mp_size_t mpn_binvert_itch (mp_size_t n) { #if WANT_FFT if (ABOVE_THRESHOLD (n, 2 * MUL_FFT_MODF_THRESHOLD)) return mpn_fft_next_size (n, mpn_fft_best_k (n, 0)); else #endif return 3 * (n - (n >> 1)); }
void mpn_binvert (mp_ptr rp, mp_srcptr up, mp_size_t n, mp_ptr scratch) { mp_ptr xp; mp_size_t rn, newrn; mp_size_t sizes[NPOWS], *sizp; mp_limb_t di; /* Compute the computation precisions from highest to lowest, leaving the base case size in 'rn'. */ sizp = sizes; for (rn = n; ABOVE_THRESHOLD (rn, BINV_NEWTON_THRESHOLD); rn = (rn + 1) >> 1) *sizp++ = rn; xp = scratch; /* Compute a base value using a low-overhead O(n^2) algorithm. FIXME: We should call some divide-and-conquer lsb division function here for an operand subrange. */ MPN_ZERO (xp, rn); xp[0] = 1; binvert_limb (di, up[0]); if (BELOW_THRESHOLD (rn, DC_BDIV_Q_THRESHOLD)) mpn_sb_bdiv_q (rp, xp, rn, up, rn, -di); else mpn_dc_bdiv_q (rp, xp, rn, up, rn, -di); /* Use Newton iterations to get the desired precision. */ for (; rn < n; rn = newrn) { newrn = *--sizp; #if WANT_FFT if (ABOVE_THRESHOLD (newrn, 2 * MUL_FFT_MODF_THRESHOLD)) { int k; mp_size_t m, i; k = mpn_fft_best_k (newrn, 0); m = mpn_fft_next_size (newrn, k); mpn_mul_fft (xp, m, up, newrn, rp, rn, k); for (i = rn - 1; i >= 0; i--) if (xp[i] > (i == 0)) { mpn_add_1 (xp + rn, xp + rn, newrn - rn, 1); break; } } else #endif mpn_mul (xp, up, newrn, rp, rn); mpn_mullow_n (rp + rn, rp, xp + rn, newrn - rn); mpn_neg_n (rp + rn, rp + rn, newrn - rn); } }
mp_size_t mpn_mulmod_bnm1_next_size (mp_size_t n) { mp_size_t nh; if (BELOW_THRESHOLD (n, MULMOD_BNM1_THRESHOLD)) return n; if (BELOW_THRESHOLD (n, 4 * (MULMOD_BNM1_THRESHOLD - 1) + 1)) return (n + (2-1)) & (-2); if (BELOW_THRESHOLD (n, 8 * (MULMOD_BNM1_THRESHOLD - 1) + 1)) return (n + (4-1)) & (-4); nh = (n + 1) >> 1; if (BELOW_THRESHOLD (nh, MUL_FFT_MODF_THRESHOLD)) return (n + (8-1)) & (-8); return 2 * mpn_fft_next_size (nh, mpn_fft_best_k (nh, 0)); }
/* Computes {rp,MIN(rn,an+bn)} <- {ap,an}*{bp,bn} Mod(B^rn-1) * * The result is expected to be ZERO if and only if one of the operand * already is. Otherwise the class [0] Mod(B^rn-1) is represented by * B^rn-1. This should not be a problem if mulmod_bnm1 is used to * combine results and obtain a natural number when one knows in * advance that the final value is less than (B^rn-1). * Moreover it should not be a problem if mulmod_bnm1 is used to * compute the full product with an+bn <= rn, because this condition * implies (B^an-1)(B^bn-1) < (B^rn-1) . * * Requires 0 < bn <= an <= rn and an + bn > rn/2 * Scratch need: rn + (need for recursive call OR rn + 4). This gives * * S(n) <= rn + MAX (rn + 4, S(n/2)) <= 2rn + 4 */ void mpn_mulmod_bnm1 (mp_ptr rp, mp_size_t rn, mp_srcptr ap, mp_size_t an, mp_srcptr bp, mp_size_t bn, mp_ptr tp) { ASSERT (0 < bn); ASSERT (bn <= an); ASSERT (an <= rn); if ((rn & 1) != 0 || BELOW_THRESHOLD (rn, MULMOD_BNM1_THRESHOLD)) { if (UNLIKELY (bn < rn)) { if (UNLIKELY (an + bn <= rn)) { mpn_mul (rp, ap, an, bp, bn); } else { mp_limb_t cy; mpn_mul (tp, ap, an, bp, bn); cy = mpn_add (rp, tp, rn, tp + rn, an + bn - rn); MPN_INCR_U (rp, rn, cy); } } else mpn_bc_mulmod_bnm1 (rp, ap, bp, rn, tp); } else { mp_size_t n; mp_limb_t cy; mp_limb_t hi; n = rn >> 1; /* We need at least an + bn >= n, to be able to fit one of the recursive products at rp. Requiring strict inequality makes the coded slightly simpler. If desired, we could avoid this restriction by initially halving rn as long as rn is even and an + bn <= rn/2. */ ASSERT (an + bn > n); /* Compute xm = a*b mod (B^n - 1), xp = a*b mod (B^n + 1) and crt together as x = -xp * B^n + (B^n + 1) * [ (xp + xm)/2 mod (B^n-1)] */ #define a0 ap #define a1 (ap + n) #define b0 bp #define b1 (bp + n) #define xp tp /* 2n + 2 */ /* am1 maybe in {xp, n} */ /* bm1 maybe in {xp + n, n} */ #define sp1 (tp + 2*n + 2) /* ap1 maybe in {sp1, n + 1} */ /* bp1 maybe in {sp1 + n + 1, n + 1} */ { mp_srcptr am1, bm1; mp_size_t anm, bnm; mp_ptr so; bm1 = b0; bnm = bn; if (LIKELY (an > n)) { am1 = xp; cy = mpn_add (xp, a0, n, a1, an - n); MPN_INCR_U (xp, n, cy); anm = n; so = xp + n; if (LIKELY (bn > n)) { bm1 = so; cy = mpn_add (so, b0, n, b1, bn - n); MPN_INCR_U (so, n, cy); bnm = n; so += n; } } else { so = xp; am1 = a0; anm = an; } mpn_mulmod_bnm1 (rp, n, am1, anm, bm1, bnm, so); } { int k; mp_srcptr ap1, bp1; mp_size_t anp, bnp; bp1 = b0; bnp = bn; if (LIKELY (an > n)) { ap1 = sp1; cy = mpn_sub (sp1, a0, n, a1, an - n); sp1[n] = 0; MPN_INCR_U (sp1, n + 1, cy); anp = n + ap1[n]; if (LIKELY (bn > n)) { bp1 = sp1 + n + 1; cy = mpn_sub (sp1 + n + 1, b0, n, b1, bn - n); sp1[2*n+1] = 0; MPN_INCR_U (sp1 + n + 1, n + 1, cy); bnp = n + bp1[n]; } } else { ap1 = a0; anp = an; } if (BELOW_THRESHOLD (n, MUL_FFT_MODF_THRESHOLD)) k=0; else { int mask; k = mpn_fft_best_k (n, 0); mask = (1<<k) - 1; while (n & mask) {k--; mask >>=1;}; } if (k >= FFT_FIRST_K) xp[n] = mpn_mul_fft (xp, n, ap1, anp, bp1, bnp, k); else if (UNLIKELY (bp1 == b0)) { ASSERT (anp + bnp <= 2*n+1); ASSERT (anp + bnp > n); ASSERT (anp >= bnp); mpn_mul (xp, ap1, anp, bp1, bnp); anp = anp + bnp - n; ASSERT (anp <= n || xp[2*n]==0); anp-= anp > n; cy = mpn_sub (xp, xp, n, xp + n, anp); xp[n] = 0; MPN_INCR_U (xp, n+1, cy); } else mpn_bc_mulmod_bnp1 (xp, ap1, bp1, n, xp); } /* Here the CRT recomposition begins. xm <- (xp + xm)/2 = (xp + xm)B^n/2 mod (B^n-1) Division by 2 is a bitwise rotation. Assumes xp normalised mod (B^n+1). The residue class [0] is represented by [B^n-1]; except when both input are ZERO. */ #if HAVE_NATIVE_mpn_rsh1add_n || HAVE_NATIVE_mpn_rsh1add_nc #if HAVE_NATIVE_mpn_rsh1add_nc cy = mpn_rsh1add_nc(rp, rp, xp, n, xp[n]); /* B^n = 1 */ hi = cy << (GMP_NUMB_BITS - 1); cy = 0; /* next update of rp[n-1] will set cy = 1 only if rp[n-1]+=hi overflows, i.e. a further increment will not overflow again. */ #else /* ! _nc */ cy = xp[n] + mpn_rsh1add_n(rp, rp, xp, n); /* B^n = 1 */ hi = (cy<<(GMP_NUMB_BITS-1))&GMP_NUMB_MASK; /* (cy&1) << ... */ cy >>= 1; /* cy = 1 only if xp[n] = 1 i.e. {xp,n} = ZERO, this implies that the rsh1add was a simple rshift: the top bit is 0. cy=1 => hi=0. */ #endif #if GMP_NAIL_BITS == 0 add_ssaaaa(cy, rp[n-1], cy, rp[n-1], 0, hi); #else cy += (hi & rp[n-1]) >> (GMP_NUMB_BITS-1); rp[n-1] ^= hi; #endif #else /* ! HAVE_NATIVE_mpn_rsh1add_n */ #if HAVE_NATIVE_mpn_add_nc cy = mpn_add_nc(rp, rp, xp, n, xp[n]); #else /* ! _nc */ cy = xp[n] + mpn_add_n(rp, rp, xp, n); /* xp[n] == 1 implies {xp,n} == ZERO */ #endif cy += (rp[0]&1); mpn_rshift(rp, rp, n, 1); ASSERT (cy <= 2); hi = (cy<<(GMP_NUMB_BITS-1))&GMP_NUMB_MASK; /* (cy&1) << ... */ cy >>= 1; /* We can have cy != 0 only if hi = 0... */ ASSERT ((rp[n-1] & GMP_NUMB_HIGHBIT) == 0); rp[n-1] |= hi; /* ... rp[n-1] + cy can not overflow, the following INCR is correct. */ #endif ASSERT (cy <= 1); /* Next increment can not overflow, read the previous comments about cy. */ ASSERT ((cy == 0) || ((rp[n-1] & GMP_NUMB_HIGHBIT) == 0)); MPN_INCR_U(rp, n, cy); /* Compute the highest half: ([(xp + xm)/2 mod (B^n-1)] - xp ) * B^n */ if (UNLIKELY (an + bn < rn)) { /* Note that in this case, the only way the result can equal zero mod B^{rn} - 1 is if one of the inputs is zero, and then the output of both the recursive calls and this CRT reconstruction is zero, not B^{rn} - 1. Which is good, since the latter representation doesn't fit in the output area.*/ cy = mpn_sub_n (rp + n, rp, xp, an + bn - n); /* FIXME: This subtraction of the high parts is not really necessary, we do it to get the carry out, and for sanity checking. */ cy = xp[n] + mpn_sub_nc (xp + an + bn - n, rp + an + bn - n, xp + an + bn - n, rn - (an + bn), cy); ASSERT (an + bn == rn - 1 || mpn_zero_p (xp + an + bn - n + 1, rn - 1 - (an + bn))); cy = mpn_sub_1 (rp, rp, an + bn, cy); ASSERT (cy == (xp + an + bn - n)[0]); } else { cy = xp[n] + mpn_sub_n (rp + n, rp, xp, n); /* cy = 1 only if {xp,n+1} is not ZERO, i.e. {rp,n} is not ZERO. DECR will affect _at most_ the lowest n limbs. */ MPN_DECR_U (rp, 2*n, cy); } #undef a0 #undef a1 #undef b0 #undef b1 #undef xp #undef sp1 } }