1 | #ifndef LLL_CC |
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2 | #define LLL_CC |
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3 | |
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4 | #include "LLL.h" |
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5 | |
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6 | // subroutines for the LLL-algorithms |
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7 | |
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8 | void REDI_KB(const short& k, const short& l, BigInt** b, |
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9 | const short& number_of_vectors, const short& vector_dimension, |
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10 | BigInt** H, BigInt* d, BigInt** lambda) |
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11 | // the REDI procedure for relations(...) (to compute the Kernel Basis, |
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12 | // algorithm 2.7.2 in Cohen's book) |
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13 | { |
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14 | #ifdef GMP |
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15 | if(abs(BigInt(2)*lambda[k][l])<=d[l+1]) |
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16 | #else // GMP |
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17 | if(labs(2*lambda[k][l])<=d[l+1]) |
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18 | // labs is the abs-function for long ints |
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19 | #endif // GMP |
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20 | return; |
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21 | |
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22 | #ifdef GMP |
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23 | BigInt q=(BigInt(2)*lambda[k][l]+d[l+1])/(BigInt(2)*d[l+1]); |
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24 | #else // GMP |
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25 | long q=(long int) floor(((float)(2*lambda[k][l]+d[l+1]))/(2*d[l+1])); |
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26 | #endif // GMP |
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27 | |
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28 | // q is the integer quotient of the division |
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29 | // (2*lambda[k][l]+d[l+1])/(2*d[l+1]). |
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30 | // Because of the rounding mode (always towards zero) of GNU C++, |
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31 | // we cannot use the built-in integer division |
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32 | // here; it causes errors when dealing with negative numbers. Therefore |
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33 | // the complicated casts: The divident is first casted to a float which |
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34 | // causes the division result to be a float. This result is explicitly |
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35 | // rounded downwards. As the floor-function returns a double (for range |
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36 | // reasons), this has to be casted to an integer again. |
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37 | |
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38 | for(short n=0;n<number_of_vectors;n++) |
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39 | H[k][n]-=q*H[l][n]; |
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40 | // H[k]=H[k]-q*H[l] |
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41 | |
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42 | for(short m=0;m<vector_dimension;m++) |
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43 | b[k][m]-=q*b[l][m]; |
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44 | // b[k]=b[k]-q*b[l] |
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45 | |
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46 | lambda[k][l]-=q*d[l+1]; |
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47 | |
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48 | for(short i=0;i<=l-1;i++) |
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49 | lambda[k][i]-=q*lambda[l][i]; |
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50 | } |
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51 | |
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52 | |
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53 | |
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54 | |
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55 | void REDI_IL(const short& k, const short& l, BigInt** b, |
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56 | const short& vector_dimension, BigInt* d, BigInt** lambda) |
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57 | // the REDI procedure for the integer LLL algorithm (algorithm 2.6.7 in |
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58 | // Cohen's book) |
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59 | { |
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60 | #ifdef GMP |
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61 | if(abs(BigInt(2)*lambda[k][l])<=d[l+1]) |
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62 | #else // GMP |
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63 | if(labs(2*lambda[k][l])<=d[l+1]) |
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64 | // labs is the abs-function for long ints |
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65 | #endif // GMP |
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66 | return; |
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67 | |
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68 | #ifdef GMP |
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69 | BigInt q=(BigInt(2)*lambda[k][l]+d[l+1])/(BigInt(2)*d[l+1]); |
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70 | #else // GMP |
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71 | long q=(long int) floor(((float)(2*lambda[k][l]+d[l+1]))/(2*d[l+1])); |
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72 | #endif // GMP |
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73 | |
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74 | // q is the integer quotient of the division |
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75 | // (2*lambda[k][l]+d[l+1])/(2*d[l+1]). |
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76 | // Because of the rounding mode (always towards zero) of GNU C++, |
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77 | // we cannot use the built-in integer division |
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78 | // here; it causes errors when dealing with negative numbers. Therefore |
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79 | // the complicated casts: The divident is first casted to a float which |
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80 | // causes the division result to be a float. This result is explicitly |
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81 | // rounded downwards. As the floor-function returns a double (for range |
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82 | // reasons), this has to be casted to an integer again. |
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83 | |
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84 | for(short m=0;m<vector_dimension;m++) |
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85 | b[k][m]-=q*b[l][m]; |
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86 | // b[k]=b[k]-q*b[l] |
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87 | |
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88 | lambda[k][l]-=q*d[l+1]; |
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89 | |
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90 | for(short i=0;i<=l-1;i++) |
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91 | lambda[k][i]-=q*lambda[l][i]; |
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92 | } |
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93 | |
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94 | |
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95 | |
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96 | |
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97 | void SWAPI(const short& k, const short& k_max, BigInt** b, BigInt* d, |
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98 | BigInt** lambda) |
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99 | // the SWAPI procedure of algorithm 2.6.7 |
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100 | { |
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101 | |
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102 | // exchange b[k] and b[k-1] |
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103 | // This can be done efficiently by swapping pointers (not entries). |
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104 | BigInt* swap=b[k]; |
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105 | b[k]=b[k-1]; |
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106 | b[k-1]=swap; |
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107 | |
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108 | if(k>1) |
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109 | for(short j=0;j<=k-2;j++) |
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110 | { |
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111 | // exchange lambda[k][j] and lambda[k-1][j] |
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112 | BigInt swap=lambda[k][j]; |
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113 | lambda[k][j]=lambda[k-1][j]; |
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114 | lambda[k-1][j]=swap; |
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115 | } |
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116 | |
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117 | BigInt _lambda=lambda[k][k-1]; |
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118 | |
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119 | BigInt B=(d[k-1]*d[k+1] + _lambda*_lambda)/d[k]; |
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120 | // It might be better to choose another evaluation order for this formula, |
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121 | // see below. |
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122 | |
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123 | for(short i=k+1;i<=k_max;i++) |
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124 | { |
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125 | BigInt t=lambda[i][k]; |
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126 | lambda[i][k]=(d[k+1]*lambda[i][k-1] - _lambda*t)/d[k]; |
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127 | lambda[i][k-1]=(B*t + _lambda*lambda[i][k])/d[k+1]; |
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128 | } |
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129 | |
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130 | d[k]=B; |
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131 | } |
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132 | |
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133 | |
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134 | |
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135 | |
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136 | void SWAPK(const short& k, const short& k_max, BigInt** b, BigInt** H, |
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137 | char *f, BigInt* d, BigInt** lambda) |
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138 | // the SWAPK procedure of algorithm 2.7.2 |
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139 | { |
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140 | // exchange H[k] and H[k-1] |
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141 | // This can be done efficiently by swapping pointers (not entries). |
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142 | BigInt *swap=H[k]; |
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143 | H[k]=H[k-1]; |
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144 | H[k-1]=swap; |
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145 | |
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146 | // exchange b[k] and b[k-1] by the same method |
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147 | swap=b[k]; |
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148 | b[k]=b[k-1]; |
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149 | b[k-1]=swap; |
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150 | |
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151 | if(k>1) |
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152 | for(short j=0;j<=k-2;j++) |
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153 | { |
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154 | // exchange lambda[k][j] and lambda[k-1][j] |
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155 | BigInt swap=lambda[k][j]; |
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156 | lambda[k][j]=lambda[k-1][j]; |
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157 | lambda[k-1][j]=swap; |
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158 | } |
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159 | |
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160 | BigInt _lambda=lambda[k][k-1]; |
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161 | |
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162 | if(_lambda==BigInt(0)) |
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163 | { |
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164 | d[k]=d[k-1]; |
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165 | f[k-1]=0; |
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166 | f[k]=1; |
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167 | lambda[k][k-1]=0; |
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168 | for(short i=k+1;i<=k_max;i++) |
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169 | { |
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170 | lambda[i][k]=lambda[i][k-1]; |
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171 | lambda[i][k-1]=0; |
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172 | } |
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173 | } |
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174 | else |
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175 | // lambda!=0 |
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176 | { |
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177 | for(short i=k+1;i<=k_max;i++) |
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178 | lambda[i][k-1]=(_lambda*lambda[i][k-1])/d[k]; |
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179 | |
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180 | // Multiplie lambda[i][k-1] by _lambda/d[k]. |
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181 | // One could also write |
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182 | // lambda[i][k-1]*=(lambda/d[k]); (*) |
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183 | // Without a BigInt class, this can prevent overflows when computing |
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184 | // _lambda*lambda[i][k-1]. |
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185 | // But examples show that lambda/d[k] is in general not an integer. |
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186 | // So (*) could lead to problems due to the inexact floating point |
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187 | // arithmetic... |
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188 | // Therefore, we chose the secure evaluation order in all such cases. |
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189 | |
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190 | BigInt t=d[k+1]; |
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191 | d[k]=(_lambda*_lambda)/d[k]; |
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192 | d[k+1]=d[k]; |
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193 | |
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194 | for(short j=k+1;j<=k_max-1;j++) |
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195 | for(short i=j+1;i<=k_max;i++) |
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196 | lambda[i][j]=(lambda[i][j]*d[k])/t; |
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197 | |
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198 | for(short j=k+1;j<=k_max;j++) |
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199 | d[j+1]=(d[j+1]*d[k])/t; |
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200 | } |
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201 | |
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202 | } |
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203 | |
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204 | typedef BigInt* BigIntP; |
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205 | |
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206 | |
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207 | short relations(BigInt **b, const short& number_of_vectors, |
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208 | const short& vector_dimension, BigInt**& H) |
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209 | { |
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210 | |
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211 | // first check arguments |
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212 | |
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213 | if(number_of_vectors<0) |
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214 | { |
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215 | cerr<<"\nWARNING: short relations(BigInt**, const short&, const short&, " |
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216 | "BigInt**):\nargument number_of_vectors out of range"<<endl; |
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217 | return -1; |
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218 | } |
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219 | |
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220 | if(vector_dimension<=0) |
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221 | { |
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222 | cerr<<"\nWARNING: short relations(BigInt**, const short&, const short&, " |
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223 | "BigInt**):\nargument vector_dimension out of range"<<endl; |
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224 | return -1; |
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225 | } |
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226 | |
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227 | |
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228 | // consider special case |
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229 | |
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230 | if(number_of_vectors==1) |
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231 | // Only one vector which has no relations if it is not zero, |
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232 | // else relation 1. |
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233 | { |
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234 | short r=1; // Suppose the only column of the matrix is zero. |
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235 | |
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236 | for(short m=0;m<vector_dimension;m++) |
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237 | if(b[0][m]!=BigInt(0)) |
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238 | // nonzero entry detected |
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239 | r=0; |
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240 | |
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241 | if(r==1) |
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242 | { |
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243 | H=new BigIntP[1]; |
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244 | H[0]=new BigInt[1]; |
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245 | H[0][0]=1; |
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246 | // This is the lattice basis of the relations... |
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247 | } |
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248 | |
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249 | return r; |
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250 | } |
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251 | |
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252 | |
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253 | // memory allocation |
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254 | |
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255 | // The names are chosen (as far as possible) according to Cohen's book. |
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256 | // However, for technical reasons, the indices do not run from 1 to |
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257 | // (e.g.) number_of_vectors, but from 0 to number_of_vectors-1. |
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258 | // Therefore all indices are shifted by -1 in comparison with this book, |
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259 | // except from the indices of the array d which has size |
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260 | // number_of_vectors+1. |
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261 | |
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262 | H=new BigIntP[number_of_vectors]; |
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263 | for(short n=0;n<number_of_vectors;n++) |
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264 | H[n]=new BigInt[number_of_vectors]; |
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265 | |
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266 | char* f=new char[number_of_vectors]; |
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267 | |
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268 | BigInt* d=new BigInt[number_of_vectors+1]; |
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269 | |
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270 | BigInt** lambda=new BigIntP[number_of_vectors]; |
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271 | for(short n=1;n<number_of_vectors;n++) |
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272 | lambda[n]=new BigInt[n]; |
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273 | // We only need lambda[n][k] for n>k. |
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274 | |
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275 | |
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276 | |
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277 | // Step 1: Initialization |
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278 | |
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279 | short k=1; |
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280 | short k_max=0; |
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281 | // for iteration |
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282 | |
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283 | d[0]=1; |
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284 | |
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285 | BigInt t=0; |
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286 | for(short m=0;m<vector_dimension;m++) |
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287 | t+=b[0][m]*b[0][m]; |
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288 | // Now, t is the scalar product of b[0] with itself. |
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289 | |
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290 | for(short n=0;n<number_of_vectors;n++) |
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291 | for(short l=0;l<number_of_vectors;l++) |
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292 | if(n==l) |
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293 | H[n][l]=1; |
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294 | else |
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295 | H[n][l]=0; |
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296 | // Now, H equals the matrix I_(number_of_vectors). |
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297 | |
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298 | if(t!=BigInt(0)) |
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299 | { |
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300 | d[1]=t; |
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301 | f[0]=1; |
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302 | } |
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303 | else |
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304 | { |
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305 | d[1]=1; |
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306 | f[0]=0; |
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307 | } |
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308 | |
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309 | |
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310 | |
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311 | // The other steps are not programmed with "goto" as in Cohen's book. |
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312 | // Instead, we enter a do-while-loop which terminates iff |
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313 | // k>=number_of_vectors. |
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314 | |
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315 | do |
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316 | { |
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317 | |
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318 | |
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319 | // Step 2: Incremental Gram-Schmidt |
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320 | |
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321 | if(k>k_max) |
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322 | // else we can immediately go to step 3. |
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323 | { |
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324 | k_max=k; |
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325 | |
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326 | for(short j=0;j<=k;j++) |
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327 | if((f[j]==0) && (j<k)) |
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328 | lambda[k][j]=0; |
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329 | else |
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330 | { |
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331 | BigInt u=0; |
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332 | |
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333 | // compute scalar product of b[k] and b[j] |
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334 | for(short m=0;m<vector_dimension;m++) |
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335 | u+=b[k][m]*b[j][m]; |
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336 | |
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337 | for(short i=0;i<=j-1;i++) |
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338 | if(f[i]!=0) |
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339 | u=(d[i+1]*u-lambda[k][i]*lambda[j][i])/d[i]; |
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340 | |
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341 | if(j<k) |
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342 | lambda[k][j]=u; |
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343 | else |
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344 | // j==k |
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345 | if(u!=BigInt(0)) |
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346 | { |
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347 | d[k+1]=u; |
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348 | f[k]=1; |
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349 | } |
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350 | else |
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351 | // u==0 |
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352 | { |
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353 | d[k+1]=d[k]; |
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354 | f[k]=0; |
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355 | } |
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356 | } |
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357 | } |
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358 | |
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359 | |
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360 | // Step 3: Test f[k]==0 and f[k-1]!=0 |
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361 | |
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362 | do |
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363 | { |
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364 | if(f[k-1]!=0) |
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365 | REDI_KB(k,k-1,b,number_of_vectors,vector_dimension,H,d,lambda); |
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366 | |
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367 | if((f[k-1]!=0) && (f[k]==0)) |
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368 | { |
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369 | SWAPK(k,k_max,b,H,f,d,lambda); |
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370 | |
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371 | if(k>1) |
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372 | k--; |
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373 | else |
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374 | k=1; |
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375 | // k=max(1,k-1) |
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376 | } |
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377 | |
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378 | else |
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379 | break; |
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380 | } |
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381 | while(1); |
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382 | |
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383 | // Now the conditions above are no longer satisfied. |
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384 | |
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385 | for(short l=k-2;l>=0;l--) |
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386 | if(f[l]!=0) |
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387 | REDI_KB(k,l,b,number_of_vectors,vector_dimension,H,d,lambda); |
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388 | k++; |
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389 | |
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390 | |
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391 | // Step 4: Finished? |
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392 | |
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393 | } |
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394 | while(k<number_of_vectors); |
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395 | |
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396 | |
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397 | |
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398 | // Now we have computed a lattice basis of the relations of the b[i]. |
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399 | // Prepare the LLL-reduction. |
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400 | |
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401 | // Compute the dimension r of the relations. |
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402 | short r=0; |
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403 | for(short n=0;n<number_of_vectors;n++) |
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404 | if(f[n]==0) // n==r!! |
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405 | r++; |
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406 | else |
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407 | break; |
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408 | |
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409 | // Delete the part of H that is no longer needed (especially the vectors |
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410 | // H[r],...,H[number_of_vectors-1]). |
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411 | BigInt **aux=H; |
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412 | if(r>0) |
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413 | H=new BigIntP[r]; |
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414 | for(short i=0;i<r;i++) |
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415 | H[i]=aux[i]; |
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416 | |
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417 | for(short i=r;i<number_of_vectors;i++) |
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418 | delete[] aux[i]; |
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419 | delete[] aux; |
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420 | |
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421 | delete[] f; |
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422 | |
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423 | delete[] d; |
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424 | |
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425 | for(short i=1;i<number_of_vectors;i++) |
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426 | delete[] lambda[i]; |
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427 | delete[] lambda; |
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428 | |
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429 | integral_LLL(H,r,number_of_vectors); |
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430 | |
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431 | return r; |
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432 | |
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433 | } |
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434 | |
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435 | |
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436 | |
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437 | |
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438 | short integral_LLL(BigInt** b, const short& number_of_vectors, |
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439 | const short& vector_dimension) |
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440 | { |
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441 | |
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442 | // first check arguments |
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443 | |
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444 | if(number_of_vectors<0) |
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445 | { |
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446 | cerr<<"\nWARNING: short integral_LL(BigInt**, const short&, const short&):" |
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447 | "\nargument number_of_vectors out of range"<<endl; |
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448 | return -1; |
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449 | } |
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450 | |
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451 | if(vector_dimension<=0) |
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452 | { |
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453 | cerr<<"\nWARNING: short integral_LLL(BigInt**, const short&, const " |
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454 | "short&):\nargument vector_dimension out of range"<<endl; |
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455 | return -1; |
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456 | } |
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457 | |
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458 | |
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459 | // consider special case |
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460 | |
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461 | if(number_of_vectors<=1) |
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462 | // 0 or 1 input vector, nothing to be done |
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463 | return 0; |
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464 | |
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465 | |
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466 | // memory allocation |
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467 | |
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468 | // The names are chosen (as far as possible) according to Cohen's book. |
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469 | // However, for technical reasons, the indices do not run from 1 to |
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470 | // (e.g.) number_of_vectors, but from 0 to number_of_vectors-1. |
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471 | // Therefore all indices are shifted by -1 in comparison with this book, |
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472 | // except from the indices of the array d which has size |
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473 | // number_of_vectors+1. |
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474 | |
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475 | BigInt* d=new BigInt[number_of_vectors+1]; |
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476 | |
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477 | BigInt** lambda=new BigIntP[number_of_vectors]; |
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478 | for(short s=1;s<number_of_vectors;s++) |
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479 | lambda[s]=new BigInt[s]; |
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480 | // We only need lambda[n][k] for n>k. |
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481 | |
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482 | |
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483 | |
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484 | // Step 1: Initialization |
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485 | |
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486 | short k=1; |
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487 | short k_max=0; |
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488 | // for iteration |
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489 | d[0]=1; |
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490 | |
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491 | d[1]=0; |
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492 | for(short n=0;n<vector_dimension;n++) |
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493 | d[1]+=b[0][n]*b[0][n]; |
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494 | // Now, d[1] is the scalar product of b[0] with itself. |
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495 | |
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496 | |
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497 | // The other steps are not programmed with "goto" as in Cohen's book. |
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498 | // Instead, we enter a do-while-loop which terminates iff k>r. |
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499 | |
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500 | do |
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501 | { |
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502 | |
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503 | |
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504 | // Step 2: Incremental Gram-Schmidt |
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505 | |
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506 | if(k>k_max) |
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507 | // else we can immediately go to step 3. |
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508 | { |
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509 | k_max=k; |
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510 | |
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511 | for(short j=0;j<=k;j++) |
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512 | { |
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513 | BigInt u=0; |
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514 | // compute scalar product of b[k] and b[j] |
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515 | for(short n=0;n<vector_dimension;n++) |
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516 | u+=b[k][n]*b[j][n]; |
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517 | |
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518 | for(short i=0;i<=j-1;i++) |
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519 | { |
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520 | u*=d[i+1]; |
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521 | u-=lambda[k][i]*lambda[j][i]; |
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522 | u/=d[i]; |
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523 | |
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524 | //u=(d[i+1]*u-lambda[k][i]*lambda[j][i])/d[i]; |
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525 | } |
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526 | |
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527 | if(j<k) |
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528 | lambda[k][j]=u; |
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529 | else |
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530 | // j==k |
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531 | d[k+1]=u; |
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532 | } |
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533 | |
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534 | if(d[k+1]==BigInt(0)) |
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535 | { |
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536 | cerr<<"\nERROR: void integral_LLL(BigInt**, const short&, const " |
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537 | "short&):\ninput vectors must be linearly independent"<<endl; |
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538 | return -1; |
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539 | } |
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540 | } |
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541 | |
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542 | |
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543 | // Step 3: Test LLL-condition |
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544 | |
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545 | do |
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546 | { |
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547 | REDI_IL(k,k-1,b,vector_dimension,d,lambda); |
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548 | |
---|
549 | //if(4*d[k+1]*d[k-1] < 3*d[k]*d[k] - lambda[k][k-1]*lambda[k][k-1]) |
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550 | if((BigInt(4))*d[k+1]*d[k-1] |
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551 | < (BigInt(3))*d[k]*d[k] - lambda[k][k-1]*lambda[k][k-1]) |
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552 | { |
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553 | SWAPI(k,k_max,b,d,lambda); |
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554 | if(k>1) |
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555 | k--; |
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556 | // k=max(1,k-1) |
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557 | } |
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558 | else |
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559 | break; |
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560 | |
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561 | } |
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562 | while(1); |
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563 | |
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564 | // Now the condition above is no longer satisfied. |
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565 | |
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566 | for(short l=k-2;l>=0;l--) |
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567 | REDI_IL(k,l,b,vector_dimension,d,lambda); |
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568 | k++; |
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569 | |
---|
570 | |
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571 | |
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572 | // Step 4: Finished? |
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573 | |
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574 | } |
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575 | while(k<number_of_vectors); |
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576 | |
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577 | |
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578 | // Now, b contains the LLL-reduced lattice basis. |
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579 | // Memory cleanup. |
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580 | |
---|
581 | delete[] d; |
---|
582 | |
---|
583 | for(short i=1;i<number_of_vectors;i++) |
---|
584 | delete[] lambda[i]; |
---|
585 | delete[] lambda; |
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586 | |
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587 | return 0; |
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588 | |
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589 | } |
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590 | #endif // LLL_CC |
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