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/* mpn_bsqrtinv, compute r such that r^2 * y = 1 (mod 2^{b+1}).
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Contributed to the GNU project by Martin Boij (as part of perfpow.c).
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Copyright 2009, 2010, 2012, 2015 Free Software Foundation, Inc.
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This file is part of the GNU MP Library.
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The GNU MP Library is free software; you can redistribute it and/or modify
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it under the terms of either:
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* the GNU Lesser General Public License as published by the Free
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Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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or
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* the GNU General Public License as published by the Free Software
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Foundation; either version 2 of the License, or (at your option) any
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later version.
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or both in parallel, as here.
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The GNU MP Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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for more details.
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You should have received copies of the GNU General Public License and the
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GNU Lesser General Public License along with the GNU MP Library. If not,
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see https://www.gnu.org/licenses/. */
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#include "gmp.h"
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#include "gmp-impl.h"
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/* Compute r such that r^2 * y = 1 (mod 2^{b+1}).
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Return non-zero if such an integer r exists.
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Iterates
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r' <-- (3r - r^3 y) / 2
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using Hensel lifting. Since we divide by two, the Hensel lifting is
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somewhat degenerates. Therefore, we lift from 2^b to 2^{b+1}-1.
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FIXME:
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(1) Simplify to do precision book-keeping in limbs rather than bits.
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(2) Rewrite iteration as
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r' <-- r - r (r^2 y - 1) / 2
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and take advantage of zero low part of r^2 y - 1.
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(3) Use wrap-around trick.
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(4) Use a small table to get starting value.
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*/
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int
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mpn_bsqrtinv (mp_ptr rp, mp_srcptr yp, mp_bitcnt_t bnb, mp_ptr tp)
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{
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mp_ptr tp2;
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mp_size_t bn, order[GMP_LIMB_BITS + 1];
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int i, d;
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ASSERT (bnb > 0);
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bn = 1 + bnb / GMP_LIMB_BITS;
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tp2 = tp + bn;
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rp[0] = 1;
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if (bnb == 1)
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{
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if ((yp[0] & 3) != 1)
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return 0;
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}
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else
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{
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if ((yp[0] & 7) != 1)
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return 0;
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d = 0;
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for (; bnb != 2; bnb = (bnb + 2) >> 1)
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order[d++] = bnb;
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for (i = d - 1; i >= 0; i--)
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{
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bnb = order[i];
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bn = 1 + bnb / GMP_LIMB_BITS;
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mpn_sqrlo (tp, rp, bn);
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mpn_mullo_n (tp2, rp, tp, bn); /* tp2 <- rp ^ 3 */
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mpn_mul_1 (tp, rp, bn, 3);
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mpn_mullo_n (rp, yp, tp2, bn);
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#if HAVE_NATIVE_mpn_rsh1sub_n
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mpn_rsh1sub_n (rp, tp, rp, bn);
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#else
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mpn_sub_n (tp2, tp, rp, bn);
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mpn_rshift (rp, tp2, bn, 1);
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#endif
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}
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}
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return 1;
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}
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