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