File: aks.c

package info (click to toggle)
libmath-prime-util-perl 0.73-1
  • links: PTS, VCS
  • area: main
  • in suites: bullseye, buster, sid
  • size: 2,800 kB
  • sloc: perl: 24,676; ansic: 11,471; python: 24; makefile: 18
file content (435 lines) | stat: -rw-r--r-- 12,483 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <float.h>

/* The AKS primality algorithm for native integers.
 *
 * There are three versions here:
 *   V6         The v6 algorithm from the latest AKS paper.
 *              https://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf
 *   BORNEMANN  Improvements from Bernstein, Voloch, and a clever r/s
 *              selection from Folkmar Bornemann.  Similar to Bornemann's
 *              2003 Pari/GP implementation:
 *              https://homepage.univie.ac.at/Dietrich.Burde/pari/aks.gp
 *   BERN41     My implementation of theorem 4.1 from Bernstein's 2003 paper.
 *              https://cr.yp.to/papers/aks.pdf
 *
 * Each one is orders of magnitude faster than the previous, and by default
 * we use Bernstein 4.1 as it is by far the fastest.
 *
 * Note that AKS is very, very slow compared to other methods.  It is, however,
 * polynomial in log(N), and log-log performance graphs show nice straight
 * lines for both implementations.  However APR-CL and ECPP both start out
 * much faster and the slope will be less for any sizes of N that we're
 * interested in.
 *
 * For native 64-bit integers this is purely a coding exercise, as BPSW is
 * a million times faster and gives proven results.
 *
 *
 * When n < 2^(wordbits/2)-1, we can do a straightforward intermediate:
 *      r = (r + a * b) % n
 * If n is larger, then these are replaced with:
 *      r = addmod( r, mulmod(a, b, n), n)
 * which is a lot more work, but keeps us correct.
 *
 * Software that does polynomial convolutions followed by a modulo can be
 * very fast, but will fail when n >= (2^wordbits)/r.
 *
 * This is all much easier in GMP.
 *
 * Copyright 2012-2016, Dana Jacobsen.
 */

#define SQRTN_SHORTCUT 1

#define IMPL_V6        0    /* From the primality_v6 paper */
#define IMPL_BORNEMANN 0    /* From Bornemann's 2002 implementation */
#define IMPL_BERN41    1    /* From Bernstein's early 2003 paper */

#include "ptypes.h"
#include "aks.h"
#define FUNC_isqrt 1
#define FUNC_gcd_ui 1
#include "util.h"
#include "cache.h"
#include "mulmod.h"
#include "factor.h"

#if IMPL_BORNEMANN || IMPL_BERN41
/* We could use lgamma, but it isn't in MSVC and not in pre-C99.  The only
 * sure way to find if it is available is test compilation (ala autoconf).
 * Instead, we'll just use our own implementation.
 * See http://mrob.com/pub/ries/lanczos-gamma.html for alternates. */
static double log_gamma(double x)
{
  static const double log_sqrt_two_pi =  0.91893853320467274178;
  static const double lanczos_coef[8+1] =
    { 0.99999999999980993, 676.5203681218851, -1259.1392167224028,
      771.32342877765313, -176.61502916214059, 12.507343278686905,
      -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 };
  double base = x + 7.5, sum = 0;
  int i;
  for (i = 8; i >= 1; i--)
    sum += lanczos_coef[i] / (x + (double)i);
  sum += lanczos_coef[0];
  sum = log_sqrt_two_pi + log(sum/x) + ( (x+0.5)*log(base) - base );
  return sum;
}

/* Note: For lgammal we need logl in the above.
 * Max error drops from 2.688466e-09 to 1.818989e-12. */
#undef lgamma
#define lgamma(x) log_gamma(x)
#endif

#if IMPL_BERN41
static double log_binomial(UV n, UV k)
{
  return log_gamma(n+1) - log_gamma(k+1) - log_gamma(n-k+1);
}
static double log_bern41_binomial(UV r, UV d, UV i, UV j, UV s)
{
  return   log_binomial( 2*s,   i)
         + log_binomial( d,     i)
         + log_binomial( 2*s-i, j)
         + log_binomial( r-2-d, j);
}
static int bern41_acceptable(UV n, UV r, UV s)
{
  double scmp = ceil(sqrt( (r-1)/3.0 )) * log(n);
  UV d = (UV) (0.5 * (r-1));
  UV i = (UV) (0.475 * (r-1));
  UV j = i;
  if (d > r-2)     d = r-2;
  if (i > d)       i = d;
  if (j > (r-2-d)) j = r-2-d;
  return (log_bern41_binomial(r,d,i,j,s) >= scmp);
}
#endif

#if 0
/* Naive znorder.  Works well if limit is small.  Note arguments.  */
static UV order(UV r, UV n, UV limit) {
  UV j;
  UV t = 1;
  for (j = 1; j <= limit; j++) {
    t = mulmod(t, n, r);
    if (t == 1)
      break;
  }
  return j;
}
static void poly_print(UV* poly, UV r)
{
  int i;
  for (i = r-1; i >= 1; i--) {
    if (poly[i] != 0)
      printf("%lux^%d + ", poly[i], i);
  }
  if (poly[0] != 0) printf("%lu", poly[0]);
  printf("\n");
}
#endif

static void poly_mod_mul(UV* px, UV* py, UV* res, UV r, UV mod)
{
  UV degpx, degpy;
  UV i, j, pxi, pyj, rindex;

  /* Determine max degree of px and py */
  for (degpx = r-1; degpx > 0 && !px[degpx]; degpx--) ; /* */
  for (degpy = r-1; degpy > 0 && !py[degpy]; degpy--) ; /* */
  /* We can sum at least j values at once */
  j = (mod >= HALF_WORD) ? 0 : (UV_MAX / ((mod-1)*(mod-1)));

  if (j >= degpx || j >= degpy) {
    /* res will be written completely, so no need to set */
    for (rindex = 0; rindex < r; rindex++) {
      UV sum = 0;
      j = rindex;
      for (i = 0; i <= degpx; i++) {
        if (j <= degpy)
          sum += px[i] * py[j];
        j = (j == 0) ? r-1 : j-1;
      }
      res[rindex] = sum % mod;
    }
  } else {
    memset(res, 0, r * sizeof(UV));  /* Zero result accumulator */
    for (i = 0; i <= degpx; i++) {
      pxi = px[i];
      if (pxi == 0)  continue;
      if (mod < HALF_WORD) {
        for (j = 0; j <= degpy; j++) {
          pyj = py[j];
          rindex = i+j;   if (rindex >= r)  rindex -= r;
          res[rindex] = (res[rindex] + (pxi*pyj) ) % mod;
        }
      } else {
        for (j = 0; j <= degpy; j++) {
          pyj = py[j];
          rindex = i+j;   if (rindex >= r)  rindex -= r;
          res[rindex] = muladdmod(pxi, pyj, res[rindex], mod);
        }
      }
    }
  }
  memcpy(px, res, r * sizeof(UV)); /* put result in px */
}
static void poly_mod_sqr(UV* px, UV* res, UV r, UV mod)
{
  UV c, d, s, sum, rindex, maxpx;
  UV degree = r-1;
  int native_sqr = (mod > isqrt(UV_MAX/(2*r))) ? 0 : 1;

  memset(res, 0, r * sizeof(UV)); /* zero out sums */
  /* Discover index of last non-zero value in px */
  for (s = degree; s > 0; s--)
    if (px[s] != 0)
      break;
  maxpx = s;
  /* 1D convolution */
  for (d = 0; d <= 2*degree; d++) {
    UV *pp1, *pp2, *ppend;
    UV s_beg = (d <= degree) ? 0 : d-degree;
    UV s_end = ((d/2) <= maxpx) ? d/2 : maxpx;
    if (s_end < s_beg) continue;
    sum = 0;
    pp1 = px + s_beg;
    pp2 = px + d - s_beg;
    ppend = px + s_end;
    if (native_sqr) {
      while (pp1 < ppend)
        sum += 2 * *pp1++  *  *pp2--;
      /* Special treatment for last point */
      c = px[s_end];
      sum += (s_end*2 == d)  ?  c*c  :  2*c*px[d-s_end];
      rindex = (d < r) ? d : d-r;  /* d % r */
      res[rindex] = (res[rindex] + sum) % mod;
#if HAVE_UINT128
    } else {
      uint128_t max = ((uint128_t)1 << 127) - 1;
      uint128_t c128, sum128 = 0;

      while (pp1 < ppend) {
        c128 = ((uint128_t)*pp1++)  *  ((uint128_t)*pp2--);
        if (c128 > max) c128 %= mod;
        c128 <<= 1;
        if (c128 > max) c128 %= mod;
        sum128 += c128;
        if (sum128 > max) sum128 %= mod;
      }
      c128 = px[s_end];
      if (s_end*2 == d) {
        c128 *= c128;
      } else {
        c128 *= px[d-s_end];
        if (c128 > max) c128 %= mod;
        c128 <<= 1;
      }
      if (c128 > max) c128 %= mod;
      sum128 += c128;
      if (sum128 > max) sum128 %= mod;
      rindex = (d < r) ? d : d-r;  /* d % r */
      res[rindex] = ((uint128_t)res[rindex] + sum128) % mod;
#else
    } else {
      while (pp1 < ppend) {
        UV p1 = *pp1++;
        UV p2 = *pp2--;
        sum = addmod(sum, mulmod(2, mulmod(p1, p2, mod), mod), mod);
      }
      c = px[s_end];
      if (s_end*2 == d)
        sum = addmod(sum, sqrmod(c, mod), mod);
      else
        sum = addmod(sum, mulmod(2, mulmod(c, px[d-s_end], mod), mod), mod);
      rindex = (d < r) ? d : d-r;  /* d % r */
      res[rindex] = addmod(res[rindex], sum, mod);
#endif
    }
  }
  memcpy(px, res, r * sizeof(UV)); /* put result in px */
}

static UV* poly_mod_pow(UV* pn, UV power, UV r, UV mod)
{
  UV *res, *temp;

  Newz(0, res, r, UV);
  New(0, temp, r, UV);
  res[0] = 1;

  while (power) {
    if (power & 1)  poly_mod_mul(res, pn, temp, r, mod);
    power >>= 1;
    if (power)      poly_mod_sqr(pn, temp, r, mod);
  }
  Safefree(temp);
  return res;
}

static int test_anr(UV a, UV n, UV r)
{
  UV* pn;
  UV* res;
  UV i;
  int retval = 1;

  Newz(0, pn, r, UV);
  a %= r;
  pn[0] = a;
  pn[1] = 1;
  res = poly_mod_pow(pn, n, r, n);
  res[n % r] = addmod(res[n % r], n - 1, n);
  res[0]     = addmod(res[0],     n - a, n);

  for (i = 0; i < r; i++)
    if (res[i] != 0)
      retval = 0;
  Safefree(res);
  Safefree(pn);
  return retval;
}

/*
 * Avanzi and Mihǎilescu, 2007
 * http://www.uni-math.gwdg.de/preda/mihailescu-papers/ouraks3.pdf
 * "As a consequence, one cannot expect the present variants of AKS to
 *  compete with the earlier primality proving methods like ECPP and
 *  cyclotomy." - conclusion regarding memory consumption
 */
int is_aks_prime(UV n)
{
  UV r, s, a, starta = 1;

  if (n < 2)
    return 0;
  if (n == 2)
    return 1;

  if (is_power(n, 0))
    return 0;

  if (n > 11 && ( !(n%2) || !(n%3) || !(n%5) || !(n%7) || !(n%11) )) return 0;
  /* if (!is_prob_prime(n)) return 0; */

#if IMPL_V6
  {
    UV sqrtn = isqrt(n);
    double log2n = log(n) / log(2);   /* C99 has a log2() function */
    UV limit = (UV) floor(log2n * log2n);

    MPUverbose(1, "# aks limit is %lu\n", (unsigned long) limit);

    for (r = 2; r < n; r++) {
      if ((n % r) == 0)
        return 0;
#if SQRTN_SHORTCUT
      if (r > sqrtn)
        return 1;
#endif
      if (znorder(n, r) > limit)
        break;
    }

    if (r >= n)
      return 1;

    s = (UV) floor(sqrt(r-1) * log2n);
  }
#endif
#if IMPL_BORNEMANN
  {
    UV fac[MPU_MAX_FACTORS+1];
    UV slim;
    double c1, c2, x;
    double const t = 48;
    double const t1 = (1.0/((t+1)*log(t+1)-t*log(t)));
    double const dlogn = log(n);
    r = next_prime( (UV) (t1*t1 * dlogn*dlogn) );
    while (!is_primitive_root(n,r,1))
      r = next_prime(r);

    slim = (UV) (2*t*(r-1));
    c1 = lgamma(r-1);
    c2 = dlogn * floor(sqrt(r));
    { /* Binary search for first s in [1,slim] where x >= 0 */
      UV i = 1;
      UV j = slim;
      while (i < j) {
        s = i + (j-i)/2;
        x = (lgamma(r-1+s) - c1 - lgamma(s+1)) / c2 - 1.0;
        if (x < 0)  i = s+1;
        else        j = s;
      }
      s = i-1;
    }
    s = (s+3) >> 1;
    /* Bornemann checks factors up to (s-1)^2, we check to max(r,s) */
    /* slim = (s-1)*(s-1); */
    slim = (r > s) ? r : s;
    MPUverbose(2, "# aks trial to %lu\n", slim);
    if (trial_factor(n, fac, 2, slim) > 1)
      return 0;
    if (slim >= HALF_WORD || (slim*slim) >= n)
      return 1;
  }
#endif
#if IMPL_BERN41
  {
    UV slim, fac[MPU_MAX_FACTORS+1];
    double const log2n = log(n) / log(2);
    /* Tuning: Initial 'r' selection.  Search limit for 's'. */
    double const r0 = ((log2n > 32) ? 0.010 : 0.003) * log2n * log2n;
    UV const rmult  =  (log2n > 32) ? 6    : 30;

    r = next_prime(r0 < 2 ? 2 : (UV)r0);  /* r must be at least 3 */
    while ( !is_primitive_root(n,r,1) || !bern41_acceptable(n,r,rmult*(r-1)) )
      r = next_prime(r);

    { /* Binary search for first s in [1,slim] where conditions met */
      UV bi = 1;
      UV bj = rmult * (r-1);
      while (bi < bj) {
        s = bi + (bj-bi)/2;
        if (!bern41_acceptable(n, r, s))  bi = s+1;
        else                              bj = s;
      }
      s = bj;
      if (!bern41_acceptable(n, r, s)) croak("AKS: bad s selected");
      /* S goes from 2 to s+1 */
      starta = 2;
      s = s+1;
    }
    /* Check divisibility to s * (s-1) to cover both gcd conditions */
    slim = s * (s-1);
    MPUverbose(2, "# aks trial to %lu\n", (unsigned long)slim);
    if (trial_factor(n, fac, 2, slim) > 1)
      return 0;
    if (slim >= HALF_WORD || (slim*slim) >= n)
      return 1;
    /* Check b^(n-1) = 1 mod n for b in [2..s] */
    for (a = 2; a <= s; a++) {
      if (powmod(a, n-1, n) != 1)
        return 0;
    }
  }
#endif

  MPUverbose(1, "# aks r = %lu  s = %lu\n", (unsigned long) r, (unsigned long) s);

  /* Almost every composite will get recognized by the first test.
   * However, we need to run 's' tests to have the result proven for all n
   * based on the theorems we have available at this time. */
  for (a = starta; a <= s; a++) {
    if (! test_anr(a, n, r) )
      return 0;
    MPUverbose(2, ".");
  }
  MPUverbose(2, "\n");
  return 1;
}