[10af64] | 1 | // -*- c++ -*- |
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| 2 | //***************************************************************************** |
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[51615d6] | 3 | /** @file cf_map_ext.cc |
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[10af64] | 4 | * |
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| 5 | * This file implements functions to map between extensions of finite fields |
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| 6 | * |
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| 7 | * @par Copyright: |
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| 8 | * (c) by The SINGULAR Team, see LICENSE file |
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| 9 | * |
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[7b4793] | 10 | * @author Martin Lee |
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| 11 | * @date 16.11.2009 |
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[10af64] | 12 | **/ |
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| 13 | //***************************************************************************** |
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| 14 | |
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[9f7665] | 15 | |
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[e4fe2b] | 16 | #include "config.h" |
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[9f7665] | 17 | |
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[10af64] | 18 | |
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[650f2d8] | 19 | #include "cf_assert.h" |
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[10af64] | 20 | #include "debug.h" |
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| 21 | |
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| 22 | #include "canonicalform.h" |
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[9c115e1] | 23 | #include "cf_util.h" |
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[d990001] | 24 | #include "imm.h" |
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| 25 | #include "cf_iter.h" |
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[10af64] | 26 | |
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| 27 | #ifdef HAVE_NTL |
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| 28 | #include "NTLconvert.h" |
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| 29 | #endif |
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| 30 | |
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[0d020e] | 31 | // cyclotomoic polys: |
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[51615d6] | 32 | #include "cf_cyclo.h" |
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[0d020e] | 33 | |
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[963057] | 34 | #include "cf_map_ext.h" |
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| 35 | |
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[10af64] | 36 | /// helper function |
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[806c18] | 37 | int findItem (const CFList& list, const CanonicalForm& item) |
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[10af64] | 38 | { |
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[806c18] | 39 | int result= 1; |
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| 40 | for (CFListIterator i= list; i.hasItem(); i++, result++) |
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[10af64] | 41 | { |
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| 42 | if (i.getItem() == item) |
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| 43 | return result; |
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| 44 | } |
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| 45 | return 0; |
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| 46 | } |
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| 47 | |
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| 48 | /// helper function |
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[806c18] | 49 | CanonicalForm getItem (const CFList& list, const int& pos) |
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[10af64] | 50 | { |
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| 51 | int j= 1; |
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[c1b9927] | 52 | if ((pos > 0) && (pos <= list.length())) |
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[10af64] | 53 | { |
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[c1b9927] | 54 | for (CFListIterator i= list; j <= pos; i++, j++) |
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| 55 | { |
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| 56 | if (j == pos) |
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| 57 | return i.getItem(); |
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| 58 | } |
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[10af64] | 59 | } |
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[c1b9927] | 60 | return 0; |
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[806c18] | 61 | } |
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[10af64] | 62 | |
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[d990001] | 63 | #ifdef HAVE_NTL |
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[806c18] | 64 | /// \f$ F_{p} (\alpha ) \subset F_{p}(\beta ) \f$ and \f$ \alpha \f$ is a |
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| 65 | /// primitive element, returns the image of \f$ \alpha \f$ |
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| 66 | static inline |
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| 67 | CanonicalForm mapUp (const Variable& alpha, const Variable& beta) |
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[10af64] | 68 | { |
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| 69 | int p= getCharacteristic (); |
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[bffe62d] | 70 | if (fac_NTL_char != p) |
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| 71 | { |
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| 72 | fac_NTL_char= p; |
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| 73 | zz_p::init (p); |
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| 74 | } |
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[806c18] | 75 | zz_pX NTL_mipo= convertFacCF2NTLzzpX (getMipo (beta)); |
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[10af64] | 76 | zz_pE::init (NTL_mipo); |
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| 77 | zz_pEX NTL_alpha_mipo= convertFacCF2NTLzz_pEX (getMipo(alpha), NTL_mipo); |
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| 78 | zz_pE root= FindRoot (NTL_alpha_mipo); |
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| 79 | return convertNTLzzpE2CF (root, beta); |
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| 80 | } |
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| 81 | |
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[d990001] | 82 | #endif |
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| 83 | |
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[806c18] | 84 | /// the CanonicalForm G is the output of map_up, returns F considered as an |
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[10af64] | 85 | /// element over \f$ F_{p}(\alpha ) \f$, WARNING: make sure coefficients of F |
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| 86 | /// are really elements of a subfield of \f$ F_{p}(\beta ) \f$ which is |
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[806c18] | 87 | /// isomorphic to \f$ F_{p}(\alpha ) \f$ |
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[51615d6] | 88 | static inline |
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[806c18] | 89 | CanonicalForm |
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[10af64] | 90 | mapDown (const CanonicalForm& F, const Variable& alpha, const |
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[806c18] | 91 | CanonicalForm& G, CFList& source, CFList& dest) |
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| 92 | { |
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[10af64] | 93 | CanonicalForm buf, buf2; |
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| 94 | int counter= 0; |
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| 95 | int pos; |
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| 96 | int p= getCharacteristic(); |
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| 97 | int d= degree(getMipo(alpha)); |
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[9c115e1] | 98 | int bound= ipower(p, d); |
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[10af64] | 99 | CanonicalForm result= 0; |
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| 100 | CanonicalForm remainder; |
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| 101 | CanonicalForm alpha_power; |
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| 102 | if (degree(F) == 0) return F; |
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[806c18] | 103 | if (F.level() < 0 && F.isUnivariate()) |
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[10af64] | 104 | { |
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| 105 | buf= F; |
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| 106 | remainder= mod (buf, G); |
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[806c18] | 107 | ASSERT (remainder.isZero(), "alpha is not primitive"); |
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[10af64] | 108 | pos= findItem (source, buf); |
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| 109 | if (pos == 0) |
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| 110 | source.append (buf); |
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| 111 | buf2= buf; |
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[806c18] | 112 | while (degree (buf) != 0 && counter < bound) |
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[10af64] | 113 | { |
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| 114 | buf /= G; |
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| 115 | counter++; |
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| 116 | if (buf == buf2) break; |
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[806c18] | 117 | } |
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| 118 | ASSERT (counter >= bound, "alpha is not primitive"); |
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| 119 | if (pos == 0) |
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[10af64] | 120 | { |
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[55608a7] | 121 | alpha_power= power (alpha, counter); |
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[10af64] | 122 | dest.append (alpha_power); |
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| 123 | } |
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| 124 | else |
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[806c18] | 125 | alpha_power= getItem (dest, pos); |
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[618da5] | 126 | result = alpha_power; |
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[10af64] | 127 | return result; |
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| 128 | } |
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[806c18] | 129 | else |
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[10af64] | 130 | { |
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[806c18] | 131 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 132 | { |
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| 133 | buf= mapDown (i.coeff(), alpha, G, source, dest); |
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| 134 | result += buf*power(F.mvar(), i.exp()); |
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| 135 | } |
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| 136 | return result; |
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| 137 | } |
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| 138 | } |
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| 139 | |
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| 140 | /// helper function |
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| 141 | static inline |
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| 142 | CanonicalForm GF2FalphaHelper (const CanonicalForm& F, const Variable& alpha) |
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| 143 | { |
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[55608a7] | 144 | if (F.isZero()) |
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| 145 | return 0; |
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[10af64] | 146 | int exp; |
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| 147 | CanonicalForm result= 0; |
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| 148 | InternalCF* buf; |
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[806c18] | 149 | if (F.inBaseDomain()) |
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[10af64] | 150 | { |
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| 151 | if (F.isOne()) return 1; |
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| 152 | buf= F.getval(); |
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| 153 | exp= imm2int(buf); |
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| 154 | result= power (alpha, exp).mapinto(); |
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| 155 | return result; |
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[806c18] | 156 | } |
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[10af64] | 157 | for (CFIterator i= F; i.hasTerms(); i++) |
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| 158 | result += GF2FalphaHelper (i.coeff(), alpha)*power (F.mvar(), i.exp()); |
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| 159 | return result; |
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| 160 | } |
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| 161 | |
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[806c18] | 162 | CanonicalForm GF2FalphaRep (const CanonicalForm& F, const Variable& alpha) |
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[10af64] | 163 | { |
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| 164 | Variable beta= rootOf (gf_mipo); |
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[03f640] | 165 | CanonicalForm result= GF2FalphaHelper (F, beta) (alpha, beta); |
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| 166 | prune (beta); |
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| 167 | return result; |
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[10af64] | 168 | } |
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| 169 | |
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[806c18] | 170 | CanonicalForm Falpha2GFRep (const CanonicalForm& F) |
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[10af64] | 171 | { |
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| 172 | CanonicalForm result= 0; |
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| 173 | InternalCF* buf; |
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| 174 | |
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[806c18] | 175 | if (F.inCoeffDomain()) |
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[10af64] | 176 | { |
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| 177 | if (F.inBaseDomain()) |
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| 178 | return F.mapinto(); |
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[806c18] | 179 | else |
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[10af64] | 180 | { |
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[806c18] | 181 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 182 | { |
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| 183 | buf= int2imm_gf (i.exp()); |
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| 184 | result += i.coeff().mapinto()*CanonicalForm (buf); |
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| 185 | } |
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| 186 | } |
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| 187 | return result; |
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[806c18] | 188 | } |
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| 189 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 190 | result += Falpha2GFRep (i.coeff())*power (F.mvar(), i.exp()); |
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| 191 | return result; |
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| 192 | } |
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| 193 | |
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| 194 | /// GF_map_up helper |
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| 195 | static inline |
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[806c18] | 196 | CanonicalForm GFPowUp (const CanonicalForm & F, int k) |
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[10af64] | 197 | { |
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| 198 | if (F.isOne()) return F; |
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| 199 | CanonicalForm result= 0; |
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[806c18] | 200 | if (F.inBaseDomain()) |
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[10af64] | 201 | return power(F, k); |
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[806c18] | 202 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 203 | result += GFPowUp (i.coeff(), k)*power (F.mvar(), i.exp()); |
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| 204 | return result; |
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| 205 | } |
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| 206 | |
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[806c18] | 207 | CanonicalForm GFMapUp (const CanonicalForm & F, int k) |
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| 208 | { |
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[10af64] | 209 | int d= getGFDegree(); |
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| 210 | ASSERT (d%k == 0, "multiple of GF degree expected"); |
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| 211 | int p= getCharacteristic(); |
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[9c115e1] | 212 | int ext_field_size= ipower (p, d); |
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| 213 | int field_size= ipower ( p, k); |
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[10af64] | 214 | int diff= (ext_field_size - 1)/(field_size - 1); |
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| 215 | return GFPowUp (F, diff); |
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| 216 | } |
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| 217 | |
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| 218 | /// GFMapDown helper |
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| 219 | static inline |
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[806c18] | 220 | CanonicalForm GFPowDown (const CanonicalForm & F, int k) |
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[10af64] | 221 | { |
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| 222 | if (F.isOne()) return F; |
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| 223 | CanonicalForm result= 0; |
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| 224 | int exp; |
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| 225 | InternalCF* buf; |
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[806c18] | 226 | if (F.inBaseDomain()) |
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[10af64] | 227 | { |
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| 228 | buf= F.getval(); |
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| 229 | exp= imm2int (buf); |
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| 230 | if ((exp % k) == 0) |
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| 231 | exp= exp/k; |
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[806c18] | 232 | else |
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[10af64] | 233 | return -1; |
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| 234 | |
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| 235 | buf= int2imm_gf (exp); |
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| 236 | return CanonicalForm (buf); |
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[806c18] | 237 | } |
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| 238 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 239 | result += GFPowDown (i.coeff(), k)*power (F.mvar(), i.exp()); |
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| 240 | return result; |
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| 241 | } |
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| 242 | |
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[806c18] | 243 | CanonicalForm GFMapDown (const CanonicalForm & F, int k) |
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[10af64] | 244 | { |
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| 245 | int d= getGFDegree(); |
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| 246 | ASSERT (d % k == 0, "multiple of GF degree expected"); |
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| 247 | int p= getCharacteristic(); |
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[9c115e1] | 248 | int ext_field_size= ipower (p, d); |
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| 249 | int field_size= ipower ( p, k); |
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[10af64] | 250 | int diff= (ext_field_size - 1)/(field_size - 1); |
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| 251 | return GFPowDown (F, diff); |
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| 252 | } |
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| 253 | |
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[98abec] | 254 | /// map F in \f$ F_{p} (\alpha ) \f$ which is generated by G into some |
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| 255 | /// \f$ F_{p}(\beta ) \f$ which is generated by H |
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[806c18] | 256 | static inline |
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| 257 | CanonicalForm mapUp (const CanonicalForm& F, const CanonicalForm& G, |
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| 258 | const Variable& alpha, const CanonicalForm& H, |
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[10af64] | 259 | CFList& source, CFList& dest) |
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[806c18] | 260 | { |
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[10af64] | 261 | CanonicalForm buf, buf2; |
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| 262 | int counter= 0; |
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| 263 | int pos; |
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| 264 | int p= getCharacteristic(); |
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| 265 | int d= degree (getMipo(alpha)); |
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[9c115e1] | 266 | int bound= ipower(p, d); |
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[10af64] | 267 | CanonicalForm result= 0; |
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| 268 | CanonicalForm remainder; |
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| 269 | CanonicalForm H_power; |
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| 270 | if (degree(F) <= 0) return F; |
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[806c18] | 271 | if (F.level() < 0 && F.isUnivariate()) |
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[10af64] | 272 | { |
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| 273 | buf= F; |
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| 274 | remainder= mod (buf, G); |
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[806c18] | 275 | ASSERT (remainder.isZero(), "alpha is not primitive"); |
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[10af64] | 276 | pos= findItem (source, buf); |
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| 277 | if (pos == 0) |
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| 278 | source.append (buf); |
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| 279 | buf2= buf; |
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[806c18] | 280 | while (degree (buf) != 0 && counter < bound) |
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[10af64] | 281 | { |
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| 282 | buf /= G; |
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| 283 | counter++; |
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| 284 | if (buf == buf2) break; |
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[806c18] | 285 | } |
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[050d1b] | 286 | ASSERT (counter <= bound, "alpha is not primitive"); |
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[806c18] | 287 | if (pos == 0) |
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[10af64] | 288 | { |
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[618da5] | 289 | H_power= buf*power (H, counter); |
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[10af64] | 290 | dest.append (H_power); |
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| 291 | } |
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| 292 | else |
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[806c18] | 293 | H_power= getItem (dest, pos); |
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[618da5] | 294 | result = H_power; |
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[10af64] | 295 | return result; |
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| 296 | } |
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[806c18] | 297 | else |
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[10af64] | 298 | { |
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[806c18] | 299 | for (CFIterator i= F; i.hasTerms(); i++) |
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[10af64] | 300 | { |
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| 301 | buf= mapUp (i.coeff(), G, alpha, H, source, dest); |
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| 302 | result += buf*power(F.mvar(), i.exp()); |
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| 303 | } |
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| 304 | return result; |
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| 305 | } |
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| 306 | } |
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| 307 | |
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[d990001] | 308 | #ifdef HAVE_NTL |
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[806c18] | 309 | CanonicalForm |
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[98abec] | 310 | primitiveElement (const Variable& alpha, Variable& beta, bool& fail) |
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[10af64] | 311 | { |
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| 312 | bool primitive= false; |
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| 313 | fail= false; |
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| 314 | primitive= isPrimitive (alpha, fail); |
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| 315 | if (fail) |
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| 316 | return 0; |
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[806c18] | 317 | if (primitive) |
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[10af64] | 318 | { |
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| 319 | beta= alpha; |
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| 320 | return alpha; |
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| 321 | } |
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[806c18] | 322 | CanonicalForm mipo= getMipo (alpha); |
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[10af64] | 323 | int d= degree (mipo); |
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| 324 | int p= getCharacteristic (); |
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[bffe62d] | 325 | if (fac_NTL_char != p) |
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| 326 | { |
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| 327 | fac_NTL_char= p; |
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| 328 | zz_p::init (p); |
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| 329 | } |
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[10af64] | 330 | zz_pX NTL_mipo; |
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| 331 | CanonicalForm mipo2; |
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| 332 | primitive= false; |
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| 333 | fail= false; |
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[03f640] | 334 | bool initialized= false; |
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[10af64] | 335 | do |
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| 336 | { |
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[806c18] | 337 | BuildIrred (NTL_mipo, d); |
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[10af64] | 338 | mipo2= convertNTLzzpX2CF (NTL_mipo, Variable (1)); |
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[03f640] | 339 | if (!initialized) |
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| 340 | beta= rootOf (mipo2); |
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| 341 | else |
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| 342 | setMipo (beta, mipo2); |
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[10af64] | 343 | primitive= isPrimitive (beta, fail); |
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| 344 | if (primitive) |
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[806c18] | 345 | break; |
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[10af64] | 346 | if (fail) |
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| 347 | return 0; |
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| 348 | } while (1); |
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[08daea] | 349 | zz_pX alpha_mipo= convertFacCF2NTLzzpX (mipo); |
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| 350 | zz_pE::init (alpha_mipo); |
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| 351 | zz_pEX NTL_beta_mipo= to_zz_pEX (NTL_mipo); |
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| 352 | zz_pE root= FindRoot (NTL_beta_mipo); |
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[10af64] | 353 | return convertNTLzzpE2CF (root, alpha); |
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| 354 | } |
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[d990001] | 355 | #endif |
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[10af64] | 356 | |
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| 357 | CanonicalForm |
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| 358 | mapDown (const CanonicalForm& F, const CanonicalForm& prim_elem, const |
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[806c18] | 359 | CanonicalForm& im_prim_elem, const Variable& alpha, CFList& source, |
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| 360 | CFList& dest) |
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| 361 | { |
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[10af64] | 362 | return mapUp (F, im_prim_elem, alpha, prim_elem, dest, source); |
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| 363 | } |
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| 364 | |
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| 365 | CanonicalForm |
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[5b8726d] | 366 | mapUp (const CanonicalForm& F, const Variable& alpha, const Variable& /*beta*/, |
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[806c18] | 367 | const CanonicalForm& prim_elem, const CanonicalForm& im_prim_elem, |
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| 368 | CFList& source, CFList& dest) |
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[10af64] | 369 | { |
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[806c18] | 370 | if (prim_elem == alpha) |
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[10af64] | 371 | return F (im_prim_elem, alpha); |
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| 372 | return mapUp (F, prim_elem, alpha, im_prim_elem, source, dest); |
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| 373 | } |
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| 374 | |
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[d990001] | 375 | #ifdef HAVE_NTL |
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[806c18] | 376 | CanonicalForm |
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[618da5] | 377 | mapPrimElem (const CanonicalForm& primElem, const Variable& alpha, |
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[51615d6] | 378 | const Variable& beta) |
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[10af64] | 379 | { |
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[618da5] | 380 | if (primElem == alpha) |
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[10af64] | 381 | return mapUp (alpha, beta); |
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| 382 | else |
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| 383 | { |
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[618da5] | 384 | CanonicalForm primElemMipo= findMinPoly (primElem, alpha); |
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| 385 | int p= getCharacteristic (); |
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[bffe62d] | 386 | if (fac_NTL_char != p) |
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| 387 | { |
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| 388 | fac_NTL_char= p; |
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| 389 | zz_p::init (p); |
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| 390 | } |
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[c1b9927] | 391 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (beta)); |
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[618da5] | 392 | zz_pE::init (NTLMipo); |
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| 393 | zz_pEX NTLPrimElemMipo= convertFacCF2NTLzz_pEX (primElemMipo, NTLMipo); |
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| 394 | zz_pE root= FindRoot (NTLPrimElemMipo); |
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| 395 | return convertNTLzzpE2CF (root, beta); |
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[806c18] | 396 | } |
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[10af64] | 397 | } |
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| 398 | |
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[55608a7] | 399 | CanonicalForm |
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| 400 | map (const CanonicalForm& primElem, const Variable& alpha, |
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| 401 | const CanonicalForm& F, const Variable& beta) |
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| 402 | { |
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| 403 | CanonicalForm G= F; |
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| 404 | int order= 0; |
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| 405 | while (!G.isOne()) |
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| 406 | { |
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| 407 | G /= primElem; |
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| 408 | order++; |
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| 409 | } |
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| 410 | int p= getCharacteristic (); |
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[bffe62d] | 411 | if (fac_NTL_char != p) |
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| 412 | { |
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| 413 | fac_NTL_char= p; |
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| 414 | zz_p::init (p); |
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| 415 | } |
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[c1b9927] | 416 | zz_pX NTL_mipo= convertFacCF2NTLzzpX (getMipo (beta)); |
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[55608a7] | 417 | zz_pE::init (NTL_mipo); |
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| 418 | zz_pEX NTL_alpha_mipo= convertFacCF2NTLzz_pEX (getMipo(alpha), NTL_mipo); |
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| 419 | zz_pE NTLBeta= to_zz_pE (convertFacCF2NTLzzpX (beta)); |
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| 420 | vec_zz_pE roots= FindRoots (NTL_alpha_mipo); |
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[c1b9927] | 421 | long ind=-1; |
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[55608a7] | 422 | for (long i= 0; i < roots.length(); i++) |
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| 423 | { |
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| 424 | if (power (roots [i], order)== NTLBeta) |
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| 425 | { |
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| 426 | ind= i; |
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| 427 | break; |
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| 428 | } |
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| 429 | } |
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| 430 | return (convertNTLzzpE2CF (roots[ind], beta)); |
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| 431 | } |
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| 432 | |
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[618da5] | 433 | CanonicalForm |
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| 434 | findMinPoly (const CanonicalForm& F, const Variable& alpha) |
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| 435 | { |
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| 436 | ASSERT (F.isUnivariate() && F.mvar()==alpha,"expected element of F_p(alpha)"); |
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| 437 | |
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[bffe62d] | 438 | if (fac_NTL_char != getCharacteristic()) |
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| 439 | { |
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| 440 | fac_NTL_char= getCharacteristic(); |
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| 441 | zz_p::init (getCharacteristic()); |
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| 442 | } |
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[618da5] | 443 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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| 444 | int d= degree (getMipo (alpha)); |
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| 445 | |
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| 446 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo(alpha)); |
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| 447 | zz_pE::init (NTLMipo); |
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| 448 | vec_zz_p pows; |
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| 449 | pows.SetLength (2*d); |
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| 450 | |
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| 451 | zz_pE powNTLF; |
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| 452 | set (powNTLF); |
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| 453 | zz_pE NTLFE= to_zz_pE (NTLF); |
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| 454 | zz_pX buf; |
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| 455 | for (int i= 0; i < 2*d; i++) |
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| 456 | { |
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| 457 | buf= rep (powNTLF); |
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| 458 | buf.rep.SetLength (d); |
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| 459 | pows [i]= buf.rep[0]; |
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| 460 | powNTLF *= NTLFE; |
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| 461 | } |
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| 462 | |
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| 463 | zz_pX NTLMinPoly; |
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| 464 | MinPolySeq (NTLMinPoly, pows, d); |
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| 465 | |
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| 466 | return convertNTLzzpX2CF (NTLMinPoly, Variable (1)); |
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| 467 | } |
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| 468 | |
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[10af64] | 469 | #endif |
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