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mpir/mpn/generic/lgcd.c

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/* lgcd.c
Lehmer gcd, based on mpn_nhgcd2.
*/
#include <stdio.h>
#include <stdlib.h>
#include "mpir.h"
#include "gmp-impl.h"
#include "longlong.h"
/* Extract one limb, shifting count bits left
________ ________
|___xh___||___xl___|
|____r____|
>count <
The count includes any nail bits, so it should work fine if
count is computed using count_leading_zeros.
*/
#define MPN_EXTRACT_LIMB(count, xh, xl) \
(((xh) << (count)) | ((xl) >> (GMP_LIMB_BITS - (count))))
/* Use binary algorithm to compute V <-- GCD (V, U) for usize, vsize == 2.
Both U and V must be odd. */
static inline mp_size_t
gcd_2 (mp_ptr vp, mp_srcptr up)
{
mp_limb_t u0, u1, v0, v1;
mp_size_t vsize;
u0 = up[0];
u1 = up[1];
v0 = vp[0];
v1 = vp[1];
while (u1 != v1 && u0 != v0)
{
unsigned long int r;
if (u1 > v1)
{
u1 -= v1 + (u0 < v0);
u0 = (u0 - v0) & GMP_NUMB_MASK;
count_trailing_zeros (r, u0);
u0 = ((u1 << (GMP_NUMB_BITS - r)) & GMP_NUMB_MASK) | (u0 >> r);
u1 >>= r;
}
else /* u1 < v1. */
{
v1 -= u1 + (v0 < u0);
v0 = (v0 - u0) & GMP_NUMB_MASK;
count_trailing_zeros (r, v0);
v0 = ((v1 << (GMP_NUMB_BITS - r)) & GMP_NUMB_MASK) | (v0 >> r);
v1 >>= r;
}
}
vp[0] = v0, vp[1] = v1, vsize = 1 + (v1 != 0);
/* If U == V == GCD, done. Otherwise, compute GCD (V, |U - V|). */
if (u1 == v1 && u0 == v0)
return vsize;
v0 = (u0 == v0) ? (u1 > v1) ? u1-v1 : v1-u1 : (u0 > v0) ? u0-v0 : v0-u0;
vp[0] = mpn_gcd_1 (vp, vsize, v0);
return 1;
}
#if 1
static void
mul_2 (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_srcptr bp)
{
mp_limb_t bh, bl;
/* Chaining variables */
mp_limb_t cy, hi;
/* Temporaries */
mp_limb_t sh, sl;
mp_limb_t th, tl;
mp_size_t i;
ASSERT (n > 1);
bl = bp[0];
umul_ppmm (hi, rp[0], bl, ap[0]);
bh = bp[1];
for (i = 1, cy = 0; i < n; i++)
{
umul_ppmm (sh, sl, bh, ap[i-1]);
umul_ppmm (th, tl, bl, ap[i]);
/* We can always add in cy and hi without overflow */
add_ssaaaa (sh, sl, sh, sl, cy, hi);
add_ssaaaa (hi, rp[i], th, tl, sh, sl);
cy = (hi < th);
}
umul_ppmm (sh, sl, bh, ap[n-1]);
add_ssaaaa (rp[n+1], rp[n], sh, sl, cy, hi);
}
#else
static void
mul_2 (mp_ptr rp, mp_srcptr ap, mp_size_t n, mp_srcptr bp)
{
rp[n] = mpn_mul_1 (rp, ap, n, bp[0]);
rp[n+1] = mpn_addmul_1 (rp + 1, ap, n, bp[1]);
}
#endif
/* Computes x . y = x_1 y_1 + x_2 y_2. No check for overflow. */
#define dotmul_ppxxyy(ph, pl, x1, x2, y1, y2) \
do { \
mp_limb_t dotmul_sh, dotmul_sl, dotmul_th, dotmul_tl; \
umul_ppmm (dotmul_sh, dotmul_sl, (x1), (y1)); \
umul_ppmm (dotmul_th, dotmul_tl, (x2), (y2)); \
add_ssaaaa ((ph), (pl), dotmul_sh, dotmul_sl, dotmul_th, dotmul_tl); \
} while (0)
struct ngcd_matrix2
{
mp_limb_t u[2][2][2];
};
/* Performs two hgcd2 steps on the five-limb numbers ap, bp.
Three possible results:
0 The first nhgcd2 call failed.
1 The first nhgcd2 call succeeded, the second failed.
ap, bp are updated, and the resulting matrix is returned in M1.
2 Both hgcd2 calls succeeded. ap, bp are updated. The resulting
matrix is returned in M.
Returns the reduced size of ap, bp, >= 3.
*/
static mp_size_t
nhgcd5 (mp_ptr ap, mp_ptr bp, int *res,
struct ngcd_matrix1 *M1, struct ngcd_matrix2 *M)
{
struct ngcd_matrix1 M2;
mp_limb_t t[5];
mp_size_t n;
mp_limb_t ah, al, bh, bl;
mp_limb_t mask;
mask = ap[4] | bp[4];
ASSERT (mask > 0);
if (mask & GMP_NUMB_HIGHBIT)
{
ah = ap[4]; al = ap[3];
bh = bp[4]; bl = bp[3];
}
else
{
int shift;
count_leading_zeros (shift, mask);
ah = MPN_EXTRACT_LIMB (shift, ap[4], ap[3]);
al = MPN_EXTRACT_LIMB (shift, ap[3], ap[2]);
bh = MPN_EXTRACT_LIMB (shift, bp[4], bp[3]);
bl = MPN_EXTRACT_LIMB (shift, bp[3], bp[2]);
}
/* Try an mpn_nhgcd2 step */
if (!mpn_nhgcd2 (ah, al, bh, bl, M1))
{
*res = 0;
return 0;
}
n = mpn_ngcd_matrix1_vector (M1, 5, ap, bp, t);
ASSERT (n >= 4);
mask = ap[n-1] | bp[n-1];
ASSERT (mask > 0);
if (mask & GMP_NUMB_HIGHBIT)
{
ah = ap[n-1]; al = ap[n-2];
bh = bp[n-1]; bl = bp[n-2];
}
else
{
int shift;
count_leading_zeros (shift, mask);
ah = MPN_EXTRACT_LIMB (shift, ap[n-1], ap[n-2]);
al = MPN_EXTRACT_LIMB (shift, ap[n-2], ap[n-3]);
bh = MPN_EXTRACT_LIMB (shift, bp[n-1], bp[n-2]);
bl = MPN_EXTRACT_LIMB (shift, bp[n-2], bp[n-3]);
}
if (!mpn_nhgcd2(ah, al, bh, bl, &M2))
{
*res = 1;
return n;
}
n = mpn_ngcd_matrix1_vector (&M2, n, ap, bp, t);
ASSERT (n >= 3);
/* Multiply M = M1 M2 */
dotmul_ppxxyy (M->u[0][0][1], M->u[0][0][0],
M1->u[0][0], M1->u[0][1], M2.u[0][0], M2.u[1][0]);
dotmul_ppxxyy (M->u[0][1][1], M->u[0][1][0],
M1->u[0][0], M1->u[0][1], M2.u[0][1], M2.u[1][1]);
dotmul_ppxxyy (M->u[1][0][1], M->u[1][0][0],
M1->u[1][0], M1->u[1][1], M2.u[0][0], M2.u[1][0]);
dotmul_ppxxyy (M->u[1][1][1], M->u[1][1][0],
M1->u[1][0], M1->u[1][1], M2.u[0][1], M2.u[1][1]);
*res = 2;
return n;
}
/* Multiplies the least significant p limbs of a,b by M^-1, and adds
to the most significant n limbs. Needs temporary space p. */
static mp_size_t
ngcd_matrix1_adjust (struct ngcd_matrix1 *M,
mp_size_t n, mp_ptr ap, mp_ptr bp,
mp_size_t p, mp_ptr tp)
{
/* M^-1 (a;b) = (r11, -r01; -r10, r00) (a ; b)
= (r11 a - r01 b; - r10 a + r00 b */
mp_limb_t ah, bh;
mp_limb_t cy;
/* Copy old value */
MPN_COPY(tp, ap, p);
/* Update a */
ah = mpn_mul_1 (ap, ap, p, M->u[1][1]);
cy = mpn_submul_1 (ap, bp, p, M->u[0][1]);
if (cy > ah)
{
MPN_DECR_U (ap + p, n, cy - ah);
ah = 0;
}
else
{
ah -= cy;
if (ah > 0)
ah = mpn_add_1 (ap + p, ap + p, n, ah);
}
/* Update b */
bh = mpn_mul_1 (bp, bp, p, M->u[0][0]);
cy = mpn_submul_1 (bp, tp, p, M->u[1][0]);
if(cy > bh)
{
MPN_DECR_U (bp + p, n, cy - bh);
bh = 0;
}
else
{
bh -= cy;
if (bh > 0)
bh = mpn_add_1 (bp + p, bp + p, n, bh);
}
n += p;
if (ah > 0 || bh > 0)
{
ap[n] = ah;
bp[n] = bh;
n++;
}
else
{
/* The subtraction can reduce the size by at most one limb. */
if (ap[n-1] == 0 && bp[n-1] == 0)
n--;
}
ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
return n;
}
/* Multiplies the least significant p limbs of a,b by M^-1, and adds
to the most significant n limbs. Needs temporary 2*(p + 2). */
static mp_size_t
ngcd_matrix2_adjust (struct ngcd_matrix2 *M,
mp_size_t n, mp_ptr ap, mp_ptr bp,
mp_size_t p, mp_ptr tp)
{
/* M^-1 (a;b) = (r11, -r01; -r10, r00) (a ; b)
= (r11 a - r01 b; - r10 a + r00 b */
mp_ptr t0 = tp;
mp_ptr t1 = tp + p + 2;
mp_limb_t ah, bh;
mp_limb_t cy;
ASSERT (n > 2);
ASSERT (p >= 2);
/* First compute the two values depending on a, before overwriting a */
mul_2 (t0, ap, p, M->u[1][1]);
mul_2 (t1, ap, p, M->u[1][0]);
/* Update a */
MPN_COPY (ap, t0, p);
ah = mpn_add (ap + p, ap + p, n, t0 + p, 2);
mpn_mul (t0, bp, p, M->u[0][1], 2);
cy = mpn_sub (ap, ap, n + p, t0, p + 2);
ASSERT (cy <= ah);
ah -= cy;
/* Update b */
mpn_mul (t0, bp, p, M->u[0][0], 2);
MPN_COPY (bp, t0, p);
bh = mpn_add (bp + p, bp + p, n, t0 + p, 2);
cy = mpn_sub (bp, bp, n + p, t1, p + 2);
ASSERT (cy <= bh);
bh -= cy;
n+= p;
if (ah > 0 || bh > 0)
{
ap[n] = ah;
bp[n] = bh;
n++;
}
else
{
/* The subtraction can reduce the size by at most one limb. */
if (ap[n-1] == 0 && bp[n-1] == 0)
n--;
}
ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
return n;
}
/* Needs temporary storage for the division */
/* If the gcd is found, stores it in gp and *gn, and returns zero.
Otherwise, compute the reduced a and b, and return the new size. */
mp_size_t
mpn_ngcd_subdiv_step (mp_ptr gp, mp_size_t *gn,
mp_ptr ap, mp_ptr bp, mp_size_t n, mp_ptr tp)
{
/* Called when nhgcd or mpn_nhgcd2 has failed. Then either one of a or b
is very small, or the difference is very small. Perform one
subtraction followed by one division. */
mp_size_t an, bn;
ASSERT (n > 0);
ASSERT (ap[n-1] > 0 || bp[n-1] > 0);
/* First, make sure that an >= bn, and subtract an -= bn */
for (an = n; an > 0; an--)
if (ap[an-1] != bp[an-1])
break;
if (an == 0)
{
/* Done */
MPN_COPY (gp, ap, n);
*gn = n;
return 0;
}
if (ap[an-1] < bp[an-1])
MP_PTR_SWAP (ap, bp);
bn = n;
MPN_NORMALIZE (bp, bn);
if (bn == 0)
{
MPN_COPY (gp, ap, n);
*gn = n;
return 0;
}
ASSERT_NOCARRY (mpn_sub_n (ap, ap, bp, an));
MPN_NORMALIZE (ap, an);
ASSERT (an > 0);
if (an < bn)
MPN_PTR_SWAP (ap, an, bp, bn);
else if (an == bn)
{
int c;
MPN_CMP (c, ap, bp, an);
if (c < 0)
MP_PTR_SWAP (ap, bp);
}
ASSERT (an >= bn);
mpn_tdiv_qr (tp, ap, 0, ap, an, bp, bn);
/* Normalizing seems to be the simplest way to test if the remainder
is zero. */
an = bn;
MPN_NORMALIZE (ap, an);
if (an == 0)
{
MPN_COPY (gp, bp, bn);
*gn = bn;
return 0;
}
return bn;
}
#define EVEN_P(x) (((x) & 1) == 0)
/* Needs n limbs of storage for ngcd_matrix1_vector, or n+1 for
division, or 2*n for ngcd_matrix2_adjust. */
mp_size_t
mpn_ngcd_lehmer (mp_ptr gp, mp_ptr ap, mp_ptr bp, mp_size_t n, mp_ptr tp)
{
mp_size_t gn;
while (ABOVE_THRESHOLD (n, NGCD_LEHMER_THRESHOLD))
{
struct ngcd_matrix1 M1;
struct ngcd_matrix2 M2;
mp_size_t p;
int res;
mp_size_t nn;
p = n - 5;
nn = nhgcd5 (ap + p, bp + p, &res, &M1, &M2);
switch (res)
{
default: abort();
case 0:
n = mpn_ngcd_subdiv_step (gp, &gn, ap, bp, n, tp);
if (n == 0)
return gn;
break;
case 1:
n = ngcd_matrix1_adjust (&M1, nn, ap, bp, p, tp);
break;
case 2:
n = ngcd_matrix2_adjust (&M2, nn, ap, bp, p, tp);
break;
}
}
while (n > 2)
{
struct ngcd_matrix1 M;
mp_limb_t ah, al, bh, bl;
mp_limb_t mask;
mask = ap[n-1] | bp[n-1];
ASSERT (mask > 0);
if (mask & GMP_NUMB_HIGHBIT)
{
ah = ap[n-1]; al = ap[n-2];
bh = bp[n-1]; bl = bp[n-2];
}
else
{
int shift;
count_leading_zeros (shift, mask);
ah = MPN_EXTRACT_LIMB (shift, ap[n-1], ap[n-2]);
al = MPN_EXTRACT_LIMB (shift, ap[n-2], ap[n-3]);
bh = MPN_EXTRACT_LIMB (shift, bp[n-1], bp[n-2]);
bl = MPN_EXTRACT_LIMB (shift, bp[n-2], bp[n-3]);
}
/* Try an mpn_nhgcd2 step */
if (mpn_nhgcd2 (ah, al, bh, bl, &M))
n = mpn_ngcd_matrix1_vector (&M, n, ap, bp, tp);
else
{
n = mpn_ngcd_subdiv_step (gp, &gn, ap, bp, n, tp);
if (n == 0)
return gn;
}
}
ASSERT (n <= 2);
if (n == 1)
{
*gp = mpn_gcd_1 (ap, 1, bp[0]);
return 1;
}
/* Due to the calling convention for mpn_gcd, at most one can be
even. */
if (EVEN_P (ap[0]))
MP_PTR_SWAP (ap, bp);
ASSERT (!EVEN_P (ap[0]));
if (bp[0] == 0)
{
*gp = mpn_gcd_1 (ap, 2, bp[1]);
return 1;
}
else if (EVEN_P (bp[0]))
{
int r;
count_trailing_zeros (r, bp[0]);
bp[0] = ((bp[1] << (GMP_NUMB_BITS - r)) & GMP_NUMB_MASK) | (bp[0] >> r);
bp[1] >>= r;
}
n = gcd_2 (ap, bp);
MPN_COPY (gp, ap, n);
return n;
}
mp_size_t
mpn_lgcd (mp_ptr gp, mp_ptr ap, mp_size_t an, mp_ptr bp, mp_size_t bn)
{
mp_size_t gn;
mp_ptr tp;
mp_size_t scratch;
TMP_DECL;
scratch = MPN_NGCD_LEHMER_ITCH (bn);
if (an >= scratch)
scratch = an + 1;
TMP_MARK;
tp = TMP_ALLOC_LIMBS (scratch);
if (an > bn)
{
mpn_tdiv_qr (tp + bn, tp, 0, ap, an, bp, bn);
an = bn;
MPN_NORMALIZE (tp, an);
if (an == 0)
{
MPN_COPY (gp, bp, bn);
TMP_FREE;
return bn;
}
else
MPN_COPY (ap, tp, bn);
}
gn = mpn_ngcd_lehmer (gp, ap, bp, bn, tp);
TMP_FREE;
return gn;
}
/*
Set (u0, u1) = (u0, u1) M
Requires temporary space un
*/
void ngcdext_cofactor1_adjust(mp_ptr u0, mp_ptr u1, mp_size_t * un, struct ngcd_matrix1 *M, mp_ptr tp)
{
/* Let M = (r00, r01)
(r10, r11)
We want u0 = u0 * r00 + u1 * r10
u1 = u0 * r01 + u1 * r11
We make a copy of u0 at tp and update u0 first
*/
mp_limb_t cy, cy2;
MPN_COPY(tp, u0, *un);
cy = mpn_mul_1(u0, u0, *un, M->u[0][0]);
cy += mpn_addmul_1(u0, u1, *un, M->u[1][0]);
cy2 = mpn_mul_1(u1, u1, *un, M->u[1][1]);
cy2 += mpn_addmul_1(u1, tp, *un, M->u[0][1]);
if ((cy) || (cy2)) /* normalise u0, u1 */
{
u0[*un] = cy;
u1[*un] = cy2;
(*un) ++;
}
}
mp_limb_t mpn_gcdext_1(mp_limb_t * a, mp_limb_t *b, mp_limb_t x, mp_limb_t y)
{
mp_limb_signed_t u1 = CNST_LIMB(1);
mp_limb_signed_t u2 = CNST_LIMB(0);
mp_limb_signed_t v1 = CNST_LIMB(0);
mp_limb_signed_t v2 = CNST_LIMB(1);
mp_limb_signed_t t1, t2;
mp_limb_t u3, v3;
mp_limb_t quot, rem;
u3 = x, v3 = y;
if (v3 > u3)
{
rem = u3;
t1 = u2; u2 = u1; u1 = t1; u3 = v3;
t2 = v2; v2 = v1; v1 = t2; v3 = rem;
}
if ((mp_limb_signed_t) (x & y) < (mp_limb_signed_t) CNST_LIMB(0)) // x and y both have top bit set
{
quot=u3-v3;
t2 = v2;
t1 = u2; u2 = u1 - u2; u1 = t1; u3 = v3;
v2 = v1 - v2; v1 = t2; v3 = quot;
}
while ((mp_limb_signed_t) (v3<<1) < (mp_limb_signed_t) CNST_LIMB(0)) // second value has second msb set
{
quot=u3-v3;
if (quot < v3)
{
t2 = v2;
t1 = u2; u2 = u1 - u2; u1 = t1; u3 = v3;
v2 = v1 - v2; v1 = t2; v3 = quot;
} else if (quot < (v3<<1))
{
t1 = u2; u2 = u1 - (u2<<1); u1 = t1; u3 = v3;
t2 = v2; v2 = v1 - (v2<<1); v1 = t2; v3 = quot-u3;
} else
{
t1 = u2; u2 = u1 - 3*u2; u1 = t1; u3 = v3;
t2 = v2; v2 = v1 - 3*v2; v1 = t2; v3 = quot-(u3<<1);
}
}
while (v3) {
quot=u3-v3;
if (u3 < (v3<<2)) // overflow not possible due to top 2 bits of v3 not being set
{
if (quot < v3)
{
t2 = v2;
t1 = u2; u2 = u1 - u2; u1 = t1; u3 = v3;
v2 = v1 - v2; v1 = t2; v3 = quot;
} else if (quot < (v3<<1))
{
t1 = u2; u2 = u1 - (u2<<1); u1 = t1; u3 = v3;
t2 = v2; v2 = v1 - (v2<<1); v1 = t2; v3 = quot-u3;
} else
{
t1 = u2; u2 = u1 - 3*u2; u1 = t1; u3 = v3;
t2 = v2; v2 = v1 - 3*v2; v1 = t2; v3 = quot-(u3<<1);
}
} else
{
quot=u3/v3;
rem = u3 - v3*quot;
t1 = u2; u2 = u1 - quot*u2; u1 = t1; u3 = v3;
t2 = v2; v2 = v1 - quot*v2; v1 = t2; v3 = rem;
}
}
// Quite remarkably, this always has |u1| < x/2 at this point, thus comparison with 0 is valid
if (u1 <= (mp_limb_signed_t) 0)
{
u1 += y;
v1 -= x;
}
*a = u1;
*b = -v1;
return u3;
}
/*
Requires temporary space MPN_GCD_LEHMER_N_ITCH(n) + 2*n+2
*/
mp_size_t
mpn_ngcdext_lehmer (mp_ptr gp, mp_ptr s0p, mp_size_t *s0size, mp_ptr ap, mp_ptr bp, mp_size_t n, mp_ptr tp)
{
mp_size_t gn, un;
mp_ptr u0, u1;
mp_limb_t cy, cy2;
mp_limb_t s, t;
MPN_ZERO(tp, 2*n+2);
u0 = tp;
u1 = tp + n + 1;
tp += 2*n + 2;
un = 1;
u0[0] = CNST_LIMB(0); /* bp = 0*ap + ?*bp, thus u0 = -0 */
u1[0] = CNST_LIMB(1); /* ap = 1*ap + ?*bp, thus u1 = 1 */
while (n >= 2)
{
struct ngcd_matrix1 M;
mp_limb_t ah, al, bh, bl;
mp_limb_t mask;
mask = ap[n-1] | bp[n-1];
ASSERT (mask > 0);
if (mask & GMP_NUMB_HIGHBIT)
{
ah = ap[n-1]; al = ap[n-2];
bh = bp[n-1]; bl = bp[n-2];
}
else
{
int shift;
count_leading_zeros (shift, mask);
ah = MPN_EXTRACT_LIMB (shift, ap[n-1], ap[n-2]);
bh = MPN_EXTRACT_LIMB (shift, bp[n-1], bp[n-2]);
if (n == 2) /* special case for n = 2 */
{
al = ap[0] << shift;
bl = bp[0] << shift;
} else
{
al = MPN_EXTRACT_LIMB (shift, ap[n-2], ap[n-3]);
bl = MPN_EXTRACT_LIMB (shift, bp[n-2], bp[n-3]);
}
}
/* Try an mpn_nhgcd2 step */
if (mpn_nhgcd2 (ah, al, bh, bl, &M))
{
n = mpn_ngcd_matrix1_vector (&M, n, ap, bp, tp);
ngcdext_cofactor1_adjust(u0, u1, &un, &M, tp);
}
else
{
n = mpn_ngcdext_subdiv_step (gp, &gn, s0p, u0, u1, &un, ap, bp, n, tp);
if (n == 0)
{
(*s0size) = un;
ASSERT(((*s0size) == 0) || (s0p[ABS(*s0size) - 1] != 0));
return gn;
}
}
}
ASSERT (n < 2);
if (!ap[0])
{
/*
If ap == 0 then gp = bp
with cofactor -u0
*/
gp[0] = bp[0];
MPN_NORMALIZE(u0, un);
MPN_COPY(s0p, u0, un);
(*s0size) = -un;
return 1;
} else if (!bp[0])
{
/*
If bp == 0 then gp = ap
with cofactor u1
*/
gp[0] = ap[0];
MPN_NORMALIZE(u1, un);
MPN_COPY(s0p, u1, un);
(*s0size) = un;
return 1;
}
gp[0] = mpn_gcdext_1 (&s, &t, ap[0], bp[0]);
/*
We want to compute s*u1 + t*u0.
*/
cy = mpn_mul_1(s0p, u0, un, t);
cy2 = mpn_addmul_1(s0p, u1, un, s);
if (cy | cy2) /* u0 is bigger than un limbs */
{
cy +=cy2;
s0p[un] = cy;
un++;
if (cy < cy2) /* overflow on addition */
{
s0p[un] = CNST_LIMB(1);
un++;
}
}
MPN_NORMALIZE(s0p, un);
(*s0size) = un;
return 1;
}
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