q; } else { mp_limb_t q; int cnt; for (cnt = 0; n1 > d1 || (n1 == d1 && n0 >= d0); cnt++) { d1 = (d1 << 1) | (d0 >> (BITS_PER_MP_LIMB - 1)); d0 = d0 << 1; } q = 0; while (cnt) { d0 = (d1 << (BITS_PER_MP_LIMB - 1)) | (d0 >> 1); d1 = d1 >> 1; q <<= 1; if (n1 > d1 || (n1 == d1 && n0 >= d0)) { sub_ddmmss (n1, n0, n1, n0, d1, d0); q |= 1; } cnt--; } *qh = 0; return q; } } mp_size_t #if EXTEND mpn_gcdext (mp_ptr gp, mp_ptr s0p, mp_size_t *s0size, mp_ptr up, mp_size_t size, mp_ptr vp, mp_size_t vsize) #else mpn_gcd (mp_ptr gp, mp_ptr up, mp_size_t size, mp_ptr vp, mp_size_t vsize) #endif { mp_limb_t A, B, C, D; int cnt; mp_ptr tp, wp; #if RECORD mp_limb_t max = 0; #endif #if EXTEND mp_ptr s1p; mp_ptr orig_s0p = s0p; mp_size_t ssize; int sign = 1; #endif int use_double_flag; TMP_DECL (mark); TMP_MARK (mark); use_double_flag = (size >= GCDEXT_THRESHOLD); tp = (mp_ptr) TMP_ALLOC ((size + 1) * BYTES_PER_MP_LIMB); wp = (mp_ptr) TMP_ALLOC ((size + 1) * BYTES_PER_MP_LIMB); #if EXTEND s1p = (mp_ptr) TMP_ALLOC ((size + 1) * BYTES_PER_MP_LIMB); #if ! WANT_GCDEXT_ONE_STEP MPN_ZERO (s0p, size); MPN_ZERO (s1p, size); #endif s0p[0] = 1; s1p[0] = 0; ssize = 1; #endif if (size > vsize) { /* Normalize V (and shift up U the same amount). */ count_leading_zeros (cnt, vp[vsize - 1]); if (cnt != 0) { mp_limb_t cy; mpn_lshift (vp, vp, vsize, cnt); cy = mpn_lshift (up, up, size, cnt); up[size] = cy; size += cy != 0; } mpn_divmod (up + vsize, up, size, vp, vsize); #if EXTEND /* This is really what it boils down to in this case... */ s0p[0] = 0; s1p[0] = 1; sign = -sign; #endif size = vsize; if (cnt != 0) { mpn_rshift (up, up, size, cnt); mpn_rshift (vp, vp, size, cnt); } MP_PTR_SWAP (up, vp); } for (;;) { mp_limb_t asign; /* Figure out exact size of V. */ vsize = size; MPN_NORMALIZE (vp, vsize); if (vsize <= 1) break; if (use_double_flag) { mp_limb_t uh, vh, ul, vl; /* Let UH,UL be the most significant limbs of U, and let VH,VL be the corresponding bits from V. */ uh = up[size - 1]; vh = vp[size - 1]; ul = up[size - 2]; vl = vp[size - 2]; count_leading_zeros (cnt, uh); if (cnt != 0) { uh = (uh << cnt) | (ul >> (BITS_PER_MP_LIMB - cnt)); vh = (vh << cnt) | (vl >> (BITS_PER_MP_LIMB - cnt)); vl <<= cnt; ul <<= cnt; if (size >= 3) { ul |= (up[size - 3] >> (BITS_PER_MP_LIMB - cnt)); vl |= (vp[size - 3] >> (BITS_PER_MP_LIMB - cnt)); } } A = 1; B = 0; C = 0; D = 1; asign = 0; for (;;) { mp_limb_t T; mp_limb_t qh, q1, q2; mp_limb_t nh, nl, dh, dl; mp_limb_t t1, t0; mp_limb_t Th, Tl; sub_ddmmss (dh, dl, vh, vl, 0, C); if ((dl | dh) == 0) break; add_ssaaaa (nh, nl, uh, ul, 0, A); q1 = div2 (&qh, nh, nl, dh, dl); if (qh != 0) break; /* could handle this */ add_ssaaaa (dh, dl, vh, vl, 0, D); if ((dl | dh) == 0) break; sub_ddmmss (nh, nl, uh, ul, 0, B); q2 = div2 (&qh, nh, nl, dh, dl); if (qh != 0) break; /* could handle this */ if (q1 != q2) break; asign = ~asign; T = A + q1 * C; A = C; C = T; T = B + q1 * D; B = D; D = T; umul_ppmm (t1, t0, q1, vl); t1 += q1 * vh; sub_ddmmss (Th, Tl, uh, ul, t1, t0); uh = vh, ul = vl; vh = Th, vl = Tl; add_ssaaaa (dh, dl, vh, vl, 0, C); sub_ddmmss (nh, nl, uh, ul, 0, A); q1 = div2 (&qh, nh, nl, dh, dl); if (qh != 0) break; /* could handle this */ sub_ddmmss (dh, dl, vh, vl, 0, D); if ((dl | dh) == 0) break; add_ssaaaa (nh, nl, uh, ul, 0, B); q2 = div2 (&qh, nh, nl, dh, dl); if (qh != 0) break; /* could handle this */ if (q1 != q2) break; asign = ~asign; T = A + q1 * C; A = C; C = T; T = B + q1 * D; B = D; D = T; umul_ppmm (t1, t0, q1, vl); t1 += q1 * vh; sub_ddmmss (Th, Tl, uh, ul, t1, t0); uh = vh, ul = vl; vh = Th, vl = Tl; } #if EXTEND if (asign) sign = -sign; #endif } else /* Same, but using single-limb calculations. */ { mp_limb_t uh, vh; /* Make UH be the most significant limb of U, and make VH be corresponding bits from V. */ uh = up[size - 1]; vh = vp[size - 1]; count_leading_zeros (cnt, uh); if (cnt != 0) { uh = (uh << cnt) | (up[size - 2] >> (BITS_PER_MP_LIMB - cnt)); vh = (vh << cnt) | (vp[size - 2] >> (BITS_PER_MP_LIMB - cnt)); } A = 1; B = 0; C = 0; D = 1; asign = 0; for (;;) { mp_limb_t q, T; if (vh - C == 0 || vh + D == 0) break; q = (uh + A) / (vh - C); if (q != (uh - B) / (vh + D)) break; asign = ~asign; T = A + q * C; A = C; C = T; T = B + q * D; B = D; D = T; T = uh - q * vh; uh = vh; vh = T; if (vh - D == 0) break; q = (uh - A) / (vh + C); if (q != (uh + B) / (vh - D)) break; asign = ~asign; T = A + q * C; A = C; C = T; T = B + q * D; B = D; D = T; T = uh - q * vh; uh = vh; vh = T; } #if EXTEND if (asign) sign = -sign; #endif } #if RECORD max = MAX (A, max); max = MAX (B, max); max = MAX (C, max); max = MAX (D, max); #endif if (B == 0) { mp_limb_t qh; mp_size_t i; /* This is quite rare. I.e., optimize something else! */ /* Normalize V (and shift up U the same amount). */ count_leading_zeros (cnt, vp[vsize - 1]); if (cnt != 0) { mp_limb_t cy; mpn_lshift (vp, vp, vsize, cnt); cy = mpn_lshift (up, up, size, cnt); up[size] = cy; size += cy != 0; } qh = mpn_divmod (up + vsize, up, size, vp, vsize); #if EXTEND MPN_COPY (tp, s0p, ssize); { mp_size_t qsize; qsize = size - vsize; /* size of stored quotient from division */ if (ssize < qsize) { MPN_ZERO (tp + ssize, qsize - ssize); MPN_ZERO (s1p + ssize, qsize); /* zero s1 too */ for (i = 0; i < ssize; i++) { mp_limb_t cy; cy = mpn_addmul_1 (tp + i, up + vsize, qsize, s1p[i]); tp[qsize + i] = cy; } } else { MPN_ZERO (s1p + ssize, qsize); /* zero s1 too */ for (i = 0; i < qsize; i++) { mp_limb_t cy; cy = mpn_addmul_1 (tp + i, s1p, ssize, up[vsize + i]); tp[ssize + i] = cy; } } if (qh != 0) { mp_limb_t cy; cy = mpn_add_n (tp + qsize, tp + qsize, s1p, ssize); if (cy != 0) { tp[qsize + ssize] = cy; s1p[qsize + ssize] = 0; ssize++; } } ssize += qsize; ssize -= tp[ssize - 1] == 0; } sign = -sign; MP_PTR_SWAP (s0p, s1p); MP_PTR_SWAP (s1p, tp); #endif size = vsize; if (cnt != 0) { mpn_rshift (up, up, size, cnt); mpn_rshift (vp, vp, size, cnt); } MP_PTR_SWAP (up, vp); } else { #if EXTEND mp_size_t tsize, wsize; #endif /* T = U*A + V*B W = U*C + V*D U = T V = W */ #if STAT { mp_limb_t x; x = A | B | C | D; count_leading_zeros (cnt, x); arr[BITS_PER_MP_LIMB - cnt]++; } #endif if (A == 0) { /* B == 1 and C == 1 (D is arbitrary) */ mp_limb_t cy; MPN_COPY (tp, vp, size); MPN_COPY (wp, up, size); mpn_submul_1 (wp, vp, size, D); MP_PTR_SWAP (tp, up); MP_PTR_SWAP (wp, vp); #if EXTEND MPN_COPY (tp, s1p, ssize); tsize = ssize; tp[ssize] = 0; /* must zero since wp might spill below */ MPN_COPY (wp, s0p, ssize); cy = mpn_addmul_1 (wp, s1p, ssize, D); wp[ssize] = cy; wsize = ssize + (cy != 0); MP_PTR_SWAP (tp, s0p); MP_PTR_SWAP (wp, s1p); ssize = MAX (wsize, tsize); #endif } else { mp_limb_t cy, cy1, cy2; if (asign) { mpn_mul_1 (tp, vp, size, B); mpn_submul_1 (tp, up, size, A); mpn_mul_1 (wp, up, size, C); mpn_submul_1 (wp, vp, size, D); } else { mpn_mul_1 (tp, up, size, A); mpn_submul_1 (tp, vp, size, B); mpn_mul_1 (wp, vp, size, D); mpn_submul_1 (wp, up, size, C); } MP_PTR_SWAP (tp, up); MP_PTR_SWAP (wp, vp); #if EXTEND /* Compute new s0 */ cy1 = mpn_mul_1 (tp, s0p, ssize, A); cy2 = mpn_addmul_1 (tp, s1p, ssize, B); cy = cy1 + cy2; tp[ssize] = cy; tsize = ssize + (cy != 0); if (cy < cy1) { abort (); /* Should not happen, even it can happen below. */ } /* Compute new s1 */ cy1 = mpn_mul_1 (wp, s1p, ssize, D); cy2 = mpn_addmul_1 (wp, s0p, ssize, C); cy = cy1 + cy2; wp[ssize] = cy; wsize = ssize + (cy != 0); if (cy < cy1) { wp[wsize] = 1; tp[wsize] = 0; /* Safe, s0 (here stored at tp) will be smaller than s1 (stored at wp), so we don't risk to overwrite s0's most significant limb. */ wsize++; } MP_PTR_SWAP (tp, s0p); MP_PTR_SWAP (wp, s1p); ssize = MAX (wsize, tsize); #endif } size -= up[size - 1] == 0; } #if WANT_GCDEXT_ONE_STEP return 0; #endif } #if RECORD printf ("max: %lx\n", max); #endif #if STAT {int i; for (i = 0; i < BITS_PER_MP_LIMB; i++) printf ("%d:%d\n", i, arr[i]);} #endif if (vsize == 0) { if (gp != up && gp != 0) MPN_COPY (gp, up, size); #if EXTEND MPN_NORMALIZE (s0p, ssize); if (orig_s0p != s0p) MPN_COPY (orig_s0p, s0p, ssize); *s0size = sign >= 0 ? ssize : -ssize; #endif TMP_FREE (mark); return size; } else { mp_limb_t vl, ul, t; #if EXTEND mp_size_t qsize, i; #endif vl = vp[0]; #if EXTEND t = mpn_divmod_1 (wp, up, size, vl); MPN_COPY (tp, s0p, ssize); qsize = size - (wp[size - 1] == 0); /* size of quotient from division */ if (ssize < qsize) { MPN_ZERO (tp + ssize, qsize - ssize); MPN_ZERO (s1p + ssize, qsize); /* zero s1 too */ for (i = 0; i < ssize; i++) { mp_limb_t cy; cy = mpn_addmul_1 (tp + i, wp, qsize, s1p[i]); tp[qsize + i] = cy; } } else { MPN_ZERO (s1p + ssize, qsize); /* zero s1 too */ for (i = 0; i < qsize; i++) { mp_limb_t cy; cy = mpn_addmul_1 (tp + i, s1p, ssize, wp[i]); tp[ssize + i] = cy; } } ssize += qsize; ssize -= tp[ssize - 1] == 0; sign = -sign; MP_PTR_SWAP (s0p, s1p); MP_PTR_SWAP (s1p, tp); #else t = mpn_mod_1 (up, size, vl); #endif ul = vl; vl = t; while (vl != 0) { mp_limb_t t; #if EXTEND mp_limb_t q; q = ul / vl; t = ul - q * vl; MPN_COPY (tp, s0p, ssize); MPN_ZERO (s1p + ssize, 1); /* zero s1 too */ { mp_limb_t cy; cy = mpn_addmul_1 (tp, s1p, ssize, q); tp[ssize] = cy; } ssize += 1; ssize -= tp[ssize - 1] == 0; sign = -sign; MP_PTR_SWAP (s0p, s1p); MP_PTR_SWAP (s1p, tp); #else t = ul % vl; #endif ul = vl; vl = t; } if (gp != 0) gp[0] = ul; #if EXTEND MPN_NORMALIZE (s0p, ssize); if (orig_s0p != s0p) MPN_COPY (orig_s0p, s0p, ssize); *s0size = sign >= 0 ? ssize : -ssize; #endif TMP_FREE (mark); return 1; } }