[10af64] | 1 | // -*- c++ -*- |
---|
| 2 | //***************************************************************************** |
---|
[51615d6] | 3 | /** @file cf_map_ext.cc |
---|
[10af64] | 4 | * |
---|
[806c18] | 5 | * @author Martin Lee |
---|
[10af64] | 6 | * @date 16.11.2009 |
---|
| 7 | * |
---|
| 8 | * This file implements functions to map between extensions of finite fields |
---|
| 9 | * |
---|
| 10 | * @par Copyright: |
---|
| 11 | * (c) by The SINGULAR Team, see LICENSE file |
---|
| 12 | * |
---|
[806c18] | 13 | * @internal |
---|
[10af64] | 14 | * @version \$Id$ |
---|
| 15 | * |
---|
| 16 | **/ |
---|
| 17 | //***************************************************************************** |
---|
| 18 | |
---|
[e4fe2b] | 19 | #include "config.h" |
---|
[10af64] | 20 | |
---|
[650f2d8] | 21 | #include "cf_assert.h" |
---|
[10af64] | 22 | #include "debug.h" |
---|
| 23 | |
---|
| 24 | #include "canonicalform.h" |
---|
[9c115e1] | 25 | #include "cf_util.h" |
---|
[d990001] | 26 | #include "imm.h" |
---|
| 27 | #include "cf_iter.h" |
---|
[10af64] | 28 | |
---|
| 29 | #ifdef HAVE_NTL |
---|
| 30 | #include <NTL/ZZ_pEXFactoring.h> |
---|
| 31 | #include "NTLconvert.h" |
---|
| 32 | #endif |
---|
| 33 | |
---|
[0d020e] | 34 | // cyclotomoic polys: |
---|
[51615d6] | 35 | #include "cf_cyclo.h" |
---|
[0d020e] | 36 | |
---|
[963057] | 37 | #include "cf_map_ext.h" |
---|
| 38 | |
---|
[10af64] | 39 | /// helper function |
---|
[806c18] | 40 | int findItem (const CFList& list, const CanonicalForm& item) |
---|
[10af64] | 41 | { |
---|
[806c18] | 42 | int result= 1; |
---|
| 43 | for (CFListIterator i= list; i.hasItem(); i++, result++) |
---|
[10af64] | 44 | { |
---|
| 45 | if (i.getItem() == item) |
---|
| 46 | return result; |
---|
| 47 | } |
---|
| 48 | return 0; |
---|
| 49 | } |
---|
| 50 | |
---|
| 51 | /// helper function |
---|
[806c18] | 52 | CanonicalForm getItem (const CFList& list, const int& pos) |
---|
[10af64] | 53 | { |
---|
| 54 | int j= 1; |
---|
[c1b9927] | 55 | if ((pos > 0) && (pos <= list.length())) |
---|
[10af64] | 56 | { |
---|
[c1b9927] | 57 | for (CFListIterator i= list; j <= pos; i++, j++) |
---|
| 58 | { |
---|
| 59 | if (j == pos) |
---|
| 60 | return i.getItem(); |
---|
| 61 | } |
---|
[10af64] | 62 | } |
---|
[c1b9927] | 63 | return 0; |
---|
[806c18] | 64 | } |
---|
[10af64] | 65 | |
---|
[d990001] | 66 | #ifdef HAVE_NTL |
---|
[806c18] | 67 | /// \f$ F_{p} (\alpha ) \subset F_{p}(\beta ) \f$ and \f$ \alpha \f$ is a |
---|
| 68 | /// primitive element, returns the image of \f$ \alpha \f$ |
---|
| 69 | static inline |
---|
| 70 | CanonicalForm mapUp (const Variable& alpha, const Variable& beta) |
---|
[10af64] | 71 | { |
---|
| 72 | int p= getCharacteristic (); |
---|
| 73 | zz_p::init (p); |
---|
[806c18] | 74 | zz_pX NTL_mipo= convertFacCF2NTLzzpX (getMipo (beta)); |
---|
[10af64] | 75 | zz_pE::init (NTL_mipo); |
---|
| 76 | zz_pEX NTL_alpha_mipo= convertFacCF2NTLzz_pEX (getMipo(alpha), NTL_mipo); |
---|
| 77 | zz_pE root= FindRoot (NTL_alpha_mipo); |
---|
| 78 | return convertNTLzzpE2CF (root, beta); |
---|
| 79 | } |
---|
| 80 | |
---|
[d990001] | 81 | #endif |
---|
| 82 | |
---|
[806c18] | 83 | /// the CanonicalForm G is the output of map_up, returns F considered as an |
---|
[10af64] | 84 | /// element over \f$ F_{p}(\alpha ) \f$, WARNING: make sure coefficients of F |
---|
| 85 | /// are really elements of a subfield of \f$ F_{p}(\beta ) \f$ which is |
---|
[806c18] | 86 | /// isomorphic to \f$ F_{p}(\alpha ) \f$ |
---|
[51615d6] | 87 | static inline |
---|
[806c18] | 88 | CanonicalForm |
---|
[10af64] | 89 | mapDown (const CanonicalForm& F, const Variable& alpha, const |
---|
[806c18] | 90 | CanonicalForm& G, CFList& source, CFList& dest) |
---|
| 91 | { |
---|
[10af64] | 92 | CanonicalForm buf, buf2; |
---|
| 93 | int counter= 0; |
---|
| 94 | int pos; |
---|
| 95 | int p= getCharacteristic(); |
---|
| 96 | int d= degree(getMipo(alpha)); |
---|
[9c115e1] | 97 | int bound= ipower(p, d); |
---|
[10af64] | 98 | CanonicalForm result= 0; |
---|
| 99 | CanonicalForm remainder; |
---|
| 100 | CanonicalForm alpha_power; |
---|
| 101 | if (degree(F) == 0) return F; |
---|
[806c18] | 102 | if (F.level() < 0 && F.isUnivariate()) |
---|
[10af64] | 103 | { |
---|
| 104 | buf= F; |
---|
| 105 | remainder= mod (buf, G); |
---|
[806c18] | 106 | ASSERT (remainder.isZero(), "alpha is not primitive"); |
---|
[10af64] | 107 | pos= findItem (source, buf); |
---|
| 108 | if (pos == 0) |
---|
| 109 | source.append (buf); |
---|
| 110 | buf2= buf; |
---|
[806c18] | 111 | while (degree (buf) != 0 && counter < bound) |
---|
[10af64] | 112 | { |
---|
| 113 | buf /= G; |
---|
| 114 | counter++; |
---|
| 115 | if (buf == buf2) break; |
---|
[806c18] | 116 | } |
---|
| 117 | ASSERT (counter >= bound, "alpha is not primitive"); |
---|
| 118 | if (pos == 0) |
---|
[10af64] | 119 | { |
---|
[55608a7] | 120 | alpha_power= power (alpha, counter); |
---|
[10af64] | 121 | dest.append (alpha_power); |
---|
| 122 | } |
---|
| 123 | else |
---|
[806c18] | 124 | alpha_power= getItem (dest, pos); |
---|
[618da5] | 125 | result = alpha_power; |
---|
[10af64] | 126 | return result; |
---|
| 127 | } |
---|
[806c18] | 128 | else |
---|
[10af64] | 129 | { |
---|
[806c18] | 130 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 131 | { |
---|
| 132 | buf= mapDown (i.coeff(), alpha, G, source, dest); |
---|
| 133 | result += buf*power(F.mvar(), i.exp()); |
---|
| 134 | } |
---|
| 135 | return result; |
---|
| 136 | } |
---|
| 137 | } |
---|
| 138 | |
---|
| 139 | /// helper function |
---|
| 140 | static inline |
---|
| 141 | CanonicalForm GF2FalphaHelper (const CanonicalForm& F, const Variable& alpha) |
---|
| 142 | { |
---|
[55608a7] | 143 | if (F.isZero()) |
---|
| 144 | return 0; |
---|
[10af64] | 145 | int exp; |
---|
| 146 | CanonicalForm result= 0; |
---|
| 147 | InternalCF* buf; |
---|
[806c18] | 148 | if (F.inBaseDomain()) |
---|
[10af64] | 149 | { |
---|
| 150 | if (F.isOne()) return 1; |
---|
| 151 | buf= F.getval(); |
---|
| 152 | exp= imm2int(buf); |
---|
| 153 | result= power (alpha, exp).mapinto(); |
---|
| 154 | return result; |
---|
[806c18] | 155 | } |
---|
[10af64] | 156 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
| 157 | result += GF2FalphaHelper (i.coeff(), alpha)*power (F.mvar(), i.exp()); |
---|
| 158 | return result; |
---|
| 159 | } |
---|
| 160 | |
---|
[806c18] | 161 | /// changes representation by primitive element to representation by residue |
---|
| 162 | /// classes modulo a Conway polynomial |
---|
| 163 | CanonicalForm GF2FalphaRep (const CanonicalForm& F, const Variable& alpha) |
---|
[10af64] | 164 | { |
---|
| 165 | Variable beta= rootOf (gf_mipo); |
---|
| 166 | return GF2FalphaHelper (F, beta) (alpha, beta); |
---|
| 167 | } |
---|
| 168 | |
---|
[806c18] | 169 | /// change representation by residue classes modulo a Conway polynomial |
---|
[10af64] | 170 | /// to representation by primitive element |
---|
[806c18] | 171 | CanonicalForm Falpha2GFRep (const CanonicalForm& F) |
---|
[10af64] | 172 | { |
---|
| 173 | CanonicalForm result= 0; |
---|
| 174 | InternalCF* buf; |
---|
| 175 | |
---|
[806c18] | 176 | if (F.inCoeffDomain()) |
---|
[10af64] | 177 | { |
---|
| 178 | if (F.inBaseDomain()) |
---|
| 179 | return F.mapinto(); |
---|
[806c18] | 180 | else |
---|
[10af64] | 181 | { |
---|
[806c18] | 182 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 183 | { |
---|
| 184 | buf= int2imm_gf (i.exp()); |
---|
| 185 | result += i.coeff().mapinto()*CanonicalForm (buf); |
---|
| 186 | } |
---|
| 187 | } |
---|
| 188 | return result; |
---|
[806c18] | 189 | } |
---|
| 190 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 191 | result += Falpha2GFRep (i.coeff())*power (F.mvar(), i.exp()); |
---|
| 192 | return result; |
---|
| 193 | } |
---|
| 194 | |
---|
| 195 | /// GF_map_up helper |
---|
| 196 | static inline |
---|
[806c18] | 197 | CanonicalForm GFPowUp (const CanonicalForm & F, int k) |
---|
[10af64] | 198 | { |
---|
| 199 | if (F.isOne()) return F; |
---|
| 200 | CanonicalForm result= 0; |
---|
[806c18] | 201 | if (F.inBaseDomain()) |
---|
[10af64] | 202 | return power(F, k); |
---|
[806c18] | 203 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 204 | result += GFPowUp (i.coeff(), k)*power (F.mvar(), i.exp()); |
---|
| 205 | return result; |
---|
| 206 | } |
---|
| 207 | |
---|
[806c18] | 208 | /// maps a polynomial over \f$ GF(p^{k}) \f$ to a polynomial over |
---|
[10af64] | 209 | /// \f$ GF(p^{d}) \f$ , d needs to be a multiple of k |
---|
[806c18] | 210 | CanonicalForm GFMapUp (const CanonicalForm & F, int k) |
---|
| 211 | { |
---|
[10af64] | 212 | int d= getGFDegree(); |
---|
| 213 | ASSERT (d%k == 0, "multiple of GF degree expected"); |
---|
| 214 | int p= getCharacteristic(); |
---|
[9c115e1] | 215 | int ext_field_size= ipower (p, d); |
---|
| 216 | int field_size= ipower ( p, k); |
---|
[10af64] | 217 | int diff= (ext_field_size - 1)/(field_size - 1); |
---|
| 218 | return GFPowUp (F, diff); |
---|
| 219 | } |
---|
| 220 | |
---|
| 221 | /// GFMapDown helper |
---|
| 222 | static inline |
---|
[806c18] | 223 | CanonicalForm GFPowDown (const CanonicalForm & F, int k) |
---|
[10af64] | 224 | { |
---|
| 225 | if (F.isOne()) return F; |
---|
| 226 | CanonicalForm result= 0; |
---|
| 227 | int exp; |
---|
| 228 | InternalCF* buf; |
---|
[806c18] | 229 | if (F.inBaseDomain()) |
---|
[10af64] | 230 | { |
---|
| 231 | buf= F.getval(); |
---|
| 232 | exp= imm2int (buf); |
---|
| 233 | if ((exp % k) == 0) |
---|
| 234 | exp= exp/k; |
---|
[806c18] | 235 | else |
---|
[10af64] | 236 | return -1; |
---|
| 237 | |
---|
| 238 | buf= int2imm_gf (exp); |
---|
| 239 | return CanonicalForm (buf); |
---|
[806c18] | 240 | } |
---|
| 241 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 242 | result += GFPowDown (i.coeff(), k)*power (F.mvar(), i.exp()); |
---|
| 243 | return result; |
---|
| 244 | } |
---|
| 245 | |
---|
[806c18] | 246 | /// maps a polynomial over \f$ GF(p^{d}) \f$ to a polynomial over |
---|
[10af64] | 247 | /// \f$ GF(p^{k})\f$ , d needs to be a multiple of k |
---|
[806c18] | 248 | CanonicalForm GFMapDown (const CanonicalForm & F, int k) |
---|
[10af64] | 249 | { |
---|
| 250 | int d= getGFDegree(); |
---|
| 251 | ASSERT (d % k == 0, "multiple of GF degree expected"); |
---|
| 252 | int p= getCharacteristic(); |
---|
[9c115e1] | 253 | int ext_field_size= ipower (p, d); |
---|
| 254 | int field_size= ipower ( p, k); |
---|
[10af64] | 255 | int diff= (ext_field_size - 1)/(field_size - 1); |
---|
| 256 | return GFPowDown (F, diff); |
---|
| 257 | } |
---|
| 258 | |
---|
[806c18] | 259 | static inline |
---|
| 260 | CanonicalForm mapUp (const CanonicalForm& F, const CanonicalForm& G, |
---|
| 261 | const Variable& alpha, const CanonicalForm& H, |
---|
[10af64] | 262 | CFList& source, CFList& dest) |
---|
[806c18] | 263 | { |
---|
[10af64] | 264 | CanonicalForm buf, buf2; |
---|
| 265 | int counter= 0; |
---|
| 266 | int pos; |
---|
| 267 | int p= getCharacteristic(); |
---|
| 268 | int d= degree (getMipo(alpha)); |
---|
[9c115e1] | 269 | int bound= ipower(p, d); |
---|
[10af64] | 270 | CanonicalForm result= 0; |
---|
| 271 | CanonicalForm remainder; |
---|
| 272 | CanonicalForm H_power; |
---|
| 273 | if (degree(F) <= 0) return F; |
---|
[806c18] | 274 | if (F.level() < 0 && F.isUnivariate()) |
---|
[10af64] | 275 | { |
---|
| 276 | buf= F; |
---|
| 277 | remainder= mod (buf, G); |
---|
[806c18] | 278 | ASSERT (remainder.isZero(), "alpha is not primitive"); |
---|
[10af64] | 279 | pos= findItem (source, buf); |
---|
| 280 | if (pos == 0) |
---|
| 281 | source.append (buf); |
---|
| 282 | buf2= buf; |
---|
[806c18] | 283 | while (degree (buf) != 0 && counter < bound) |
---|
[10af64] | 284 | { |
---|
| 285 | buf /= G; |
---|
| 286 | counter++; |
---|
| 287 | if (buf == buf2) break; |
---|
[806c18] | 288 | } |
---|
| 289 | ASSERT (counter >= bound, "alpha is not primitive"); |
---|
| 290 | if (pos == 0) |
---|
[10af64] | 291 | { |
---|
[618da5] | 292 | H_power= buf*power (H, counter); |
---|
[10af64] | 293 | dest.append (H_power); |
---|
| 294 | } |
---|
| 295 | else |
---|
[806c18] | 296 | H_power= getItem (dest, pos); |
---|
[618da5] | 297 | result = H_power; |
---|
[10af64] | 298 | return result; |
---|
| 299 | } |
---|
[806c18] | 300 | else |
---|
[10af64] | 301 | { |
---|
[806c18] | 302 | for (CFIterator i= F; i.hasTerms(); i++) |
---|
[10af64] | 303 | { |
---|
| 304 | buf= mapUp (i.coeff(), G, alpha, H, source, dest); |
---|
| 305 | result += buf*power(F.mvar(), i.exp()); |
---|
| 306 | } |
---|
| 307 | return result; |
---|
| 308 | } |
---|
| 309 | } |
---|
| 310 | |
---|
[d990001] | 311 | #ifdef HAVE_NTL |
---|
[806c18] | 312 | /// determine a primitive element of \f$ F_{p} (\alpha ) \f$, |
---|
| 313 | /// \f$ \beta \f$ is a primitive element of a field which is isomorphic to |
---|
[10af64] | 314 | /// \f$ F_{p}(\alpha ) \f$ |
---|
[806c18] | 315 | CanonicalForm |
---|
| 316 | primitiveElement (const Variable& alpha, Variable& beta, bool fail) |
---|
[10af64] | 317 | { |
---|
| 318 | bool primitive= false; |
---|
| 319 | fail= false; |
---|
| 320 | primitive= isPrimitive (alpha, fail); |
---|
| 321 | if (fail) |
---|
| 322 | return 0; |
---|
[806c18] | 323 | if (primitive) |
---|
[10af64] | 324 | { |
---|
| 325 | beta= alpha; |
---|
| 326 | return alpha; |
---|
| 327 | } |
---|
[806c18] | 328 | CanonicalForm mipo= getMipo (alpha); |
---|
[10af64] | 329 | int d= degree (mipo); |
---|
| 330 | int p= getCharacteristic (); |
---|
| 331 | zz_p::init (p); |
---|
| 332 | zz_pX NTL_mipo; |
---|
| 333 | CanonicalForm mipo2; |
---|
| 334 | primitive= false; |
---|
| 335 | fail= false; |
---|
| 336 | do |
---|
| 337 | { |
---|
[806c18] | 338 | BuildIrred (NTL_mipo, d); |
---|
[10af64] | 339 | mipo2= convertNTLzzpX2CF (NTL_mipo, Variable (1)); |
---|
| 340 | beta= rootOf (mipo2); |
---|
| 341 | primitive= isPrimitive (beta, fail); |
---|
| 342 | if (primitive) |
---|
[806c18] | 343 | break; |
---|
[10af64] | 344 | if (fail) |
---|
| 345 | return 0; |
---|
| 346 | } while (1); |
---|
[08daea] | 347 | zz_pX alpha_mipo= convertFacCF2NTLzzpX (mipo); |
---|
| 348 | zz_pE::init (alpha_mipo); |
---|
| 349 | zz_pEX NTL_beta_mipo= to_zz_pEX (NTL_mipo); |
---|
| 350 | zz_pE root= FindRoot (NTL_beta_mipo); |
---|
[10af64] | 351 | return convertNTLzzpE2CF (root, alpha); |
---|
| 352 | } |
---|
[d990001] | 353 | #endif |
---|
[10af64] | 354 | |
---|
| 355 | CanonicalForm |
---|
| 356 | mapDown (const CanonicalForm& F, const CanonicalForm& prim_elem, const |
---|
[806c18] | 357 | CanonicalForm& im_prim_elem, const Variable& alpha, CFList& source, |
---|
| 358 | CFList& dest) |
---|
| 359 | { |
---|
[10af64] | 360 | return mapUp (F, im_prim_elem, alpha, prim_elem, dest, source); |
---|
| 361 | } |
---|
| 362 | |
---|
| 363 | CanonicalForm |
---|
[5b8726d] | 364 | mapUp (const CanonicalForm& F, const Variable& alpha, const Variable& /*beta*/, |
---|
[806c18] | 365 | const CanonicalForm& prim_elem, const CanonicalForm& im_prim_elem, |
---|
| 366 | CFList& source, CFList& dest) |
---|
[10af64] | 367 | { |
---|
[806c18] | 368 | if (prim_elem == alpha) |
---|
[10af64] | 369 | return F (im_prim_elem, alpha); |
---|
| 370 | return mapUp (F, prim_elem, alpha, im_prim_elem, source, dest); |
---|
| 371 | } |
---|
| 372 | |
---|
[d990001] | 373 | #ifdef HAVE_NTL |
---|
[806c18] | 374 | CanonicalForm |
---|
[618da5] | 375 | mapPrimElem (const CanonicalForm& primElem, const Variable& alpha, |
---|
[51615d6] | 376 | const Variable& beta) |
---|
[10af64] | 377 | { |
---|
[618da5] | 378 | if (primElem == alpha) |
---|
[10af64] | 379 | return mapUp (alpha, beta); |
---|
| 380 | else |
---|
| 381 | { |
---|
[618da5] | 382 | CanonicalForm primElemMipo= findMinPoly (primElem, alpha); |
---|
| 383 | int p= getCharacteristic (); |
---|
| 384 | zz_p::init (p); |
---|
[c1b9927] | 385 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (beta)); |
---|
[618da5] | 386 | zz_pE::init (NTLMipo); |
---|
| 387 | zz_pEX NTLPrimElemMipo= convertFacCF2NTLzz_pEX (primElemMipo, NTLMipo); |
---|
| 388 | zz_pE root= FindRoot (NTLPrimElemMipo); |
---|
| 389 | return convertNTLzzpE2CF (root, beta); |
---|
[806c18] | 390 | } |
---|
[10af64] | 391 | } |
---|
| 392 | |
---|
[55608a7] | 393 | CanonicalForm |
---|
| 394 | map (const CanonicalForm& primElem, const Variable& alpha, |
---|
| 395 | const CanonicalForm& F, const Variable& beta) |
---|
| 396 | { |
---|
| 397 | CanonicalForm G= F; |
---|
| 398 | int order= 0; |
---|
| 399 | while (!G.isOne()) |
---|
| 400 | { |
---|
| 401 | G /= primElem; |
---|
| 402 | order++; |
---|
| 403 | } |
---|
| 404 | int p= getCharacteristic (); |
---|
| 405 | zz_p::init (p); |
---|
[c1b9927] | 406 | zz_pX NTL_mipo= convertFacCF2NTLzzpX (getMipo (beta)); |
---|
[55608a7] | 407 | zz_pE::init (NTL_mipo); |
---|
| 408 | zz_pEX NTL_alpha_mipo= convertFacCF2NTLzz_pEX (getMipo(alpha), NTL_mipo); |
---|
| 409 | zz_pE NTLBeta= to_zz_pE (convertFacCF2NTLzzpX (beta)); |
---|
| 410 | vec_zz_pE roots= FindRoots (NTL_alpha_mipo); |
---|
[c1b9927] | 411 | long ind=-1; |
---|
[55608a7] | 412 | for (long i= 0; i < roots.length(); i++) |
---|
| 413 | { |
---|
| 414 | if (power (roots [i], order)== NTLBeta) |
---|
| 415 | { |
---|
| 416 | ind= i; |
---|
| 417 | break; |
---|
| 418 | } |
---|
| 419 | } |
---|
| 420 | return (convertNTLzzpE2CF (roots[ind], beta)); |
---|
| 421 | } |
---|
| 422 | |
---|
[618da5] | 423 | CanonicalForm |
---|
| 424 | findMinPoly (const CanonicalForm& F, const Variable& alpha) |
---|
| 425 | { |
---|
| 426 | ASSERT (F.isUnivariate() && F.mvar()==alpha,"expected element of F_p(alpha)"); |
---|
| 427 | |
---|
| 428 | zz_p::init (getCharacteristic()); |
---|
| 429 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
---|
| 430 | int d= degree (getMipo (alpha)); |
---|
| 431 | |
---|
| 432 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo(alpha)); |
---|
| 433 | zz_pE::init (NTLMipo); |
---|
| 434 | vec_zz_p pows; |
---|
| 435 | pows.SetLength (2*d); |
---|
| 436 | |
---|
| 437 | zz_pE powNTLF; |
---|
| 438 | set (powNTLF); |
---|
| 439 | zz_pE NTLFE= to_zz_pE (NTLF); |
---|
| 440 | zz_pX buf; |
---|
| 441 | for (int i= 0; i < 2*d; i++) |
---|
| 442 | { |
---|
| 443 | buf= rep (powNTLF); |
---|
| 444 | buf.rep.SetLength (d); |
---|
| 445 | pows [i]= buf.rep[0]; |
---|
| 446 | powNTLF *= NTLFE; |
---|
| 447 | } |
---|
| 448 | |
---|
| 449 | zz_pX NTLMinPoly; |
---|
| 450 | MinPolySeq (NTLMinPoly, pows, d); |
---|
| 451 | |
---|
| 452 | return convertNTLzzpX2CF (NTLMinPoly, Variable (1)); |
---|
| 453 | } |
---|
| 454 | |
---|
[10af64] | 455 | #endif |
---|