1 | // -*- c++ -*- |
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2 | //***************************************************************************** |
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3 | /** @file cf_gcd_smallp.cc |
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4 | * |
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5 | * @author Martin Lee |
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6 | * @date 22.10.2009 |
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7 | * |
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8 | * This file implements the GCD of two polynomials over \f$ F_{p} \f$ , |
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9 | * \f$ F_{p}(\alpha ) \f$ or GF based on Alg. 7.2. as described in "Algorithms |
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10 | * for Computer Algebra" by Geddes, Czapor, Labahnn |
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11 | * |
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12 | * @par Copyright: |
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13 | * (c) by The SINGULAR Team, see LICENSE file |
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14 | * |
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15 | * @internal |
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16 | * @version \$Id$ |
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17 | * |
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18 | **/ |
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19 | //***************************************************************************** |
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20 | |
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21 | #include <config.h> |
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22 | |
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23 | #include "assert.h" |
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24 | #include "debug.h" |
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25 | #include "timing.h" |
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26 | |
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27 | #include "canonicalform.h" |
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28 | #include "cf_map.h" |
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29 | #include "cf_util.h" |
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30 | #include "ftmpl_functions.h" |
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31 | #include "cf_random.h" |
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32 | |
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33 | // inline helper functions: |
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34 | #include "cf_map_ext.h" |
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35 | |
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36 | #ifdef HAVE_NTL |
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37 | #include <NTL/ZZ_pEX.h> |
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38 | #endif |
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39 | |
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40 | #ifdef HAVE_NTL |
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41 | |
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42 | TIMING_DEFINE_PRINT(gcd_recursion); |
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43 | TIMING_DEFINE_PRINT(newton_interpolation); |
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44 | |
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45 | /// compressing two polynomials F and G, M is used for compressing, |
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46 | /// N to reverse the compression |
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47 | int myCompress (const CanonicalForm& F, const CanonicalForm& G, CFMap & M, |
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48 | CFMap & N, bool& top_level) |
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49 | { |
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50 | int n= tmax (F.level(), G.level()); |
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51 | int * degsf= new int [n + 1]; |
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52 | int * degsg= new int [n + 1]; |
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53 | |
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54 | for (int i = 0; i <= n; i++) |
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55 | degsf[i]= degsg[i]= 0; |
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56 | |
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57 | degsf= degrees (F, degsf); |
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58 | degsg= degrees (G, degsg); |
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59 | |
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60 | int both_non_zero= 0; |
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61 | int f_zero= 0; |
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62 | int g_zero= 0; |
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63 | int both_zero= 0; |
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64 | |
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65 | if (top_level) |
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66 | { |
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67 | for (int i= 1; i <= n; i++) |
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68 | { |
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69 | if (degsf[i] != 0 && degsg[i] != 0) |
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70 | { |
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71 | both_non_zero++; |
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72 | continue; |
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73 | } |
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74 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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75 | { |
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76 | f_zero++; |
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77 | continue; |
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78 | } |
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79 | if (degsg[i] == 0 && degsf[i] && i <= F.level()) |
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80 | { |
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81 | g_zero++; |
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82 | continue; |
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83 | } |
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84 | } |
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85 | |
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86 | if (both_non_zero == 0) return 0; |
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87 | |
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88 | // map Variables which do not occur in both polynomials to higher levels |
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89 | int k= 1; |
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90 | int l= 1; |
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91 | for (int i= 1; i <= n; i++) |
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92 | { |
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93 | if (degsf[i] != 0 && degsg[i] == 0 && i <= F.level()) |
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94 | { |
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95 | if (k + both_non_zero != i) |
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96 | { |
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97 | M.newpair (Variable (i), Variable (k + both_non_zero)); |
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98 | N.newpair (Variable (k + both_non_zero), Variable (i)); |
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99 | } |
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100 | k++; |
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101 | } |
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102 | if (degsf[i] == 0 && degsg[i] != 0 && i <= G.level()) |
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103 | { |
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104 | if (l + g_zero + both_non_zero != i) |
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105 | { |
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106 | M.newpair (Variable (i), Variable (l + g_zero + both_non_zero)); |
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107 | N.newpair (Variable (l + g_zero + both_non_zero), Variable (i)); |
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108 | } |
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109 | l++; |
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110 | } |
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111 | } |
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112 | |
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113 | // sort Variables x_{i} in increasing order of |
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114 | // min(deg_{x_{i}}(f),deg_{x_{i}}(g)) |
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115 | int m= tmin (F.level(), G.level()); |
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116 | int max_min_deg; |
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117 | k= both_non_zero; |
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118 | l= 0; |
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119 | int i= 1; |
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120 | while (k > 0) |
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121 | { |
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122 | max_min_deg= tmin (degsf[i], degsg[i]); |
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123 | while (max_min_deg == 0) |
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124 | { |
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125 | i++; |
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126 | max_min_deg= tmin (degsf[i], degsg[i]); |
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127 | } |
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128 | for (int j= i + 1; j <= m; j++) |
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129 | { |
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130 | if (tmin (degsf[j],degsg[j]) >= max_min_deg) |
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131 | { |
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132 | max_min_deg= tmin (degsf[j], degsg[j]); |
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133 | l= j; |
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134 | } |
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135 | } |
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136 | if (l != 0) |
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137 | { |
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138 | if (l != k) |
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139 | { |
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140 | M.newpair (Variable (l), Variable(k)); |
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141 | N.newpair (Variable (k), Variable(l)); |
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142 | degsf[l]= 0; |
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143 | degsg[l]= 0; |
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144 | l= 0; |
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145 | } |
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146 | else |
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147 | { |
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148 | degsf[l]= 0; |
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149 | degsg[l]= 0; |
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150 | l= 0; |
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151 | } |
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152 | } |
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153 | else if (l == 0) |
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154 | { |
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155 | if (i != k) |
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156 | { |
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157 | M.newpair (Variable (i), Variable (k)); |
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158 | N.newpair (Variable (k), Variable (i)); |
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159 | degsf[i]= 0; |
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160 | degsg[i]= 0; |
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161 | } |
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162 | else |
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163 | { |
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164 | degsf[i]= 0; |
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165 | degsg[i]= 0; |
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166 | } |
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167 | i++; |
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168 | } |
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169 | k--; |
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170 | } |
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171 | } |
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172 | else |
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173 | { |
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174 | //arrange Variables such that no gaps occur |
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175 | for (int i= 1; i <= n; i++) |
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176 | { |
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177 | if (degsf[i] == 0 && degsg[i] == 0) |
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178 | { |
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179 | both_zero++; |
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180 | continue; |
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181 | } |
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182 | else |
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183 | { |
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184 | if (both_zero != 0) |
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185 | { |
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186 | M.newpair (Variable (i), Variable (i - both_zero)); |
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187 | N.newpair (Variable (i - both_zero), Variable (i)); |
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188 | } |
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189 | } |
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190 | } |
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191 | } |
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192 | |
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193 | delete [] degsf; |
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194 | delete [] degsg; |
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195 | |
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196 | return 1; |
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197 | } |
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198 | |
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199 | /// compute the content of F, where F is considered as an element of |
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200 | /// \f$ R[x_{1}][x_{2},\ldots ,x_{n}] \f$ |
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201 | static inline CanonicalForm |
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202 | uni_content (const CanonicalForm & F) |
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203 | { |
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204 | if (F.inBaseDomain()) |
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205 | return F.genOne(); |
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206 | if (F.level() == 1 && F.isUnivariate()) |
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207 | return F; |
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208 | if (F.level() != 1 && F.isUnivariate()) |
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209 | return F.genOne(); |
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210 | if (degree (F,1) == 0) return F.genOne(); |
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211 | |
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212 | int l= F.level(); |
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213 | if (l == 2) |
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214 | return content(F); |
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215 | else |
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216 | { |
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217 | CanonicalForm pol, c = 0; |
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218 | CFIterator i = F; |
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219 | for (; i.hasTerms(); i++) |
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220 | { |
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221 | pol= i.coeff(); |
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222 | pol= uni_content (pol); |
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223 | c= gcd (c, pol); |
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224 | if (c.isOne()) |
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225 | return c; |
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226 | } |
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227 | return c; |
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228 | } |
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229 | } |
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230 | |
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231 | /// compute the leading coefficient of F, where F is considered as an element |
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232 | /// of \f$ R[x_{1}][x_{2},\ldots ,x_{n}] \f$, order on Mon (x_{2},\ldots ,x_{n}) |
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233 | /// is dp. |
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234 | static inline |
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235 | CanonicalForm uni_lcoeff (const CanonicalForm& F) |
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236 | { |
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237 | if (F.level() <= 1) |
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238 | return F; |
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239 | else |
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240 | { |
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241 | Variable x= Variable (2); |
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242 | int deg= totaldegree (F, x, F.mvar()); |
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243 | for (CFIterator i= F; i.hasTerms(); i++) |
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244 | { |
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245 | if (i.exp() + totaldegree (i.coeff(), x, i.coeff().mvar()) == deg) |
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246 | return uni_lcoeff (i.coeff()); |
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247 | } |
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248 | } |
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249 | } |
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250 | |
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251 | /// Newton interpolation - Incremental algorithm. |
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252 | /// Given a list of values alpha_i and a list of polynomials u_i, 1 <= i <= n, |
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253 | /// computes the interpolation polynomial assuming that |
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254 | /// the polynomials in u are the results of evaluating the variabe x |
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255 | /// of the unknown polynomial at the alpha values. |
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256 | /// This incremental version receives only the values of alpha_n and u_n and |
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257 | /// the previous interpolation polynomial for points 1 <= i <= n-1, and computes |
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258 | /// the polynomial interpolating in all the points. |
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259 | /// newtonPoly must be equal to (x - alpha_1) * ... * (x - alpha_{n-1}) |
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260 | static inline CanonicalForm |
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261 | newtonInterp(const CanonicalForm alpha, const CanonicalForm u, const CanonicalForm newtonPoly, const CanonicalForm oldInterPoly, const Variable & x) |
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262 | { |
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263 | CanonicalForm interPoly; |
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264 | |
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265 | interPoly = oldInterPoly + ((u - oldInterPoly(alpha, x)) / newtonPoly(alpha, x)) * newtonPoly; |
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266 | return interPoly; |
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267 | } |
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268 | |
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269 | /// compute a random element a of \f$ F_{p}(\alpha ) \f$ , |
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270 | /// s.t. F(a) \f$ \neq 0 \f$ , F is a univariate polynomial, returns |
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271 | /// fail if there are no field elements left which have not been used before |
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272 | static inline CanonicalForm |
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273 | randomElement (const CanonicalForm & F, const Variable & alpha, CFList & list, |
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274 | bool & fail) |
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275 | { |
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276 | fail= false; |
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277 | Variable x= F.mvar(); |
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278 | AlgExtRandomF genAlgExt (alpha); |
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279 | FFRandom genFF; |
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280 | CanonicalForm random, mipo; |
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281 | mipo= getMipo (alpha); |
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282 | int p= getCharacteristic (); |
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283 | int d= degree (mipo); |
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284 | int bound= ipower (p, d); |
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285 | do |
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286 | { |
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287 | if (list.length() == bound) |
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288 | { |
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289 | fail= true; |
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290 | break; |
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291 | } |
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292 | if (list.length() < p) |
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293 | { |
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294 | random= genFF.generate(); |
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295 | while (find (list, random)) |
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296 | random= genFF.generate(); |
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297 | } |
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298 | else |
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299 | { |
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300 | random= genAlgExt.generate(); |
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301 | while (find (list, random)) |
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302 | random= genAlgExt.generate(); |
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303 | } |
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304 | if (F (random, x) == 0) |
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305 | { |
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306 | list.append (random); |
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307 | continue; |
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308 | } |
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309 | } while (find (list, random)); |
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310 | return random; |
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311 | } |
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312 | |
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313 | /// chooses a suitable extension of \f$ F_{p}(\alpha ) \f$ |
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314 | /// we do not extend \f$ F_{p}(\alpha ) \f$ itself but extend \f$ F_{p} \f$ , |
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315 | /// s.t. \f$ F_{p}(\alpha ) \subset F_{p}(\beta ) \f$ |
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316 | static inline |
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317 | void choose_extension (const int& d, const int& num_vars, Variable& beta) |
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318 | { |
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319 | int p= getCharacteristic(); |
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320 | ZZ NTLp= to_ZZ (p); |
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321 | ZZ_p::init (NTLp); |
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322 | ZZ_pX NTLirredpoly; |
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323 | //TODO: replace d by max_{i} (deg_x{i}(f)) |
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324 | int i= (int) (log ((double) ipower (d + 1, num_vars))/log ((double) p)); |
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325 | int m= degree (getMipo (beta)); |
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326 | if (i <= 1) |
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327 | i= 2; |
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328 | BuildIrred (NTLirredpoly, i*m); |
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329 | CanonicalForm mipo= convertNTLZZpX2CF (NTLirredpoly, Variable(1)); |
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330 | beta= rootOf (mipo); |
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331 | } |
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332 | |
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333 | /// GCD of F and G over \f$ F_{p}(\alpha ) \f$ , |
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334 | /// l and top_level are only used internally, output is monic |
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335 | /// based on Alg. 7.2. as described in "Algorithms for |
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336 | /// Computer Algebra" by Geddes, Czapor, Labahn |
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337 | CanonicalForm |
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338 | GCD_Fp_extension (const CanonicalForm& F, const CanonicalForm& G, |
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339 | Variable & alpha, CFList& l, bool& top_level) |
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340 | { |
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341 | CanonicalForm A= F; |
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342 | CanonicalForm B= G; |
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343 | if (F.isZero() && degree(G) > 0) return G/Lc(G); |
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344 | else if (G.isZero() && degree (F) > 0) return F/Lc(F); |
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345 | else if (F.isZero() && G.isZero()) return F.genOne(); |
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346 | if (F.inBaseDomain() || G.inBaseDomain()) return F.genOne(); |
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347 | if (F.isUnivariate() && fdivides(F, G)) return F/Lc(F); |
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348 | if (G.isUnivariate() && fdivides(G, F)) return G/Lc(G); |
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349 | if (F == G) return F/Lc(F); |
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350 | |
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351 | CFMap M,N; |
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352 | int best_level= myCompress (A, B, M, N, top_level); |
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353 | |
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354 | if (best_level == 0) return B.genOne(); |
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355 | |
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356 | A= M(A); |
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357 | B= M(B); |
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358 | |
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359 | Variable x= Variable(1); |
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360 | |
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361 | //univariate case |
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362 | if (A.isUnivariate() && B.isUnivariate()) |
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363 | return N (gcd(A,B)); |
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364 | |
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365 | CanonicalForm cA, cB; // content of A and B |
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366 | CanonicalForm ppA, ppB; // primitive part of A and B |
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367 | CanonicalForm gcdcAcB; |
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368 | |
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369 | cA = uni_content (A); |
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370 | cB = uni_content (B); |
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371 | |
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372 | gcdcAcB= gcd (cA, cB); |
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373 | |
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374 | ppA= A/cA; |
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375 | ppB= B/cB; |
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376 | |
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377 | CanonicalForm lcA, lcB; // leading coefficients of A and B |
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378 | CanonicalForm gcdlcAlcB; |
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379 | |
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380 | lcA= uni_lcoeff (ppA); |
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381 | lcB= uni_lcoeff (ppB); |
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382 | |
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383 | if (fdivides (lcA, lcB)) { |
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384 | if (fdivides (A, B)) |
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385 | return F/Lc(F); |
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386 | } |
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387 | if (fdivides (lcB, lcA)) |
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388 | { |
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389 | if (fdivides (B, A)) |
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390 | return G/Lc(G); |
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391 | } |
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392 | |
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393 | gcdlcAlcB= gcd (lcA, lcB); |
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394 | |
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395 | int d= totaldegree (ppA, Variable(2), Variable (ppA.level())); |
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396 | |
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397 | if (d == 0) return N(gcdcAcB); |
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398 | int d0= totaldegree (ppB, Variable(2), Variable (ppB.level())); |
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399 | if (d0 < d) |
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400 | d= d0; |
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401 | if (d == 0) return N(gcdcAcB); |
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402 | |
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403 | CanonicalForm m, random_element, G_m, G_random_element, H, cH, ppH; |
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404 | CanonicalForm newtonPoly; |
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405 | |
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406 | newtonPoly= 1; |
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407 | m= gcdlcAlcB; |
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408 | G_m= 0; |
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409 | H= 0; |
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410 | bool fail= false; |
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411 | top_level= false; |
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412 | bool inextension= false; |
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413 | Variable V_buf= alpha; |
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414 | CanonicalForm prim_elem, im_prim_elem; |
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415 | CFList source, dest; |
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416 | do |
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417 | { |
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418 | random_element= randomElement (m, V_buf, l, fail); |
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419 | if (fail) |
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420 | { |
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421 | source= CFList(); |
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422 | dest= CFList(); |
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423 | int num_vars= tmin (getNumVars(A), getNumVars(B)); |
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424 | num_vars--; |
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425 | |
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426 | choose_extension (d, num_vars, V_buf); |
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427 | bool prim_fail= false; |
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428 | Variable V_buf2; |
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429 | prim_elem= primitiveElement (alpha, V_buf2, prim_fail); |
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430 | |
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431 | ASSERT (!prim_fail, "failure in integer factorizer"); |
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432 | if (prim_fail) |
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433 | ; //ERROR |
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434 | else |
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435 | im_prim_elem= mapPrimElem (prim_elem, alpha, V_buf); |
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436 | |
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437 | DEBOUTLN (cerr, "getMipo (alpha)= " << getMipo (alpha)); |
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438 | DEBOUTLN (cerr, "getMipo (alpha)= " << getMipo (V_buf2)); |
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439 | inextension= true; |
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440 | for (CFListIterator i= l; i.hasItem(); i++) |
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441 | i.getItem()= mapUp (i.getItem(), alpha, V_buf, prim_elem, |
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442 | im_prim_elem, source, dest); |
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443 | m= mapUp (m, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
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444 | G_m= mapUp (G_m, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
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445 | newtonPoly= mapUp (newtonPoly, alpha, V_buf, prim_elem, im_prim_elem, |
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446 | source, dest); |
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447 | ppA= mapUp (ppA, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
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448 | ppB= mapUp (ppB, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
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449 | gcdlcAlcB= mapUp (gcdlcAlcB, alpha, V_buf, prim_elem, im_prim_elem, |
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450 | source, dest); |
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451 | |
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452 | fail= false; |
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453 | random_element= randomElement (m, V_buf, l, fail ); |
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454 | DEBOUTLN (cerr, "fail= " << fail); |
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455 | CFList list; |
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456 | TIMING_START (gcd_recursion); |
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457 | G_random_element= |
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458 | GCD_Fp_extension (ppA (random_element, x), ppB (random_element, x), V_buf, |
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459 | list, top_level); |
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460 | TIMING_END_AND_PRINT (gcd_recursion, |
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461 | "time for recursive call: "); |
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462 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
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463 | } |
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464 | else |
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465 | { |
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466 | CFList list; |
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467 | TIMING_START (gcd_recursion); |
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468 | G_random_element= |
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469 | GCD_Fp_extension (ppA(random_element, x), ppB(random_element, x), V_buf, |
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470 | list, top_level); |
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471 | TIMING_END_AND_PRINT (gcd_recursion, |
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472 | "time for recursive call: "); |
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473 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
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474 | } |
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475 | |
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476 | d0= totaldegree (G_random_element, Variable(2), |
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477 | Variable (G_random_element.level())); |
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478 | if (d0 == 0) return N(gcdcAcB); |
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479 | if (d0 > d) |
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480 | { |
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481 | if (!find (l, random_element)) |
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482 | l.append (random_element); |
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483 | continue; |
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484 | } |
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485 | |
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486 | G_random_element= |
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487 | (gcdlcAlcB(random_element, x)/uni_lcoeff (G_random_element)) |
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488 | * G_random_element; |
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489 | |
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490 | d0= totaldegree (G_random_element, Variable(2), |
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491 | Variable(G_random_element.level())); |
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492 | if (d0 < d) |
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493 | { |
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494 | m= gcdlcAlcB; |
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495 | newtonPoly= 1; |
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496 | G_m= 0; |
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497 | d= d0; |
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498 | } |
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499 | |
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500 | TIMING_START (newton_interpolation); |
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501 | H= newtonInterp (random_element, G_random_element, newtonPoly, G_m, x); |
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502 | TIMING_END_AND_PRINT (newton_interpolation, |
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503 | "time for newton interpolation: "); |
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504 | |
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505 | //termination test |
---|
506 | if (uni_lcoeff (H) == gcdlcAlcB) |
---|
507 | { |
---|
508 | cH= uni_content (H); |
---|
509 | ppH= H/cH; |
---|
510 | if (inextension) |
---|
511 | { |
---|
512 | CFList u, v; |
---|
513 | //maybe it's better to test if ppH is an element of F(\alpha) before |
---|
514 | //mapping down |
---|
515 | DEBOUTLN (cerr, "ppH before mapDown= " << ppH); |
---|
516 | ppH= mapDown (ppH, prim_elem, im_prim_elem, alpha, u, v); |
---|
517 | ppH /= Lc(ppH); |
---|
518 | DEBOUTLN (cerr, "ppH after mapDown= " << ppH); |
---|
519 | if (fdivides (ppH, A) && fdivides (ppH, B)) |
---|
520 | return N(gcdcAcB*ppH); |
---|
521 | } |
---|
522 | else if (fdivides (ppH, A) && fdivides (ppH, B)) |
---|
523 | return N(gcdcAcB*(ppH/Lc(ppH))); |
---|
524 | } |
---|
525 | |
---|
526 | G_m= H; |
---|
527 | newtonPoly= newtonPoly*(x - random_element); |
---|
528 | m= m*(x - random_element); |
---|
529 | if (!find (l, random_element)) |
---|
530 | l.append (random_element); |
---|
531 | } while (1); |
---|
532 | } |
---|
533 | |
---|
534 | /// compute a random element a of GF, s.t. F(a) \f$ \neq 0 \f$ , F is a |
---|
535 | /// univariate polynomial, returns fail if there are no field elements left |
---|
536 | /// which have not been used before |
---|
537 | CanonicalForm |
---|
538 | GFRandomElement (const CanonicalForm& F, CFList& list, bool& fail) |
---|
539 | { |
---|
540 | fail= false; |
---|
541 | Variable x= F.mvar(); |
---|
542 | GFRandom genGF; |
---|
543 | CanonicalForm random; |
---|
544 | int p= getCharacteristic(); |
---|
545 | int d= getGFDegree(); |
---|
546 | int bound= ipower (p, d); |
---|
547 | do |
---|
548 | { |
---|
549 | if (list.length() == bound) |
---|
550 | { |
---|
551 | fail= true; |
---|
552 | break; |
---|
553 | } |
---|
554 | if (list.length() < 1) |
---|
555 | random= 0; |
---|
556 | else |
---|
557 | { |
---|
558 | random= genGF.generate(); |
---|
559 | while (find (list, random)) |
---|
560 | random= genGF.generate(); |
---|
561 | } |
---|
562 | if (F (random, x) == 0) |
---|
563 | { |
---|
564 | list.append (random); |
---|
565 | continue; |
---|
566 | } |
---|
567 | } while (find (list, random)); |
---|
568 | return random; |
---|
569 | } |
---|
570 | |
---|
571 | /// GCD of F and G over GF, based on Alg. 7.2. as described in "Algorithms for |
---|
572 | /// Computer Algebra" by Geddes, Czapor, Labahn |
---|
573 | /// Usually this algorithm will be faster than GCD_Fp_extension since GF has |
---|
574 | /// faster field arithmetics, however it might fail if the input is large since |
---|
575 | /// the size of the base field is bounded by 2^16, output is monic |
---|
576 | CanonicalForm |
---|
577 | GCD_GF (const CanonicalForm& F, const CanonicalForm& G, CFList& l, |
---|
578 | bool& top_level) |
---|
579 | { |
---|
580 | CanonicalForm A= F; |
---|
581 | CanonicalForm B= G; |
---|
582 | if (F.isZero() && degree(G) > 0) return G/Lc(G); |
---|
583 | else if (G.isZero() && degree (F) > 0) return F/Lc(F); |
---|
584 | else if (F.isZero() && G.isZero()) return F.genOne(); |
---|
585 | if (F.inBaseDomain() || G.inBaseDomain()) return F.genOne(); |
---|
586 | if (F.isUnivariate() && fdivides(F, G)) return F/Lc(F); |
---|
587 | if (G.isUnivariate() && fdivides(G, F)) return G/Lc(G); |
---|
588 | if (F == G) return F/Lc(F); |
---|
589 | |
---|
590 | CFMap M,N; |
---|
591 | int best_level= myCompress (A, B, M, N, top_level); |
---|
592 | |
---|
593 | if (best_level == 0) return B.genOne(); |
---|
594 | |
---|
595 | A= M(A); |
---|
596 | B= M(B); |
---|
597 | |
---|
598 | Variable x= Variable(1); |
---|
599 | |
---|
600 | //univariate case |
---|
601 | if (A.isUnivariate() && B.isUnivariate()) |
---|
602 | return N (gcd (A, B)); |
---|
603 | |
---|
604 | CanonicalForm cA, cB; // content of A and B |
---|
605 | CanonicalForm ppA, ppB; // primitive part of A and B |
---|
606 | CanonicalForm gcdcAcB; |
---|
607 | |
---|
608 | cA = uni_content (A); |
---|
609 | cB = uni_content (B); |
---|
610 | |
---|
611 | gcdcAcB= gcd (cA, cB); |
---|
612 | |
---|
613 | ppA= A/cA; |
---|
614 | ppB= B/cB; |
---|
615 | |
---|
616 | CanonicalForm lcA, lcB; // leading coefficients of A and B |
---|
617 | CanonicalForm gcdlcAlcB; |
---|
618 | |
---|
619 | lcA= uni_lcoeff (ppA); |
---|
620 | lcB= uni_lcoeff (ppB); |
---|
621 | |
---|
622 | if (fdivides (lcA, lcB)) |
---|
623 | { |
---|
624 | if (fdivides (A, B)) |
---|
625 | return F; |
---|
626 | } |
---|
627 | if (fdivides (lcB, lcA)) |
---|
628 | { |
---|
629 | if (fdivides (B, A)) |
---|
630 | return G; |
---|
631 | } |
---|
632 | |
---|
633 | gcdlcAlcB= gcd (lcA, lcB); |
---|
634 | |
---|
635 | int d= totaldegree (ppA, Variable(2), Variable (ppA.level())); |
---|
636 | if (d == 0) return N(gcdcAcB); |
---|
637 | int d0= totaldegree (ppB, Variable(2), Variable (ppB.level())); |
---|
638 | if (d0 < d) |
---|
639 | d= d0; |
---|
640 | if (d == 0) return N(gcdcAcB); |
---|
641 | |
---|
642 | CanonicalForm m, random_element, G_m, G_random_element, H, cH, ppH; |
---|
643 | CanonicalForm newtonPoly; |
---|
644 | |
---|
645 | newtonPoly= 1; |
---|
646 | m= gcdlcAlcB; |
---|
647 | G_m= 0; |
---|
648 | H= 0; |
---|
649 | bool fail= false; |
---|
650 | top_level= false; |
---|
651 | bool inextension= false; |
---|
652 | int p; |
---|
653 | int k= getGFDegree(); |
---|
654 | int kk; |
---|
655 | int expon; |
---|
656 | char gf_name_buf= gf_name; |
---|
657 | do |
---|
658 | { |
---|
659 | random_element= GFRandomElement (m, l, fail); |
---|
660 | if (fail) |
---|
661 | { |
---|
662 | int num_vars= tmin (getNumVars(A), getNumVars(B)); |
---|
663 | num_vars--; |
---|
664 | p= getCharacteristic(); |
---|
665 | expon= floor ((log ((double) ipower (d + 1, num_vars))/log ((double) p))); |
---|
666 | if (expon < 2) |
---|
667 | expon= 2; |
---|
668 | kk= getGFDegree(); |
---|
669 | if (ipower (p, kk*expon) < (1 << 16)) |
---|
670 | setCharacteristic (p, kk*(int)expon, 'b'); |
---|
671 | else |
---|
672 | { |
---|
673 | expon= floor((log ((double)((1<<16) - 1)))/(log((double)p)*kk)); |
---|
674 | ASSERT (expon >= 2, "not enough points in GCD_GF"); |
---|
675 | setCharacteristic (p, (int)(kk*expon), 'b'); |
---|
676 | } |
---|
677 | inextension= true; |
---|
678 | fail= false; |
---|
679 | for (CFListIterator i= l; i.hasItem(); i++) |
---|
680 | i.getItem()= GFMapUp (i.getItem(), kk); |
---|
681 | m= GFMapUp (m, kk); |
---|
682 | G_m= GFMapUp (G_m, kk); |
---|
683 | newtonPoly= GFMapUp (newtonPoly, kk); |
---|
684 | ppA= GFMapUp (ppA, kk); |
---|
685 | ppB= GFMapUp (ppB, kk); |
---|
686 | gcdlcAlcB= GFMapUp (gcdlcAlcB, kk); |
---|
687 | random_element= GFRandomElement (m, l, fail); |
---|
688 | DEBOUTLN (cerr, "fail= " << fail); |
---|
689 | CFList list; |
---|
690 | TIMING_START (gcd_recursion); |
---|
691 | G_random_element= GCD_GF (ppA(random_element, x), ppB(random_element, x), |
---|
692 | list, top_level); |
---|
693 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
694 | "time for recursive call: "); |
---|
695 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
696 | } |
---|
697 | else |
---|
698 | { |
---|
699 | CFList list; |
---|
700 | TIMING_START (gcd_recursion); |
---|
701 | G_random_element= GCD_GF (ppA(random_element, x), ppB(random_element, x), |
---|
702 | list, top_level); |
---|
703 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
704 | "time for recursive call: "); |
---|
705 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
706 | } |
---|
707 | |
---|
708 | d0= totaldegree (G_random_element, Variable(2), |
---|
709 | Variable (G_random_element.level())); |
---|
710 | if (d0 == 0) |
---|
711 | { |
---|
712 | if (inextension) |
---|
713 | { |
---|
714 | setCharacteristic (p, k, gf_name_buf); |
---|
715 | return N(gcdcAcB); |
---|
716 | } |
---|
717 | else |
---|
718 | return N(gcdcAcB); |
---|
719 | } |
---|
720 | if (d0 > d) |
---|
721 | { |
---|
722 | if (!find (l, random_element)) |
---|
723 | l.append (random_element); |
---|
724 | continue; |
---|
725 | } |
---|
726 | |
---|
727 | G_random_element= |
---|
728 | (gcdlcAlcB(random_element, x)/uni_lcoeff(G_random_element)) * |
---|
729 | G_random_element; |
---|
730 | d0= totaldegree (G_random_element, Variable(2), |
---|
731 | Variable (G_random_element.level())); |
---|
732 | |
---|
733 | if (d0 < d) |
---|
734 | { |
---|
735 | m= gcdlcAlcB; |
---|
736 | newtonPoly= 1; |
---|
737 | G_m= 0; |
---|
738 | d= d0; |
---|
739 | } |
---|
740 | |
---|
741 | TIMING_START (newton_interpolation); |
---|
742 | H= newtonInterp (random_element, G_random_element, newtonPoly, G_m, x); |
---|
743 | TIMING_END_AND_PRINT (newton_interpolation, "time for newton interpolation: "); |
---|
744 | |
---|
745 | //termination test |
---|
746 | if (uni_lcoeff (H) == gcdlcAlcB) |
---|
747 | { |
---|
748 | cH= uni_content (H); |
---|
749 | ppH= H/cH; |
---|
750 | if (inextension) |
---|
751 | { |
---|
752 | if (fdivides(ppH, GFMapUp(A, k)) && fdivides(ppH, GFMapUp(B,k))) |
---|
753 | { |
---|
754 | DEBOUTLN (cerr, "ppH before mapDown= " << ppH); |
---|
755 | ppH= GFMapDown (ppH, k); |
---|
756 | DEBOUTLN (cerr, "ppH after mapDown= " << ppH); |
---|
757 | setCharacteristic (p, k, gf_name_buf); |
---|
758 | return N(gcdcAcB*ppH); |
---|
759 | } |
---|
760 | } |
---|
761 | else |
---|
762 | { |
---|
763 | if (fdivides (ppH, A) && fdivides (ppH, B)) |
---|
764 | return N(gcdcAcB*ppH); |
---|
765 | } |
---|
766 | } |
---|
767 | |
---|
768 | G_m= H; |
---|
769 | newtonPoly= newtonPoly*(x - random_element); |
---|
770 | m= m*(x - random_element); |
---|
771 | if (!find (l, random_element)) |
---|
772 | l.append (random_element); |
---|
773 | } while (1); |
---|
774 | } |
---|
775 | |
---|
776 | /// F is assumed to be an univariate polynomial in x, |
---|
777 | /// computes a random monic irreducible univariate polynomial of random |
---|
778 | /// degree < i in x which does not divide F |
---|
779 | CanonicalForm |
---|
780 | randomIrredpoly (int i, const Variable & x, |
---|
781 | CFList & list) |
---|
782 | { |
---|
783 | int random; |
---|
784 | int p= getCharacteristic(); |
---|
785 | ZZ NTLp= to_ZZ (p); |
---|
786 | ZZ_p::init (NTLp); |
---|
787 | ZZ_pX NTLirredpoly; |
---|
788 | CanonicalForm CFirredpoly; |
---|
789 | int buf= ceil((log((double) i) / log((double) p))); |
---|
790 | buf= buf + 1; |
---|
791 | int bound= 0; |
---|
792 | //lower bound on the number of monic irreducibles of degree less than buf |
---|
793 | for (int j= 2; j < buf; j++) |
---|
794 | bound += ((ipower (p, j) - 2*ipower(p, j/2)) / j); |
---|
795 | if (list.length() > bound) |
---|
796 | { |
---|
797 | if (buf > 1) |
---|
798 | buf--; |
---|
799 | buf *= 2; |
---|
800 | buf++; |
---|
801 | } |
---|
802 | for (int j= ((buf - 1)/2) + 1; j < buf; j++) |
---|
803 | bound += ((ipower (p, j) - 2*ipower(p, j/2)) / j); |
---|
804 | do |
---|
805 | { |
---|
806 | if (list.length() <= bound) |
---|
807 | { |
---|
808 | random= factoryrandom ((int) buf); |
---|
809 | if (random <= 1) |
---|
810 | random= 2; |
---|
811 | } |
---|
812 | else |
---|
813 | { |
---|
814 | if (buf > 1) |
---|
815 | buf--; |
---|
816 | buf *= 2; |
---|
817 | buf++; |
---|
818 | for (int j= ((int) (buf - 1)/2) + 1; j < buf; j++) |
---|
819 | bound += ((ipower (p, j) - 2*ipower(p, j/2)) / j); |
---|
820 | random= factoryrandom ((int) buf); |
---|
821 | if (random <= 1) |
---|
822 | random= 2; |
---|
823 | } |
---|
824 | BuildIrred (NTLirredpoly, random); |
---|
825 | CFirredpoly= convertNTLZZpX2CF (NTLirredpoly, x); |
---|
826 | } while (find (list, CFirredpoly)); |
---|
827 | return CFirredpoly; |
---|
828 | } |
---|
829 | |
---|
830 | CanonicalForm |
---|
831 | FpRandomElement (const CanonicalForm& F, CFList& list, bool& fail) |
---|
832 | { |
---|
833 | fail= false; |
---|
834 | Variable x= F.mvar(); |
---|
835 | FFRandom genFF; |
---|
836 | CanonicalForm random; |
---|
837 | int p= getCharacteristic(); |
---|
838 | int bound= p; |
---|
839 | do |
---|
840 | { |
---|
841 | if (list.length() == bound) |
---|
842 | { |
---|
843 | fail= true; |
---|
844 | break; |
---|
845 | } |
---|
846 | if (list.length() < 1) |
---|
847 | random= 0; |
---|
848 | else |
---|
849 | { |
---|
850 | random= genFF.generate(); |
---|
851 | while (find (list, random)) |
---|
852 | random= genFF.generate(); |
---|
853 | } |
---|
854 | if (F (random, x) == 0) |
---|
855 | { |
---|
856 | list.append (random); |
---|
857 | continue; |
---|
858 | } |
---|
859 | } while (find (list, random)); |
---|
860 | return random; |
---|
861 | } |
---|
862 | |
---|
863 | CanonicalForm GCD_small_p (const CanonicalForm& F, const CanonicalForm& G, |
---|
864 | bool& top_level, CFList& l) |
---|
865 | { |
---|
866 | CanonicalForm A= F; |
---|
867 | CanonicalForm B= G; |
---|
868 | if (F.isZero() && degree(G) > 0) return G/Lc(G); |
---|
869 | else if (G.isZero() && degree (F) > 0) return F/Lc(F); |
---|
870 | else if (F.isZero() && G.isZero()) return F.genOne(); |
---|
871 | if (F.inBaseDomain() || G.inBaseDomain()) return F.genOne(); |
---|
872 | if (F.isUnivariate() && fdivides(F, G)) return F/Lc(F); |
---|
873 | if (G.isUnivariate() && fdivides(G, F)) return G/Lc(G); |
---|
874 | if (F == G) return F/Lc(F); |
---|
875 | |
---|
876 | CFMap M,N; |
---|
877 | int best_level= myCompress (A, B, M, N, top_level); |
---|
878 | |
---|
879 | if (best_level == 0) return B.genOne(); |
---|
880 | |
---|
881 | A= M(A); |
---|
882 | B= M(B); |
---|
883 | |
---|
884 | Variable x= Variable (1); |
---|
885 | |
---|
886 | //univariate case |
---|
887 | if (A.isUnivariate() && B.isUnivariate()) |
---|
888 | return N (gcd (A, B)); |
---|
889 | |
---|
890 | CanonicalForm cA, cB; // content of A and B |
---|
891 | CanonicalForm ppA, ppB; // primitive part of A and B |
---|
892 | CanonicalForm gcdcAcB; |
---|
893 | cA = uni_content (A); |
---|
894 | cB = uni_content (B); |
---|
895 | gcdcAcB= gcd (cA, cB); |
---|
896 | ppA= A/cA; |
---|
897 | ppB= B/cB; |
---|
898 | |
---|
899 | CanonicalForm lcA, lcB; // leading coefficients of A and B |
---|
900 | CanonicalForm gcdlcAlcB; |
---|
901 | lcA= uni_lcoeff (ppA); |
---|
902 | lcB= uni_lcoeff (ppB); |
---|
903 | |
---|
904 | if (fdivides (lcA, lcB)) |
---|
905 | { |
---|
906 | if (fdivides (A, B)) |
---|
907 | return F/Lc(F); |
---|
908 | } |
---|
909 | if (fdivides (lcB, lcA)) |
---|
910 | { |
---|
911 | if (fdivides (B, A)) |
---|
912 | return G/Lc(G); |
---|
913 | } |
---|
914 | |
---|
915 | gcdlcAlcB= gcd (lcA, lcB); |
---|
916 | |
---|
917 | int d= totaldegree (ppA, Variable (2), Variable (ppA.level())); |
---|
918 | int d0; |
---|
919 | |
---|
920 | if (d == 0) return N(gcdcAcB); |
---|
921 | d0= totaldegree (ppB, Variable (2), Variable (ppB.level())); |
---|
922 | |
---|
923 | if (d0 < d) |
---|
924 | d= d0; |
---|
925 | |
---|
926 | if (d == 0) return N(gcdcAcB); |
---|
927 | |
---|
928 | CanonicalForm m, random_element, G_m, G_random_element, H, cH, ppH; |
---|
929 | CanonicalForm newtonPoly= 1; |
---|
930 | m= gcdlcAlcB; |
---|
931 | H= 0; |
---|
932 | G_m= 0; |
---|
933 | Variable alpha, V_buf; |
---|
934 | int p, k; |
---|
935 | bool fail= false; |
---|
936 | bool inextension= false; |
---|
937 | bool inextensionextension= false; |
---|
938 | top_level= false; |
---|
939 | CanonicalForm CF_buf; |
---|
940 | CFList source, dest; |
---|
941 | CanonicalForm gcdcheck; |
---|
942 | do |
---|
943 | { |
---|
944 | if (inextension) |
---|
945 | random_element= randomElement (m, alpha, l, fail); |
---|
946 | else |
---|
947 | random_element= FpRandomElement (m, l, fail); |
---|
948 | |
---|
949 | if (!fail && !inextension) |
---|
950 | { |
---|
951 | CFList list; |
---|
952 | TIMING_START (gcd_recursion); |
---|
953 | G_random_element= |
---|
954 | GCD_small_p (ppA (random_element,x), ppB (random_element,x), top_level, |
---|
955 | list); |
---|
956 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
957 | "time for recursive call: "); |
---|
958 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
959 | } |
---|
960 | else if (!fail && inextension) |
---|
961 | { |
---|
962 | CFList list; |
---|
963 | TIMING_START (gcd_recursion); |
---|
964 | G_random_element= |
---|
965 | GCD_Fp_extension (ppA (random_element, x), ppB (random_element, x), alpha, |
---|
966 | list, top_level); |
---|
967 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
968 | "time for recursive call: "); |
---|
969 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
970 | } |
---|
971 | else if (fail && !inextension) |
---|
972 | { |
---|
973 | source= CFList(); |
---|
974 | dest= CFList(); |
---|
975 | CFList list; |
---|
976 | CanonicalForm mipo; |
---|
977 | int deg= 3; |
---|
978 | do { |
---|
979 | mipo= randomIrredpoly (deg, x, list); |
---|
980 | alpha= rootOf (mipo); |
---|
981 | inextension= true; |
---|
982 | fail= false; |
---|
983 | random_element= randomElement (m, alpha, l, fail); |
---|
984 | deg++; |
---|
985 | } while (fail); |
---|
986 | list= CFList(); |
---|
987 | TIMING_START (gcd_recursion); |
---|
988 | G_random_element= |
---|
989 | GCD_Fp_extension (ppA (random_element, x), ppB (random_element, x), alpha, |
---|
990 | list, top_level); |
---|
991 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
992 | "time for recursive call: "); |
---|
993 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
994 | } |
---|
995 | else if (fail && inextension) |
---|
996 | { |
---|
997 | source= CFList(); |
---|
998 | dest= CFList(); |
---|
999 | int num_vars= tmin (getNumVars(A), getNumVars(B)); |
---|
1000 | num_vars--; |
---|
1001 | V_buf= alpha; |
---|
1002 | choose_extension (d, num_vars, V_buf); |
---|
1003 | bool prim_fail= false; |
---|
1004 | Variable V_buf2; |
---|
1005 | CanonicalForm prim_elem, im_prim_elem; |
---|
1006 | prim_elem= primitiveElement (alpha, V_buf2, prim_fail); |
---|
1007 | |
---|
1008 | ASSERT (!prim_fail, "failure in integer factorizer"); |
---|
1009 | if (prim_fail) |
---|
1010 | ; //ERROR |
---|
1011 | else |
---|
1012 | im_prim_elem= mapPrimElem (prim_elem, alpha, V_buf); |
---|
1013 | |
---|
1014 | DEBOUTLN (cerr, "getMipo (alpha)= " << getMipo (alpha)); |
---|
1015 | DEBOUTLN (cerr, "getMipo (alpha)= " << getMipo (V_buf2)); |
---|
1016 | |
---|
1017 | inextensionextension= true; |
---|
1018 | for (CFListIterator i= l; i.hasItem(); i++) |
---|
1019 | i.getItem()= mapUp (i.getItem(), alpha, V_buf, prim_elem, |
---|
1020 | im_prim_elem, source, dest); |
---|
1021 | m= mapUp (m, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
---|
1022 | G_m= mapUp (G_m, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
---|
1023 | newtonPoly= mapUp (newtonPoly, alpha, V_buf, prim_elem, im_prim_elem, |
---|
1024 | source, dest); |
---|
1025 | ppA= mapUp (ppA, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
---|
1026 | ppB= mapUp (ppB, alpha, V_buf, prim_elem, im_prim_elem, source, dest); |
---|
1027 | gcdlcAlcB= mapUp (gcdlcAlcB, alpha, V_buf, prim_elem, im_prim_elem, |
---|
1028 | source, dest); |
---|
1029 | fail= false; |
---|
1030 | random_element= randomElement (m, V_buf, l, fail ); |
---|
1031 | DEBOUTLN (cerr, "fail= " << fail); |
---|
1032 | CFList list; |
---|
1033 | TIMING_START (gcd_recursion); |
---|
1034 | G_random_element= |
---|
1035 | GCD_Fp_extension (ppA (random_element, x), ppB (random_element, x), V_buf, |
---|
1036 | list, top_level); |
---|
1037 | TIMING_END_AND_PRINT (gcd_recursion, |
---|
1038 | "time for recursive call: "); |
---|
1039 | DEBOUTLN (cerr, "G_random_element= " << G_random_element); |
---|
1040 | alpha= V_buf; |
---|
1041 | } |
---|
1042 | |
---|
1043 | d0= totaldegree (G_random_element, Variable(2), |
---|
1044 | Variable (G_random_element.level())); |
---|
1045 | |
---|
1046 | if (d0 == 0) |
---|
1047 | { |
---|
1048 | return N(gcdcAcB); |
---|
1049 | } |
---|
1050 | if (d0 > d) |
---|
1051 | { |
---|
1052 | if (!find (l, random_element)) |
---|
1053 | l.append (random_element); |
---|
1054 | continue; |
---|
1055 | } |
---|
1056 | |
---|
1057 | G_random_element= (gcdlcAlcB(random_element,x)/uni_lcoeff(G_random_element)) |
---|
1058 | *G_random_element; |
---|
1059 | |
---|
1060 | |
---|
1061 | d0= totaldegree (G_random_element, Variable(2), |
---|
1062 | Variable(G_random_element.level())); |
---|
1063 | |
---|
1064 | if (d0 < d) |
---|
1065 | { |
---|
1066 | m= gcdlcAlcB; |
---|
1067 | newtonPoly= 1; |
---|
1068 | G_m= 0; |
---|
1069 | d= d0; |
---|
1070 | } |
---|
1071 | |
---|
1072 | TIMING_START (newton_interpolation); |
---|
1073 | H= newtonInterp (random_element, G_random_element, newtonPoly, G_m, x); |
---|
1074 | TIMING_END_AND_PRINT (newton_interpolation, |
---|
1075 | "time for newton_interpolation: "); |
---|
1076 | |
---|
1077 | //termination test |
---|
1078 | if (uni_lcoeff (H) == gcdlcAlcB) |
---|
1079 | { |
---|
1080 | cH= uni_content (H); |
---|
1081 | ppH= H/cH; |
---|
1082 | ppH /= Lc (ppH); |
---|
1083 | DEBOUTLN (cerr, "ppH= " << ppH); |
---|
1084 | if (fdivides (ppH, A) && fdivides (ppH, B)) |
---|
1085 | return N(gcdcAcB*ppH); |
---|
1086 | } |
---|
1087 | |
---|
1088 | G_m= H; |
---|
1089 | newtonPoly= newtonPoly*(x - random_element); |
---|
1090 | m= m*(x - random_element); |
---|
1091 | if (!find (l, random_element)) |
---|
1092 | l.append (random_element); |
---|
1093 | } while (1); |
---|
1094 | } |
---|
1095 | |
---|
1096 | #endif |
---|