1 /* mpn_perfect_square_p(u,usize) -- Return non-zero if U is a perfect square,
4 Copyright 1991, 1993, 1994, 1996, 1997, 2000, 2001, 2002, 2005 Free Software
7 This file is part of the GNU MP Library.
9 The GNU MP Library is free software; you can redistribute it and/or modify
10 it under the terms of the GNU Lesser General Public License as published by
11 the Free Software Foundation; either version 3 of the License, or (at your
12 option) any later version.
14 The GNU MP Library is distributed in the hope that it will be useful, but
15 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
16 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
17 License for more details.
19 You should have received a copy of the GNU Lesser General Public License
20 along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */
22 #include <stdio.h> /* for NULL */
30 /* change this to "#define TRACE(x) x" for diagnostics */
35 /* PERFSQR_MOD_* detects non-squares using residue tests.
37 A macro PERFSQR_MOD_TEST is setup by gen-psqr.c in perfsqr.h. It takes
38 {up,usize} modulo a selected modulus to get a remainder r. For 32-bit or
39 64-bit limbs this modulus will be 2^24-1 or 2^48-1 using PERFSQR_MOD_34,
40 or for other limb or nail sizes a PERFSQR_PP is chosen and PERFSQR_MOD_PP
41 used. PERFSQR_PP_NORM and PERFSQR_PP_INVERTED are pre-calculated in this
44 PERFSQR_MOD_TEST then makes various calls to PERFSQR_MOD_1 or
45 PERFSQR_MOD_2 with divisors d which are factors of the modulus, and table
46 data indicating residues and non-residues modulo those divisors. The
47 table data is in 1 or 2 limbs worth of bits respectively, per the size of
50 A "modexact" style remainder is taken to reduce r modulo d.
51 PERFSQR_MOD_IDX implements this, producing an index "idx" for use with
52 the table data. Notice there's just one multiplication by a constant
55 The modexact doesn't produce a true r%d remainder, instead idx satisfies
56 "-(idx<<PERFSQR_MOD_BITS) == r mod d". Because d is odd, this factor
57 -2^PERFSQR_MOD_BITS is a one-to-one mapping between r and idx, and is
58 accounted for by having the table data suitably permuted.
60 The remainder r fits within PERFSQR_MOD_BITS which is less than a limb.
61 In fact the GMP_LIMB_BITS - PERFSQR_MOD_BITS spare bits are enough to fit
62 each divisor d meaning the modexact multiply can take place entirely
63 within one limb, giving the compiler the chance to optimize it, in a way
64 that say umul_ppmm would not give.
66 There's no need for the divisors d to be prime, in fact gen-psqr.c makes
67 a deliberate effort to combine factors so as to reduce the number of
68 separate tests done on r. But such combining is limited to d <=
69 2*GMP_LIMB_BITS so that the table data fits in at most 2 limbs.
73 It'd be possible to use bigger divisors d, and more than 2 limbs of table
74 data, but this doesn't look like it would be of much help to the prime
75 factors in the usual moduli 2^24-1 or 2^48-1.
77 The moduli 2^24-1 or 2^48-1 are nothing particularly special, they're
78 just easy to calculate (see mpn_mod_34lsub1) and have a nice set of prime
79 factors. 2^32-1 and 2^64-1 would be equally easy to calculate, but have
82 The nails case usually ends up using mpn_mod_1, which is a lot slower
83 than mpn_mod_34lsub1. Perhaps other such special moduli could be found
84 for the nails case. Two-term things like 2^30-2^15-1 might be
85 candidates. Or at worst some on-the-fly de-nailing would allow the plain
86 2^24-1 to be used. Currently nails are too preliminary to be worried
91 #define PERFSQR_MOD_MASK ((CNST_LIMB(1) << PERFSQR_MOD_BITS) - 1)
93 #define MOD34_BITS (GMP_NUMB_BITS / 4 * 3)
94 #define MOD34_MASK ((CNST_LIMB(1) << MOD34_BITS) - 1)
96 #define PERFSQR_MOD_34(r, up, usize) \
98 (r) = mpn_mod_34lsub1 (up, usize); \
99 (r) = ((r) & MOD34_MASK) + ((r) >> MOD34_BITS); \
102 /* FIXME: The %= here isn't good, and might destroy any savings from keeping
103 the PERFSQR_MOD_IDX stuff within a limb (rather than needing umul_ppmm).
104 Maybe a new sort of mpn_preinv_mod_1 could accept an unnormalized divisor
105 and a shift count, like mpn_preinv_divrem_1. But mod_34lsub1 is our
106 normal case, so lets not worry too much about mod_1. */
107 #define PERFSQR_MOD_PP(r, up, usize) \
109 if (BELOW_THRESHOLD (usize, PREINV_MOD_1_TO_MOD_1_THRESHOLD)) \
111 (r) = mpn_preinv_mod_1 (up, usize, PERFSQR_PP_NORM, \
112 PERFSQR_PP_INVERTED); \
117 (r) = mpn_mod_1 (up, usize, PERFSQR_PP); \
121 #define PERFSQR_MOD_IDX(idx, r, d, inv) \
124 ASSERT ((r) <= PERFSQR_MOD_MASK); \
125 ASSERT ((((inv) * (d)) & PERFSQR_MOD_MASK) == 1); \
126 ASSERT (MP_LIMB_T_MAX / (d) >= PERFSQR_MOD_MASK); \
128 q = ((r) * (inv)) & PERFSQR_MOD_MASK; \
129 ASSERT (r == ((q * (d)) & PERFSQR_MOD_MASK)); \
130 (idx) = (q * (d)) >> PERFSQR_MOD_BITS; \
133 #define PERFSQR_MOD_1(r, d, inv, mask) \
136 ASSERT ((d) <= GMP_LIMB_BITS); \
137 PERFSQR_MOD_IDX(idx, r, d, inv); \
138 TRACE (printf (" PERFSQR_MOD_1 d=%u r=%lu idx=%u\n", \
140 if ((((mask) >> idx) & 1) == 0) \
142 TRACE (printf (" non-square\n")); \
147 /* The expression "(int) idx - GMP_LIMB_BITS < 0" lets the compiler use the
148 sign bit from "idx-GMP_LIMB_BITS", which might help avoid a branch. */
149 #define PERFSQR_MOD_2(r, d, inv, mhi, mlo) \
153 ASSERT ((d) <= 2*GMP_LIMB_BITS); \
155 PERFSQR_MOD_IDX (idx, r, d, inv); \
156 TRACE (printf (" PERFSQR_MOD_2 d=%u r=%lu idx=%u\n", \
158 m = ((int) idx - GMP_LIMB_BITS < 0 ? (mlo) : (mhi)); \
159 idx %= GMP_LIMB_BITS; \
160 if (((m >> idx) & 1) == 0) \
162 TRACE (printf (" non-square\n")); \
169 mpn_perfect_square_p (mp_srcptr up, mp_size_t usize)
173 TRACE (gmp_printf ("mpn_perfect_square_p %Nd\n", up, usize));
175 /* The first test excludes 212/256 (82.8%) of the perfect square candidates
178 unsigned idx = up[0] % 0x100;
179 if (((sq_res_0x100[idx / GMP_LIMB_BITS]
180 >> (idx % GMP_LIMB_BITS)) & 1) == 0)
185 /* Check that we have even multiplicity of 2, and then check that the rest is
186 a possible perfect square. Leave disabled until we can determine this
187 really is an improvement. It it is, it could completely replace the
188 simple probe above, since this should through out more non-squares, but at
189 the expense of somewhat more cycles. */
195 up++, lo = up[0], usize--;
196 count_trailing_zeros (cnt, lo);
198 return 0; /* return of not even multiplicity of 2 */
199 lo >>= cnt; /* shift down to align lowest non-zero bit */
200 lo >>= 1; /* shift away lowest non-zero bit */
207 /* The second test uses mpn_mod_34lsub1 or mpn_mod_1 to detect non-squares
208 according to their residues modulo small primes (or powers of
209 primes). See perfsqr.h. */
210 PERFSQR_MOD_TEST (up, usize);
213 /* For the third and last test, we finally compute the square root,
214 to make sure we've really got a perfect square. */
221 root_ptr = TMP_ALLOC_LIMBS ((usize + 1) / 2);
223 /* Iff mpn_sqrtrem returns zero, the square is perfect. */
224 res = ! mpn_sqrtrem (root_ptr, NULL, up, usize);