1 /* mpn_perfect_square_p(u,usize) -- Return non-zero if U is a perfect square,
2 zero otherwise.
3
4 Copyright 1991, 1993, 1994, 1996, 1997, 2000-2002, 2005, 2012 Free Software
5 Foundation, Inc.
6
7 This file is part of the GNU MP Library.
8
9 The GNU MP Library is free software; you can redistribute it and/or modify
10 it under the terms of either:
11
12 * the GNU Lesser General Public License as published by the Free
13 Software Foundation; either version 3 of the License, or (at your
14 option) any later version.
15
16 or
17
18 * the GNU General Public License as published by the Free Software
19 Foundation; either version 2 of the License, or (at your option) any
20 later version.
21
22 or both in parallel, as here.
23
24 The GNU MP Library is distributed in the hope that it will be useful, but
25 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
26 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
27 for more details.
28
29 You should have received copies of the GNU General Public License and the
30 GNU Lesser General Public License along with the GNU MP Library. If not,
31 see https://www.gnu.org/licenses/. */
32
33 #include <stdio.h> /* for NULL */
34 #include "gmp-impl.h"
35 #include "longlong.h"
36
37 #include "perfsqr.h"
38
39
40 /* change this to "#define TRACE(x) x" for diagnostics */
41 #define TRACE(x)
42
43
44
45 /* PERFSQR_MOD_* detects non-squares using residue tests.
46
47 A macro PERFSQR_MOD_TEST is setup by gen-psqr.c in perfsqr.h. It takes
48 {up,usize} modulo a selected modulus to get a remainder r. For 32-bit or
49 64-bit limbs this modulus will be 2^24-1 or 2^48-1 using PERFSQR_MOD_34,
50 or for other limb or nail sizes a PERFSQR_PP is chosen and PERFSQR_MOD_PP
51 used. PERFSQR_PP_NORM and PERFSQR_PP_INVERTED are pre-calculated in this
52 case too.
53
54 PERFSQR_MOD_TEST then makes various calls to PERFSQR_MOD_1 or
55 PERFSQR_MOD_2 with divisors d which are factors of the modulus, and table
56 data indicating residues and non-residues modulo those divisors. The
57 table data is in 1 or 2 limbs worth of bits respectively, per the size of
58 each d.
59
60 A "modexact" style remainder is taken to reduce r modulo d.
61 PERFSQR_MOD_IDX implements this, producing an index "idx" for use with
62 the table data. Notice there's just one multiplication by a constant
63 "inv", for each d.
64
65 The modexact doesn't produce a true r%d remainder, instead idx satisfies
66 "-(idx<<PERFSQR_MOD_BITS) == r mod d". Because d is odd, this factor
67 -2^PERFSQR_MOD_BITS is a one-to-one mapping between r and idx, and is
68 accounted for by having the table data suitably permuted.
69
70 The remainder r fits within PERFSQR_MOD_BITS which is less than a limb.
71 In fact the GMP_LIMB_BITS - PERFSQR_MOD_BITS spare bits are enough to fit
72 each divisor d meaning the modexact multiply can take place entirely
73 within one limb, giving the compiler the chance to optimize it, in a way
74 that say umul_ppmm would not give.
75
76 There's no need for the divisors d to be prime, in fact gen-psqr.c makes
77 a deliberate effort to combine factors so as to reduce the number of
78 separate tests done on r. But such combining is limited to d <=
79 2*GMP_LIMB_BITS so that the table data fits in at most 2 limbs.
80
81 Alternatives:
82
83 It'd be possible to use bigger divisors d, and more than 2 limbs of table
84 data, but this doesn't look like it would be of much help to the prime
85 factors in the usual moduli 2^24-1 or 2^48-1.
86
87 The moduli 2^24-1 or 2^48-1 are nothing particularly special, they're
88 just easy to calculate (see mpn_mod_34lsub1) and have a nice set of prime
89 factors. 2^32-1 and 2^64-1 would be equally easy to calculate, but have
90 fewer prime factors.
91
92 The nails case usually ends up using mpn_mod_1, which is a lot slower
93 than mpn_mod_34lsub1. Perhaps other such special moduli could be found
94 for the nails case. Two-term things like 2^30-2^15-1 might be
95 candidates. Or at worst some on-the-fly de-nailing would allow the plain
96 2^24-1 to be used. Currently nails are too preliminary to be worried
97 about.
98
99 */
100
101 #define PERFSQR_MOD_MASK ((CNST_LIMB(1) << PERFSQR_MOD_BITS) - 1)
102
103 #define MOD34_BITS (GMP_NUMB_BITS / 4 * 3)
104 #define MOD34_MASK ((CNST_LIMB(1) << MOD34_BITS) - 1)
105
106 #define PERFSQR_MOD_34(r, up, usize) \
107 do { \
108 (r) = mpn_mod_34lsub1 (up, usize); \
109 (r) = ((r) & MOD34_MASK) + ((r) >> MOD34_BITS); \
110 } while (0)
111
112 /* FIXME: The %= here isn't good, and might destroy any savings from keeping
113 the PERFSQR_MOD_IDX stuff within a limb (rather than needing umul_ppmm).
114 Maybe a new sort of mpn_preinv_mod_1 could accept an unnormalized divisor
115 and a shift count, like mpn_preinv_divrem_1. But mod_34lsub1 is our
116 normal case, so lets not worry too much about mod_1. */
117 #define PERFSQR_MOD_PP(r, up, usize) \
118 do { \
119 if (BELOW_THRESHOLD (usize, PREINV_MOD_1_TO_MOD_1_THRESHOLD)) \
120 { \
121 (r) = mpn_preinv_mod_1 (up, usize, PERFSQR_PP_NORM, \
122 PERFSQR_PP_INVERTED); \
123 (r) %= PERFSQR_PP; \
124 } \
125 else \
126 { \
127 (r) = mpn_mod_1 (up, usize, PERFSQR_PP); \
128 } \
129 } while (0)
130
131 #define PERFSQR_MOD_IDX(idx, r, d, inv) \
132 do { \
133 mp_limb_t q; \
134 ASSERT ((r) <= PERFSQR_MOD_MASK); \
135 ASSERT ((((inv) * (d)) & PERFSQR_MOD_MASK) == 1); \
136 ASSERT (MP_LIMB_T_MAX / (d) >= PERFSQR_MOD_MASK); \
137 \
138 q = ((r) * (inv)) & PERFSQR_MOD_MASK; \
139 ASSERT (r == ((q * (d)) & PERFSQR_MOD_MASK)); \
140 (idx) = (q * (d)) >> PERFSQR_MOD_BITS; \
141 } while (0)
142
143 #define PERFSQR_MOD_1(r, d, inv, mask) \
144 do { \
145 unsigned idx; \
146 ASSERT ((d) <= GMP_LIMB_BITS); \
147 PERFSQR_MOD_IDX(idx, r, d, inv); \
148 TRACE (printf (" PERFSQR_MOD_1 d=%u r=%lu idx=%u\n", \
149 d, r%d, idx)); \
150 if ((((mask) >> idx) & 1) == 0) \
151 { \
152 TRACE (printf (" non-square\n")); \
153 return 0; \
154 } \
155 } while (0)
156
157 /* The expression "(int) idx - GMP_LIMB_BITS < 0" lets the compiler use the
158 sign bit from "idx-GMP_LIMB_BITS", which might help avoid a branch. */
159 #define PERFSQR_MOD_2(r, d, inv, mhi, mlo) \
160 do { \
161 mp_limb_t m; \
162 unsigned idx; \
163 ASSERT ((d) <= 2*GMP_LIMB_BITS); \
164 \
165 PERFSQR_MOD_IDX (idx, r, d, inv); \
166 TRACE (printf (" PERFSQR_MOD_2 d=%u r=%lu idx=%u\n", \
167 d, r%d, idx)); \
168 m = ((int) idx - GMP_LIMB_BITS < 0 ? (mlo) : (mhi)); \
169 idx %= GMP_LIMB_BITS; \
170 if (((m >> idx) & 1) == 0) \
171 { \
172 TRACE (printf (" non-square\n")); \
173 return 0; \
174 } \
175 } while (0)
176
177
178 int
mpn_perfect_square_p(mp_srcptr up,mp_size_t usize)179 mpn_perfect_square_p (mp_srcptr up, mp_size_t usize)
180 {
181 ASSERT (usize >= 1);
182
183 TRACE (gmp_printf ("mpn_perfect_square_p %Nd\n", up, usize));
184
185 /* The first test excludes 212/256 (82.8%) of the perfect square candidates
186 in O(1) time. */
187 {
188 unsigned idx = up[0] % 0x100;
189 if (((sq_res_0x100[idx / GMP_LIMB_BITS]
190 >> (idx % GMP_LIMB_BITS)) & 1) == 0)
191 return 0;
192 }
193
194 #if 0
195 /* Check that we have even multiplicity of 2, and then check that the rest is
196 a possible perfect square. Leave disabled until we can determine this
197 really is an improvement. It it is, it could completely replace the
198 simple probe above, since this should throw out more non-squares, but at
199 the expense of somewhat more cycles. */
200 {
201 mp_limb_t lo;
202 int cnt;
203 lo = up[0];
204 while (lo == 0)
205 up++, lo = up[0], usize--;
206 count_trailing_zeros (cnt, lo);
207 if ((cnt & 1) != 0)
208 return 0; /* return of not even multiplicity of 2 */
209 lo >>= cnt; /* shift down to align lowest non-zero bit */
210 lo >>= 1; /* shift away lowest non-zero bit */
211 if ((lo & 3) != 0)
212 return 0;
213 }
214 #endif
215
216
217 /* The second test uses mpn_mod_34lsub1 or mpn_mod_1 to detect non-squares
218 according to their residues modulo small primes (or powers of
219 primes). See perfsqr.h. */
220 PERFSQR_MOD_TEST (up, usize);
221
222
223 /* For the third and last test, we finally compute the square root,
224 to make sure we've really got a perfect square. */
225 {
226 mp_ptr root_ptr;
227 int res;
228 TMP_DECL;
229
230 TMP_MARK;
231 root_ptr = TMP_ALLOC_LIMBS ((usize + 1) / 2);
232
233 /* Iff mpn_sqrtrem returns zero, the square is perfect. */
234 res = ! mpn_sqrtrem (root_ptr, NULL, up, usize);
235 TMP_FREE;
236
237 return res;
238 }
239 }
240