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