1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: AtkinsTest.lib,v 1.6 2006-12-14 17:30:46 Singular Exp $"; |
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3 | category="Teaching"; |
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4 | info=" |
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5 | LIBRARY: AtkinsTest.lib Procedures for teaching cryptography |
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6 | AUTHOR: Stefan Steidel, Stefan.Steidel@gmx.de |
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7 | |
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8 | NOTE: The library contains auxiliary procedures to compute the elliptic |
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9 | curve primality test of Atkin and the Atkin's Test itself. |
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10 | The library is intended to be used for teaching purposes but not |
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11 | for serious computations. Sufficiently high printLevel allows to |
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12 | control each step, thus illustrating the algorithms at work. |
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13 | |
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14 | |
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15 | PROCEDURES: |
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16 | new(L,D) checks if number D already exists in list L |
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17 | bubblesort(L) sorts elements (out of Z) of the list L in decreasing order |
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18 | disc(N,k) generates a sequence of negative discriminants D with |D|<4N, sort in decreasing order |
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19 | Cornacchia(d,p) computes solution (x,y) for the Diophantine equation x^2+d*y^2=p with p prime and 0<d<p |
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20 | CornacchiaModified(D,p) computes solution (x,y) for the Diophantine equation x^2+|D|*y^2=4p with p prime |
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21 | pFactor1(n,B,P) Pollard's p-factorization |
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22 | maximum(L) computes the maximal number contained in list L |
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23 | cmod(x,y) computes x mod y while working in the complex numbers, e.g. ring C=(complex,30,i),x,dp; |
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24 | sqr(w,k) computes the square root of w |
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25 | e(z,k) computes e^z, i.e. the exponential function of z to the order k |
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26 | jot(t,k) computes the j-invariant of the complex number t |
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27 | round(r) rounds r to the nearest number out of Z |
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28 | HilbertClassPolynomial(D,k) computes the monic polynomial of degree h(D) in Z[X] of which jot((D+sqr(D))/2) is a root |
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29 | RootsModp(p,P) computes roots of the polynomial P modulo p with p prime and p>=3 |
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30 | w(D) computes the number of roots of unity in the quadratic order of discriminant D |
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31 | Atkin(N,K,B) tries to prove that N is prime |
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32 | "; |
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33 | |
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34 | LIB "krypto.lib"; |
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35 | LIB "general.lib"; |
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36 | LIB "ntsolve.lib"; |
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37 | LIB "inout.lib"; |
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38 | |
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39 | /////////////////////////////////////////////////////////////////////////////// |
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40 | |
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41 | proc new(list L, number D) |
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42 | "USAGE: new(L,D); |
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43 | RETURN: 1, if D does not already exist in L, |
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44 | -1, if D does already exist in L |
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45 | EXAMPLE:example new; shows an example |
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46 | " |
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47 | { |
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48 | number a=1; // a=1 bedeutet: D noch nicht in L vorhanden |
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49 | int i; |
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50 | for(i=1;i<=size(L);i++) |
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51 | { |
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52 | if(D==L[i]) |
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53 | { |
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54 | a=-1; // a=-1 bedeutet: D bereits in L vorhanden |
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55 | break; |
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56 | } |
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57 | } |
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58 | return(a); |
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59 | } |
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60 | example |
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61 | { "EXAMPLE:"; echo = 2; |
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62 | ring r = 0,x,dp; |
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63 | list L=8976,-223456,556,-778,3,-55603,45,766677; |
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64 | number D=-55603; |
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65 | new(L,D); |
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66 | } |
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67 | |
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68 | proc bubblesort(list L) |
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69 | "USAGE: bubblesort(L); |
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70 | RETURN: list L, sort in decreasing order |
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71 | EXAMPLE:example bubblesort; shows an example |
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72 | " |
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73 | { |
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74 | number b; |
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75 | int n,i,j; |
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76 | while(j==0) |
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77 | { |
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78 | i=i+1; |
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79 | j=1; |
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80 | for(n=1;n<=size(L)-i;n++) |
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81 | { |
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82 | if(L[n]<L[n+1]) |
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83 | { |
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84 | b=L[n]; |
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85 | L[n]=L[n+1]; |
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86 | L[n+1]=b; |
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87 | j=0; |
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88 | } |
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89 | } |
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90 | } |
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91 | return(L); |
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92 | } |
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93 | example |
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94 | { "EXAMPLE:"; echo = 2; |
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95 | ring r = 0,x,dp; |
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96 | list L=-567,-233,446,12,-34,8907; |
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97 | bubblesort(L); |
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98 | } |
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99 | |
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100 | |
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101 | |
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102 | proc disc(number N, int k) |
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103 | "USAGE: disc(N,k); |
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104 | RETURN: list L of negative discriminants D, sort in decreasing order |
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105 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4N |
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106 | NOTE: D=b^2-4*a, where 0<=b<=k and intPart((b^2)/4)+1<=a<=k for each b |
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107 | EXAMPLE:example disc; shows an example |
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108 | " |
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109 | { |
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110 | list L=-3,-4,-7; |
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111 | number D; |
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112 | number B; |
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113 | int a,b; |
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114 | for(b=0;b<=k;b++) |
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115 | { |
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116 | B=b^2; |
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117 | for(a=int(intPart(B/4))+1;a<=k;a++) |
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118 | { |
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119 | D=-4*a+B; |
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120 | if((D<0)&&((D mod 4)!=2)&&((D mod 4)!=3)&&(absValue(D)<4*N)&&(new(L,D)==1)) |
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121 | { |
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122 | L[size(L)+1]=D; |
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123 | } |
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124 | } |
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125 | } |
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126 | L=bubblesort(L); |
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127 | return(L); |
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128 | } |
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129 | example |
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130 | { "EXAMPLE:"; echo = 2; |
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131 | ring R = 0,x,dp; |
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132 | disc(2003,50); |
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133 | } |
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134 | |
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135 | |
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136 | |
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137 | proc Cornacchia(number d, number p) |
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138 | "USAGE: Cornacchia(d,p); |
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139 | RETURN: x,y such that x^2+d*y^2=p with p prime, |
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140 | -1, if the Diophantine equation has no solution, |
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141 | 0, if the parameters are wrong selected |
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142 | ASSUME: 0<d<p |
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143 | EXAMPLE:example Cornacchia; shows an example |
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144 | " |
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145 | { |
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146 | if((d<0)||(p<d)) // (0)[Test if assumptions well-defined] |
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147 | { |
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148 | return(0); |
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149 | // ERROR("Parameters wrong selected! It has to be 0<d<p!"); |
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150 | } |
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151 | else |
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152 | { |
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153 | number k,x(0),a,b,l,r,c,i; |
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154 | |
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155 | k=Jacobi(-d,p); // (1)[Test if residue] |
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156 | if(k==-1) |
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157 | { |
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158 | return(-1); |
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159 | // ERROR("The Diophantine equation has no solution!"); |
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160 | } |
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161 | else |
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162 | { |
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163 | x(0)=squareRoot(-d,p); // (2)[Compute square root] |
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164 | if((p/2>=x(0))||(p<=x(0))) |
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165 | { |
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166 | x(0)=-x(0) mod p; |
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167 | } |
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168 | a=p; |
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169 | b=x(0); |
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170 | l=intRoot(p); |
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171 | while(b>l) // (3)[Euclidean algorithm] |
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172 | { |
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173 | r=a mod b; |
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174 | a=b; |
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175 | b=r; |
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176 | } |
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177 | c=(p-b^2)/d; // (4)[Test solution] |
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178 | i=intRoot(c); |
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179 | if((((p-b^2) mod d)!=0)||(c!=i^2)) |
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180 | { |
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181 | return(-1); |
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182 | } |
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183 | else |
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184 | { |
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185 | list L=b,i; |
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186 | return(L); |
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187 | } |
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188 | } |
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189 | } |
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190 | } |
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191 | example |
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192 | { "EXAMPLE:"; echo = 2; |
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193 | ring R = 0,x,dp; |
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194 | Cornacchia(55,9551); |
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195 | } |
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196 | |
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197 | |
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198 | |
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199 | proc CornacchiaModified(number D, number p) |
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200 | "USAGE: CornacchiaModified(D,p); |
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201 | RETURN: x,y such that x^2+|D|*y^2=p with p prime, |
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202 | -1, if the Diophantine equation has no solution, |
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203 | 0, if the parameters are wrong selected |
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204 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4p |
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205 | EXAMPLE:example CornacchiaModified; shows an example |
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206 | " |
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207 | { |
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208 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)||(absValue(D)>=4*p)) |
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209 | {// (0)[Test if assumptions well-defined] |
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210 | return(0); |
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211 | // ERROR("Parameters wrong selected!"); |
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212 | } |
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213 | else |
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214 | { |
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215 | if(p==2) // (1)[Case p=2] |
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216 | { |
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217 | if((D+8)==intRoot(D+8)^2) |
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218 | { |
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219 | return(intRoot(D+8),1); |
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220 | } |
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221 | else |
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222 | { |
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223 | return(-1); |
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224 | // ERROR("The Diophantine equation has no solution!"); |
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225 | } |
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226 | } |
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227 | else |
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228 | { |
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229 | number k,x(0),a,b,l,r,c; |
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230 | k=Jacobi(D,p); // (2)[Test if residue] |
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231 | if(k==-1) |
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232 | { |
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233 | return(-1); |
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234 | // ERROR("The Diophantine equation has no solution!"); |
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235 | } |
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236 | else |
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237 | { |
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238 | x(0)=squareRoot(D,p); // (3)[Compute square root] |
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239 | if((x(0) mod 2)!=(D mod 2)) |
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240 | { |
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241 | x(0)=p-x(0); |
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242 | } |
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243 | a=2*p; |
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244 | b=x(0); |
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245 | l=intRoot(4*p); |
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246 | while(b>l) // (4)[Euclidean algorithm] |
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247 | { |
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248 | r=a mod b; |
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249 | a=b; |
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250 | b=r; |
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251 | } |
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252 | c=(4*p-b^2)/absValue(D);// (5)[Test solution] |
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253 | if((((4*p-b^2) mod absValue(D))!=0)||(c!=intRoot(c)^2)) |
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254 | { |
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255 | return(-1); |
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256 | // ERROR("The Diophantine equation has no solution!"); |
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257 | } |
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258 | else |
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259 | { |
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260 | list L=b,intRoot(c); |
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261 | return(L); |
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262 | } |
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263 | } |
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264 | } |
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265 | } |
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266 | } |
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267 | example |
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268 | { "EXAMPLE:"; echo = 2; |
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269 | ring R = 0,x,dp; |
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270 | CornacchiaModified(-107,1319); |
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271 | } |
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272 | |
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273 | |
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274 | |
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275 | proc pFactor1(number n,int B, list P) |
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276 | "USAGE: pFactor1(n,B,P); n to be factorized, B a bound , P a list of primes |
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277 | RETURN: a list of factors of n or the message: no factor found |
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278 | NOTE: Pollard's p-factorization |
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279 | creates the product k of powers of primes (bounded by B) from |
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280 | the list P with the idea that for a prime divisor p of n p-1|k |
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281 | then p devides gcd(a^k-1,n) for some random a |
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282 | EXAMPLE:example pFactor1; shows an example |
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283 | " |
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284 | { |
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285 | int i; |
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286 | number k=1; |
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287 | number w; |
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288 | while(i<size(P)) |
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289 | { |
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290 | i++; |
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291 | w=P[i]; |
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292 | if(w>B) {break;} |
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293 | while(w*P[i]<=B) |
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294 | { |
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295 | w=w*P[i]; |
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296 | } |
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297 | k=k*w; |
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298 | } |
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299 | number a=random(2,2147483629); |
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300 | number d=gcdN(powerN(a,k,n)-1,n); |
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301 | if((d>1)&&(d<n)){return(d);} |
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302 | return(n); |
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303 | } |
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304 | example |
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305 | { "EXAMPLE:"; echo = 2; |
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306 | ring R = 0,z,dp; |
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307 | list L=primList(1000); |
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308 | pFactor1(1241143,13,L); |
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309 | number h=10; |
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310 | h=h^30+25; |
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311 | pFactor1(h,20,L); |
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312 | } |
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313 | |
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314 | |
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315 | |
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316 | proc maximum(list L) |
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317 | "USAGE: maximum(list L); |
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318 | RETURN: the maximal number contained in list L |
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319 | EXAMPLE:example maximum; shows an example |
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320 | " |
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321 | { |
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322 | number max=L[1]; |
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323 | |
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324 | int i; |
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325 | for(i=2;i<=size(L);i++) |
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326 | { |
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327 | if(L[i]>max) |
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328 | { |
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329 | max=L[i]; |
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330 | } |
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331 | } |
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332 | |
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333 | return(max); |
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334 | } |
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335 | example |
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336 | { "EXAMPLE:"; echo = 2; |
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337 | ring r = 0,x,dp; |
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338 | list L=465,867,1233,4567,776544,233445,2334,556; |
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339 | maximum(L); |
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340 | } |
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341 | |
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342 | |
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343 | |
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344 | proc cmod(number x, number y) |
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345 | "USAGE: cmod(x,y); |
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346 | RETURN: x mod y |
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347 | ASSUME: x,y out of Z and x,y<=2147483647 |
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348 | NOTE: this algorithm is a helping procedure to be able to calculate |
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349 | x mod y with x,y out of Z while working in the complex field |
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350 | EXAMPLE:example cmod; shows an example |
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351 | " |
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352 | { |
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353 | int rest=int(x-y*int(x/y)); |
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354 | if(rest<0) |
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355 | { |
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356 | rest=rest+int(y); |
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357 | } |
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358 | |
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359 | return(rest); |
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360 | } |
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361 | example |
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362 | { "EXAMPLE:"; echo = 2; |
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363 | ring r = (complex,30,i),x,dp; |
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364 | number x=-1004456; |
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365 | number y=1233; |
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366 | cmod(x,y); |
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367 | } |
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368 | |
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369 | |
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370 | |
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371 | proc sqr(number w, int k) |
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372 | "USAGE: sqr(w,k); |
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373 | RETURN: the square root of w |
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374 | ASSUME: w>=0 |
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375 | NOTE: k describes the number of decimals being calculated in the real numbers, |
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376 | k, intPart(k/5) are inputs for the procedure "nt_solve" |
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377 | EXAMPLE:example sqr; shows an example |
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378 | " |
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379 | { |
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380 | poly f=var(1)^2-w; |
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381 | def S=basering; |
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382 | ring R=(real,k),var(1),dp; |
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383 | poly f=imap(S,f); |
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384 | ideal I=nt_solve(f,1.1,list(k,int(intPart(k/5)))); |
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385 | number c=leadcoef(I[1]); |
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386 | setring S; |
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387 | number c=imap(R,c); |
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388 | return(c); |
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389 | } |
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390 | example |
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391 | { "EXAMPLE:"; echo = 2; |
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392 | ring R = (real,60),x,dp; |
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393 | number ww=288469650108669535726081; |
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394 | sqr(ww,60); |
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395 | } |
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396 | |
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397 | |
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398 | |
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399 | proc e(number z, int k) |
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400 | "USAGE: e(z,k); |
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401 | RETURN: e^z to the order k |
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402 | NOTE: k describes the number of summands being calculated in the exponential power series |
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403 | EXAMPLE:example e; shows an example |
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404 | " |
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405 | { |
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406 | number q=1; |
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407 | number e=1; |
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408 | |
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409 | int n; |
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410 | for(n=1;n<=k;n++) |
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411 | { |
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412 | q=q*z/n; |
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413 | e=e+q; |
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414 | } |
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415 | return(e); |
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416 | } |
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417 | |
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418 | example |
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419 | { "EXAMPLE:"; echo = 2; |
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420 | ring r = (real,30),x,dp; |
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421 | number z=40.35; |
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422 | e(z,1000); |
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423 | } |
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424 | |
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425 | |
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426 | |
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427 | proc jot(number t, int k) |
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428 | "USAGE: jot(t,k); |
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429 | RETURN: the j-invariant of t |
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430 | ASSUME: t is a complex number with positive imaginary part |
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431 | NOTE: k describes the number of summands being calculated in the power series, |
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432 | 10*k is input for the procedure "e" |
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433 | EXAMPLE:example jot; shows an example |
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434 | " |
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435 | { |
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436 | number q1,q2,qr1,qi1,tr,ti,m1,m2,f,j; |
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437 | |
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438 | number pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989; |
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439 | |
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440 | tr=repart(t); |
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441 | ti=impart(t); |
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442 | if(tr==-1/2){qr1=-1;} |
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443 | if(tr==0){qr1=1;} |
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444 | if((tr!=-1/2)&&(tr!=0)) |
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445 | { |
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446 | tr=tr-round(tr); |
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447 | qr1=e(2*i*pi*tr,10*k); |
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448 | } |
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449 | |
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450 | qi1=e(-pi*ti,10*k); |
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451 | q1=qr1*qi1^2; |
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452 | q2=q1^2; |
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453 | |
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454 | int n=1; |
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455 | while(n<=k) |
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456 | { |
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457 | m1=m1+(-1)^n*(q1^(n*(3*n-1)/2)+q1^(n*(3*n+1)/2)); |
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458 | m2=m2+(-1)^n*(q2^(n*(3*n-1)/2)+q2^(n*(3*n+1)/2)); |
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459 | n=n+1; |
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460 | } |
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461 | |
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462 | f=q1*((1+m2)/(1+m1))^24; |
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463 | |
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464 | j=(256*f+1)^3/f; |
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465 | return(j); |
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466 | } |
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467 | |
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468 | example |
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469 | { "EXAMPLE:"; echo = 2; |
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470 | ring r = (complex,30,i),x,dp; |
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471 | number t=(-7+i*sqr(7,250))/2; |
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472 | jot(t,50); |
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473 | } |
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474 | |
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475 | |
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476 | |
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477 | proc round(number r) |
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478 | "USAGE: round(r); |
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479 | RETURN: the nearest number to r out of Z |
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480 | ASSUME: r should be a rational or a real number |
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481 | EXAMPLE:example round; shows an example |
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482 | " |
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483 | { |
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484 | number a=absValue(r); |
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485 | number v=r/a; |
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486 | |
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487 | number d=10; |
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488 | int e; |
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489 | while(1) |
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490 | { |
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491 | e=e+1; |
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492 | if(a-d^e<0) |
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493 | { |
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494 | e=e-1; |
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495 | break; |
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496 | } |
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497 | } |
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498 | |
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499 | number b=a; |
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500 | int k; |
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501 | for(k=0;k<=e;k++) |
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502 | { |
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503 | while(1) |
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504 | { |
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505 | b=b-d^(e-k); |
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506 | if(b<0) |
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507 | { |
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508 | b=b+d^(e-k); |
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509 | break; |
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510 | } |
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511 | } |
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512 | } |
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513 | |
---|
514 | if(b<1/2) |
---|
515 | { |
---|
516 | return(v*(a-b)); |
---|
517 | } |
---|
518 | else |
---|
519 | { |
---|
520 | return(v*(a+1-b)); |
---|
521 | } |
---|
522 | } |
---|
523 | example |
---|
524 | { "EXAMPLE:"; echo = 2; |
---|
525 | ring R = (real,50),x,dp; |
---|
526 | number r=7357683445788723456321.6788643224; |
---|
527 | round(r); |
---|
528 | } |
---|
529 | |
---|
530 | |
---|
531 | |
---|
532 | proc HilbertClassPolynomial(number D, int k) |
---|
533 | "USAGE: HilbertClassPolynomial(D,k); |
---|
534 | RETURN: the monic polynomial of degree h(D) in Z[X] of which jot((D+sqr(D))/2) is a root |
---|
535 | ASSUME: D is a negative discriminant |
---|
536 | NOTE: k is input for the procedure "jot", |
---|
537 | 5*k is input for the procedure "sqr", |
---|
538 | 10*k describes the number of decimals being calculated in the complex numbers |
---|
539 | EXAMPLE:example HilbertClassPolynomial; shows an example |
---|
540 | " |
---|
541 | { |
---|
542 | if(D>=0) // (0)[Test if assumptions well-defined] |
---|
543 | { |
---|
544 | ERROR("Parameter wrong selected!"); |
---|
545 | } |
---|
546 | |
---|
547 | else |
---|
548 | { |
---|
549 | def S=basering; |
---|
550 | ring R=0,x,dp; |
---|
551 | |
---|
552 | string s1,s2,s3; |
---|
553 | number a1,b1,t1,g1; |
---|
554 | number D=imap(S,D); |
---|
555 | number B=intRoot(absValue(D)/3); |
---|
556 | |
---|
557 | ring C=(complex,10*k,i),x,dp; |
---|
558 | number D=imap(S,D); |
---|
559 | |
---|
560 | poly P=1; // (1)[Initialize] |
---|
561 | number b=cmod(D,2); |
---|
562 | number B=imap(R,B); |
---|
563 | |
---|
564 | number t,a,g,tau,j; |
---|
565 | list L; |
---|
566 | |
---|
567 | int step=2; |
---|
568 | while(1) |
---|
569 | { |
---|
570 | if(step==2) // (2)[Initialize a] |
---|
571 | { |
---|
572 | t=(b^2-D)/4; |
---|
573 | L=b,1; |
---|
574 | a=maximum(L); |
---|
575 | step=3; |
---|
576 | } |
---|
577 | |
---|
578 | if(step==3) // (3)[Test] |
---|
579 | { |
---|
580 | if((cmod(t,a)!=0)) |
---|
581 | { |
---|
582 | step=4; |
---|
583 | } |
---|
584 | |
---|
585 | else |
---|
586 | { |
---|
587 | s1=string(a); |
---|
588 | s2=string(b); |
---|
589 | s3=string(t); |
---|
590 | |
---|
591 | setring R; |
---|
592 | execute("a1="+s1+";"); |
---|
593 | execute("b1="+s2+";"); |
---|
594 | execute("t1="+s3+";"); |
---|
595 | g1=gcd(gcd(a1,b1),t1/a1); |
---|
596 | setring C; |
---|
597 | g=imap(R,g1); |
---|
598 | |
---|
599 | if(g!=1) |
---|
600 | { |
---|
601 | step=4; |
---|
602 | } |
---|
603 | |
---|
604 | else |
---|
605 | { |
---|
606 | tau=(-b+i*sqr(absValue(D),5*k))/(2*a); |
---|
607 | j=jot(tau,k); |
---|
608 | if((a==b)||(a^2==t)||(b==0)) |
---|
609 | { |
---|
610 | P=P*(var(1)-repart(j)); |
---|
611 | step=4; |
---|
612 | } |
---|
613 | |
---|
614 | else |
---|
615 | { |
---|
616 | P=P*(var(1)^2-2*repart(j)*var(1)+repart(j)^2+impart(j)^2); |
---|
617 | step=4; |
---|
618 | } |
---|
619 | } |
---|
620 | } |
---|
621 | } |
---|
622 | |
---|
623 | if(step==4) // (4)[Loop on a] |
---|
624 | { |
---|
625 | a=a+1; |
---|
626 | if(a^2<=t) |
---|
627 | { |
---|
628 | step=3; |
---|
629 | continue; |
---|
630 | } |
---|
631 | |
---|
632 | else |
---|
633 | { |
---|
634 | step=5; |
---|
635 | } |
---|
636 | } |
---|
637 | |
---|
638 | if(step==5) // (5)[Loop on b] |
---|
639 | { |
---|
640 | b=b+2; |
---|
641 | if(b<=B) |
---|
642 | { |
---|
643 | step=2; |
---|
644 | } |
---|
645 | |
---|
646 | else |
---|
647 | { |
---|
648 | break; |
---|
649 | } |
---|
650 | } |
---|
651 | } |
---|
652 | |
---|
653 | matrix M=coeffs(P,var(1)); |
---|
654 | |
---|
655 | list liste; |
---|
656 | int n; |
---|
657 | for(n=1;n<=nrows(M);n++) |
---|
658 | { |
---|
659 | liste[n]=round(repart(number(M[n,1]))); |
---|
660 | } |
---|
661 | |
---|
662 | poly Q; |
---|
663 | int m; |
---|
664 | for(m=1;m<=size(liste);m++) |
---|
665 | { |
---|
666 | Q=Q+liste[m]*var(1)^(m-1); |
---|
667 | } |
---|
668 | |
---|
669 | string s=string(Q); |
---|
670 | setring S; |
---|
671 | execute("poly Q="+s+";"); |
---|
672 | return(Q); |
---|
673 | } |
---|
674 | } |
---|
675 | example |
---|
676 | { "EXAMPLE:"; echo = 2; |
---|
677 | ring r = 0,x,dp; |
---|
678 | number D=-23; |
---|
679 | HilbertClassPolynomial(D,50); |
---|
680 | } |
---|
681 | |
---|
682 | |
---|
683 | |
---|
684 | proc RootsModp(int p, poly P) |
---|
685 | "USAGE: RootsModp(p,P); |
---|
686 | RETURN: list of roots of the polynomial P modulo p with p prime |
---|
687 | ASSUME: p>=3 |
---|
688 | NOTE: this algorithm will be called recursively, and it is understood |
---|
689 | that all the operations are done in Z/pZ (excepting sqareRoot(d,p)) |
---|
690 | EXAMPLE:example RootsModp; shows an example |
---|
691 | " |
---|
692 | { |
---|
693 | if(p<3) // (0)[Test if assumptions well-defined] |
---|
694 | { |
---|
695 | ERROR("Parameter wrong selected, since p<3!"); |
---|
696 | } |
---|
697 | |
---|
698 | else |
---|
699 | { |
---|
700 | def S=basering; |
---|
701 | ring R=p,var(1),dp; |
---|
702 | |
---|
703 | poly P=imap(S,P); |
---|
704 | number d; |
---|
705 | int a; |
---|
706 | list L; |
---|
707 | |
---|
708 | poly A=gcd(var(1)^p-var(1),P); // (1)[Isolate roots in Z/pZ] |
---|
709 | if(subst(A,var(1),0)==0) |
---|
710 | { |
---|
711 | L[1]=0; |
---|
712 | A=A/var(1); |
---|
713 | } |
---|
714 | |
---|
715 | if(deg(A)==0) // (2)[Small degree?] |
---|
716 | { |
---|
717 | return(L); |
---|
718 | } |
---|
719 | |
---|
720 | if(deg(A)==1) |
---|
721 | { |
---|
722 | matrix M=coeffs(A,var(1)); |
---|
723 | L[size(L)+1]=-leadcoef(M[1,1])/leadcoef(M[2,1]); |
---|
724 | setring S; |
---|
725 | list L=imap(R,L); |
---|
726 | return(L); |
---|
727 | } |
---|
728 | |
---|
729 | if(deg(A)==2) |
---|
730 | { |
---|
731 | matrix M=coeffs(A,var(1)); |
---|
732 | d=leadcoef(M[2,1])^2-4*leadcoef(M[1,1])*leadcoef(M[3,1]); |
---|
733 | |
---|
734 | ring T=0,var(1),dp; |
---|
735 | number d=imap(R,d); |
---|
736 | number e=squareRoot(d,p); |
---|
737 | setring R; |
---|
738 | number e=imap(T,e); |
---|
739 | |
---|
740 | L[size(L)+1]=(-leadcoef(M[2,1])+e)/(2*leadcoef(M[3,1])); |
---|
741 | L[size(L)+1]=(-leadcoef(M[2,1])-e)/(2*leadcoef(M[3,1])); |
---|
742 | setring S; |
---|
743 | list L=imap(R,L); |
---|
744 | return(L); |
---|
745 | } |
---|
746 | |
---|
747 | poly B=1; // (3)[Random splitting] |
---|
748 | poly C; |
---|
749 | while((deg(B)==0)||(deg(B)==deg(A))) |
---|
750 | { |
---|
751 | a=random(0,p-1); |
---|
752 | B=gcd((var(1)+a)^((p-1)/2)-1,A); |
---|
753 | C=A/B; |
---|
754 | } |
---|
755 | |
---|
756 | setring S; // (4)[Recurse] |
---|
757 | poly B=imap(R,B); |
---|
758 | poly C=imap(R,C); |
---|
759 | list l=L+RootsModp(p,B)+RootsModp(p,C); |
---|
760 | return(l); |
---|
761 | } |
---|
762 | } |
---|
763 | example |
---|
764 | { "EXAMPLE:"; echo = 2; |
---|
765 | ring r = 0,x,dp; |
---|
766 | poly f=x4+2x3-5x2+x; |
---|
767 | RootsModp(7,f); |
---|
768 | poly g=x5+112x4+655x3+551x2+1129x+831; |
---|
769 | RootsModp(1223,g); |
---|
770 | } |
---|
771 | |
---|
772 | |
---|
773 | |
---|
774 | proc w(number D) |
---|
775 | "USAGE: w(D); |
---|
776 | RETURN: the number of roots of unity in the quadratic order of discriminant D |
---|
777 | ASSUME: D<0 a discriminant kongruent to 0 or 1 modulo 4 |
---|
778 | EXAMPLE:example w; shows an example |
---|
779 | " |
---|
780 | { |
---|
781 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)) |
---|
782 | { |
---|
783 | ERROR("Parameter wrong selected!"); |
---|
784 | } |
---|
785 | else |
---|
786 | { |
---|
787 | if(D<-4) {return(2);} |
---|
788 | if(D==-4){return(4);} |
---|
789 | if(D==-3){return(6);} |
---|
790 | } |
---|
791 | } |
---|
792 | example |
---|
793 | { "EXAMPLE:"; echo = 2; |
---|
794 | ring r = 0,x,dp; |
---|
795 | number D=-3; |
---|
796 | w(D); |
---|
797 | } |
---|
798 | |
---|
799 | |
---|
800 | |
---|
801 | proc Atkin(number N, int K, int B) |
---|
802 | "USAGE: Atkin(N,K,B); |
---|
803 | RETURN: 1, if N is prime, |
---|
804 | -1, if N is not prime, |
---|
805 | 0, if the algorithm is not applicable, since there are too little discriminants |
---|
806 | ASSUME: N is coprime to 6 and different from 1 |
---|
807 | NOTE: - K/2 is input for the procedure "disc", |
---|
808 | K is input for the procedure "HilbertClassPolynomial", |
---|
809 | B describes the number of recursions being calculated |
---|
810 | - The basis of the the algorithm is the following theorem: |
---|
811 | Let N be an integer coprime to 6 and different from 1 and E be an ellipic curve modulo N. |
---|
812 | Assume that we know an integer m and a point P of E(Z/NZ) satisfying the following conditions. |
---|
813 | (1) There exists a prime divisor q of m such that q>(4-th root(N)+1)^2. |
---|
814 | (2) m*P=O(E)=(0:1:0). |
---|
815 | (3) (m/q)*P=(x:y:t) with t element of (Z/NZ)*. |
---|
816 | Then N is prime. |
---|
817 | EXAMPLE:example Atkin; shows an example |
---|
818 | " |
---|
819 | { |
---|
820 | if(N==1) {return(-1);} |
---|
821 | if((N==2)||(N==3)) {return(1);} |
---|
822 | if(gcdN(N,6)!=1) |
---|
823 | { |
---|
824 | if(printlevel>=1) |
---|
825 | { |
---|
826 | "ggT(N,6)="+string(gcdN(N,6)); |
---|
827 | pause(); |
---|
828 | } |
---|
829 | return(-1); |
---|
830 | } |
---|
831 | else |
---|
832 | { |
---|
833 | int i; // (1)[Initialize] |
---|
834 | int n(i); |
---|
835 | number N(i)=N; |
---|
836 | if(printlevel>=1) |
---|
837 | { |
---|
838 | "Setze i=0, n=0 und N(i)=N(0)="+string(N(i))+"."; |
---|
839 | pause(); |
---|
840 | } |
---|
841 | |
---|
842 | // declarations: |
---|
843 | int j(0),j(1),j(2),j(3),j(4),k; // running indices |
---|
844 | list L; // all primes smaller than 1000 |
---|
845 | list H; // sequence of negative discriminants |
---|
846 | number D; // discriminant out of H |
---|
847 | list L1,L2,S,S1,S2,R; // lists of relevant elements |
---|
848 | list P,P1,P2; // elliptic points on E(Z/N(i)Z) |
---|
849 | number m,q; // m=|E(Z/N(i)Z)| and q|m |
---|
850 | number a,b,j,c; // characterize E(Z/N(i)Z) |
---|
851 | number g,u; // g out of Z/N(i)Z, u=Jacobi(g,N(i)) |
---|
852 | poly T; // T=HilbertClassPolynomial(D,K) |
---|
853 | matrix M; // M contains the coefficients of T |
---|
854 | |
---|
855 | if(printlevel>=1) |
---|
856 | { |
---|
857 | "Liste H der moeglichen geeigneten Diskriminanten wird berechnet."; |
---|
858 | } |
---|
859 | H=disc(N,K/2); |
---|
860 | if(printlevel>=1) {"H="+string(H);pause();} |
---|
861 | |
---|
862 | int step=2; |
---|
863 | while(1) |
---|
864 | { |
---|
865 | if(step==2) |
---|
866 | { |
---|
867 | L=5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997; |
---|
868 | for(j(0)=1;j(0)<=size(L);j(0)++) // (2)[Is N(i) small??] |
---|
869 | { |
---|
870 | if(((N(i) mod L[j(0)])==0)&&(N(i)!=L[j(0)])) |
---|
871 | { |
---|
872 | if(printlevel>=1) |
---|
873 | { |
---|
874 | "N("+string(i)+")="+string(N(i))+" ist durch "+string(L[j(0)])+" teilbar.";pause(); |
---|
875 | } |
---|
876 | step=14; |
---|
877 | break; |
---|
878 | } |
---|
879 | } |
---|
880 | |
---|
881 | if(step==2) |
---|
882 | { |
---|
883 | step=3; |
---|
884 | } |
---|
885 | } |
---|
886 | |
---|
887 | if(step==3) // (3)[Choose next discriminant] |
---|
888 | { |
---|
889 | n(i)=n(i)+1; |
---|
890 | if(n(i)==size(H)+1) |
---|
891 | { |
---|
892 | if(printlevel>=1) |
---|
893 | { |
---|
894 | "Algorithmus nicht anwendbar, da zu wenige geeignete Diskriminanten existieren."; |
---|
895 | "Erhoehe den Genauigkeitsparameter K und starte den Algorithmus erneut.";pause(); |
---|
896 | } |
---|
897 | return(0); |
---|
898 | } |
---|
899 | |
---|
900 | D=H[n(i)]; |
---|
901 | if(printlevel>=1) {"Naechste Diskriminante D wird gewaehlt. D="+string(D)+".";pause();} |
---|
902 | |
---|
903 | if(Jacobi(D,N(i))!=1) |
---|
904 | { |
---|
905 | if(printlevel>=1) {"Jacobi(D,N("+string(i)+"))="+string(Jacobi(D,N(i)));pause();} |
---|
906 | continue; |
---|
907 | } |
---|
908 | |
---|
909 | else |
---|
910 | { |
---|
911 | L1=CornacchiaModified(D,N(i)); |
---|
912 | if(size(L1)>1) |
---|
913 | { |
---|
914 | if(printlevel>=1) |
---|
915 | { |
---|
916 | "Die Loesung (x,y) der Gleichung x^2+|D|y^2=4N("+string(i)+") lautet"; |
---|
917 | L1; |
---|
918 | pause(); |
---|
919 | } |
---|
920 | step=4; |
---|
921 | } |
---|
922 | else |
---|
923 | { |
---|
924 | if(L1[1]==-1) |
---|
925 | { |
---|
926 | if(printlevel>=1) |
---|
927 | { |
---|
928 | "Die Gleichung x^2+|D|y^2=4N("+string(i)+") hat keine Loesung."; |
---|
929 | pause(); |
---|
930 | } |
---|
931 | continue; |
---|
932 | } |
---|
933 | |
---|
934 | if(L1[1]==0) |
---|
935 | { |
---|
936 | if(printLevel>=1) |
---|
937 | { |
---|
938 | "Algorithmus fuer N("+string(i)+")="+string(N(i))+" nicht anwendbar, da zu wenige geeignete Diskriminanten existieren."; |
---|
939 | pause(); |
---|
940 | } |
---|
941 | step=14; |
---|
942 | } |
---|
943 | } |
---|
944 | } |
---|
945 | } |
---|
946 | |
---|
947 | if(step==4) // (4)[Factor m] |
---|
948 | { |
---|
949 | if(printlevel>=1) {"Die Liste L2 der moeglichen m=|E(Z/N("+string(i)+")Z)| wird berechnet.";} |
---|
950 | if(absValue(L1[1])^2<=4*N(i)) {L2=N(i)+1+L1[1],N(i)+1-L1[1];} |
---|
951 | if(D==-4) |
---|
952 | { |
---|
953 | if(absValue(2*L1[2])^2<=4*N(i)) |
---|
954 | { |
---|
955 | L2[size(L2)+1]=N(i)+1+2*L1[2]; |
---|
956 | L2[size(L2)+1]=N(i)+1-2*L1[2]; |
---|
957 | } |
---|
958 | } |
---|
959 | |
---|
960 | if(D==-3) |
---|
961 | { |
---|
962 | if(absValue(L1[1]+3*L1[2])^2<=4*N(i)) |
---|
963 | { |
---|
964 | L2[size(L2)+1]=N(i)+1+(L1[1]+3*L1[2])/2; |
---|
965 | L2[size(L2)+1]=N(i)+1-(L1[1]+3*L1[2])/2; |
---|
966 | } |
---|
967 | if(absValue(L1[1]-3*L1[2])^2<=4*N(i)) |
---|
968 | { |
---|
969 | L2[size(L2)+1]=N(i)+1+(L1[1]-3*L1[2])/2; |
---|
970 | L2[size(L2)+1]=N(i)+1-(L1[1]-3*L1[2])/2; |
---|
971 | } |
---|
972 | } |
---|
973 | |
---|
974 | if(size(L2)==0) |
---|
975 | { |
---|
976 | if(printlevel>=1) |
---|
977 | { |
---|
978 | "Nach dem Satz von Hasse wurden keine moeglichen m=|E(Z/N("+string(i)+")Z)|"; |
---|
979 | "fuer D="+string(D)+" gefunden."; |
---|
980 | } |
---|
981 | step=3; |
---|
982 | continue; |
---|
983 | } |
---|
984 | else |
---|
985 | { |
---|
986 | if(printlevel>=1) |
---|
987 | { |
---|
988 | "L2=";L2; |
---|
989 | pause(); |
---|
990 | } |
---|
991 | } |
---|
992 | |
---|
993 | if(printlevel>=1) {"Die Liste S der Faktoren aller moeglichen m wird berechnet.";} |
---|
994 | S=list(); |
---|
995 | for(j(1)=1;j(1)<=size(L2);j(1)++) |
---|
996 | { |
---|
997 | m=L2[j(1)]; |
---|
998 | if(m!=0) |
---|
999 | { |
---|
1000 | S1=PollardRho(m,10000,1,L); |
---|
1001 | S2=pFactor1(m,100,L); |
---|
1002 | S[size(S)+1]=list(m,S1+S2); |
---|
1003 | } |
---|
1004 | } |
---|
1005 | if(printlevel>=1) |
---|
1006 | { |
---|
1007 | "S=";S; |
---|
1008 | pause(); |
---|
1009 | } |
---|
1010 | step=5; |
---|
1011 | } |
---|
1012 | |
---|
1013 | if(step==5) // (5)[Does a suitable m exist??] |
---|
1014 | { |
---|
1015 | for(j(2)=1;j(2)<=size(S);j(2)++) |
---|
1016 | { |
---|
1017 | m=L2[j(2)]; |
---|
1018 | for(j(3)=1;j(3)<=size(S[j(2)][2]);j(3)++) |
---|
1019 | { |
---|
1020 | q=S[j(2)][2][j(3)]; |
---|
1021 | if((q>(intRoot(intRoot(N(i)))+1)^2) && (MillerRabin(q,5)==1)) |
---|
1022 | { |
---|
1023 | step=6; |
---|
1024 | break; |
---|
1025 | } |
---|
1026 | } |
---|
1027 | |
---|
1028 | if(step==6) |
---|
1029 | { |
---|
1030 | if(printlevel>=1) |
---|
1031 | { |
---|
1032 | "Geeignetes Paar (m,q) gefunden, so dass q|m,"; |
---|
1033 | "q>(4-th root(N("+string(i)+"))+1)^2 und q den Miller-Rabin-Test passiert."; |
---|
1034 | "m="+string(m)+",";"q="+string(q); |
---|
1035 | pause(); |
---|
1036 | } |
---|
1037 | break; |
---|
1038 | } |
---|
1039 | else |
---|
1040 | { |
---|
1041 | step=3; |
---|
1042 | } |
---|
1043 | } |
---|
1044 | |
---|
1045 | if(step==3) |
---|
1046 | { |
---|
1047 | if(printlevel>=1) |
---|
1048 | { |
---|
1049 | "Kein geeignetes Paar (m,q), so dass q|m,"; |
---|
1050 | "q>(4-th root(N("+string(i)+"))+1)^2 und q den Miller-Rabin-Test passiert, gefunden."; |
---|
1051 | pause(); |
---|
1052 | } |
---|
1053 | continue; |
---|
1054 | } |
---|
1055 | } |
---|
1056 | |
---|
1057 | if(step==6) // (6)[Compute elliptic curve] |
---|
1058 | { |
---|
1059 | if(D==-4) |
---|
1060 | { |
---|
1061 | a=-1; |
---|
1062 | b=0; |
---|
1063 | if(printlevel>=1) |
---|
1064 | { |
---|
1065 | "Da D=-4, setze a=-1 und b=0."; |
---|
1066 | pause(); |
---|
1067 | } |
---|
1068 | } |
---|
1069 | |
---|
1070 | if(D==-3) |
---|
1071 | { |
---|
1072 | a=0; |
---|
1073 | b=-1; |
---|
1074 | if(printlevel>=1) |
---|
1075 | { |
---|
1076 | "Da D=-3, setze a=0 und b=-1."; |
---|
1077 | pause(); |
---|
1078 | } |
---|
1079 | } |
---|
1080 | |
---|
1081 | if(D<-4) |
---|
1082 | { |
---|
1083 | if(printlevel>=1) |
---|
1084 | { |
---|
1085 | "Das Minimalpolynom T von j((D+sqr(D))/2) aus Z[X] fuer D="+string(D)+" wird berechnet."; |
---|
1086 | } |
---|
1087 | T=HilbertClassPolynomial(D,K); |
---|
1088 | if(printlevel>=1) |
---|
1089 | { |
---|
1090 | "T="+string(T); |
---|
1091 | pause(); |
---|
1092 | } |
---|
1093 | |
---|
1094 | M=coeffs(T,var(1)); |
---|
1095 | T=0; |
---|
1096 | |
---|
1097 | for(j(4)=1;j(4)<=nrows(M);j(4)++) |
---|
1098 | { |
---|
1099 | M[j(4),1]=leadcoef(M[j(4),1]) mod N(i); |
---|
1100 | T=T+M[j(4),1]*var(1)^(j(4)-1); |
---|
1101 | } |
---|
1102 | if(printlevel>=1) |
---|
1103 | { |
---|
1104 | "Setze T=T mod N("+string(i)+").";"T="+string(T); |
---|
1105 | pause(); |
---|
1106 | } |
---|
1107 | |
---|
1108 | R=RootsModp(int(N(i)),T); |
---|
1109 | if(deg(T)>size(R)) |
---|
1110 | { |
---|
1111 | ERROR("Das Polynom T zerfaellt modulo N("+string(i)+") nicht vollstaendig in Linearfaktoren."+ |
---|
1112 | "Erhoehe den Genauigkeitsparameter K und starte den Algorithmus erneut."); |
---|
1113 | } |
---|
1114 | if(printlevel>=1) |
---|
1115 | { |
---|
1116 | if(deg(T)>1) |
---|
1117 | { |
---|
1118 | "Die "+string(deg(T))+" Nullstellen von T modulo N("+string(i)+") sind"; |
---|
1119 | R; |
---|
1120 | pause(); |
---|
1121 | } |
---|
1122 | if(deg(T)==1) |
---|
1123 | { |
---|
1124 | "Die Nullstelle von T modulo N("+string(i)+") ist"; |
---|
1125 | R; |
---|
1126 | pause(); |
---|
1127 | } |
---|
1128 | } |
---|
1129 | |
---|
1130 | j=R[1]; |
---|
1131 | c=j*exgcdN(j-1728,N(i))[1]; |
---|
1132 | a=-3*c mod N(i); |
---|
1133 | b=2*c mod N(i); |
---|
1134 | if(printlevel>=1) |
---|
1135 | { |
---|
1136 | "Waehle die Nullstelle j="+string(j)+" aus und setze";"c=j/(j-1728) mod N("+string(i)+"), a=-3c mod N("+string(i)+"), b=2c mod N("+string(i)+")."; |
---|
1137 | "a="+string(a)+",";"b="+string(b); |
---|
1138 | pause(); |
---|
1139 | } |
---|
1140 | } |
---|
1141 | |
---|
1142 | step=7; |
---|
1143 | } |
---|
1144 | |
---|
1145 | if(step==7) // (7)[Find g] |
---|
1146 | { |
---|
1147 | if(D==-3) |
---|
1148 | { |
---|
1149 | while(1) |
---|
1150 | { |
---|
1151 | g=random(1,2147483647) mod N(i); |
---|
1152 | u=Jacobi(g,N(i)); |
---|
1153 | if((u==-1)&&(powerN(g,(N(i)-1)/3,N(i))!=1)) |
---|
1154 | { |
---|
1155 | if(printlevel>=1) |
---|
1156 | { |
---|
1157 | "g="+string(g); |
---|
1158 | pause(); |
---|
1159 | } |
---|
1160 | break; |
---|
1161 | } |
---|
1162 | } |
---|
1163 | } |
---|
1164 | else |
---|
1165 | { |
---|
1166 | while(1) |
---|
1167 | { |
---|
1168 | g=random(1,2147483647) mod N(i); |
---|
1169 | u=Jacobi(g,N(i)); |
---|
1170 | if(u==-1) |
---|
1171 | { |
---|
1172 | if(printlevel>=1) |
---|
1173 | { |
---|
1174 | "g="+string(g); |
---|
1175 | pause(); |
---|
1176 | } |
---|
1177 | break; |
---|
1178 | } |
---|
1179 | } |
---|
1180 | } |
---|
1181 | |
---|
1182 | step=8; |
---|
1183 | } |
---|
1184 | |
---|
1185 | if(step==8) // (8)[Find P] |
---|
1186 | { |
---|
1187 | if(printlevel>=1) |
---|
1188 | { |
---|
1189 | "Ein zufaelliger Punkt P auf der Elliptischen Kurve"; |
---|
1190 | "mit der Gleichung y^2=x^3+ax+b fuer";"N("+string(i)+")="+string(N(i))+",";" a="+string(a)+",";" b="+string(b);"wird gewaehlt."; |
---|
1191 | } |
---|
1192 | P=ellipticRandomPoint(N(i),a,b); |
---|
1193 | if(printlevel>=1) {"P=("+string(P)+")";pause();} |
---|
1194 | |
---|
1195 | if(size(P)==1) |
---|
1196 | { |
---|
1197 | step=14; |
---|
1198 | } |
---|
1199 | else |
---|
1200 | { |
---|
1201 | step=9; |
---|
1202 | } |
---|
1203 | } |
---|
1204 | |
---|
1205 | if(step==9) // (9)[Find right curve] |
---|
1206 | { |
---|
1207 | if(printlevel>=1) {"Die Punkte P2=(m/q)*P und P1=q*P2 auf der Kurve werden berechnet.";} |
---|
1208 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
1209 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
1210 | if(printlevel>=1) {"P1=("+string(P1)+"),";"P2=("+string(P2)+")";pause();} |
---|
1211 | |
---|
1212 | if((P1[1]==0)&&(P1[2]==1)&&(P1[3]==0)) |
---|
1213 | { |
---|
1214 | step=12; |
---|
1215 | } |
---|
1216 | else |
---|
1217 | { |
---|
1218 | if(printlevel>=1) {"Da P1!=(0:1:0), ist fuer die Koeffizienten a="+string(a)+" und b="+string(b)+" m!=|E(Z/N("+string(i)+")Z)|."; |
---|
1219 | "Waehle daher neue Koeffizienten a und b.";pause();} |
---|
1220 | step=10; |
---|
1221 | } |
---|
1222 | } |
---|
1223 | |
---|
1224 | if(step==10) |
---|
1225 | { |
---|
1226 | k=k+1; |
---|
1227 | if(k>=w(D)) |
---|
1228 | { |
---|
1229 | if(printlevel>=1) {"Da k=w(D)="+string(k)+", ist N("+string(i)+")="+string(N(i))+" nicht prim.";pause();} |
---|
1230 | step=14; |
---|
1231 | } |
---|
1232 | |
---|
1233 | else |
---|
1234 | { |
---|
1235 | if(D<-4) {a=a*g^2 mod N(i); b=b*g^3 mod N(i); |
---|
1236 | if(printlevel>=1) {"Da D<-4, setze a=a*g^2 mod N("+string(i)+") und b=b*g^3 mod N("+string(i)+").";"a="+string(a)+",";"b="+string(b)+",";"k="+string(k);pause();}} |
---|
1237 | if(D==-4){a=a*g mod N(i); |
---|
1238 | if(printlevel>=1) {"Da D=-4, setze a=a*g mod N("+string(i)+").";"a="+string(a)+",";"b="+string(b)+",";"k="+string(k);pause();}} |
---|
1239 | if(D==-3){b=b*g mod N(i); |
---|
1240 | if(printlevel>=1) {"Da D=-3, setze b=b*g mod N("+string(i)+").";"a="+string(a)+",";"b="+string(b)+",";"k="+string(k);pause();}} |
---|
1241 | step=8; |
---|
1242 | continue; |
---|
1243 | } |
---|
1244 | } |
---|
1245 | |
---|
1246 | if(step==11) // (11)[Find a new P] |
---|
1247 | { |
---|
1248 | if(printlevel>=1) |
---|
1249 | { |
---|
1250 | "Ein neuer zufaelliger Punkt P auf der Elliptischen Kurve wird gewaehlt,"; |
---|
1251 | "da auch P2=(0:1:0)."; |
---|
1252 | } |
---|
1253 | P=ellipticRandomPoint(N(i),a,b); |
---|
1254 | if(printlevel>=1) |
---|
1255 | { |
---|
1256 | "P=("+string(P)+")"; |
---|
1257 | pause(); |
---|
1258 | } |
---|
1259 | |
---|
1260 | if(size(P)==1) |
---|
1261 | { |
---|
1262 | step=14; |
---|
1263 | } |
---|
1264 | else |
---|
1265 | { |
---|
1266 | if(printlevel>=1) |
---|
1267 | { |
---|
1268 | "Die Punkte P2=(m/q)*P und P1=q*P2 auf der Kurve werden berechnet."; |
---|
1269 | } |
---|
1270 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
1271 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
1272 | if(printlevel>=1) |
---|
1273 | { |
---|
1274 | "P1=("+string(P1)+"),";"P2=("+string(P2)+")"; |
---|
1275 | pause(); |
---|
1276 | } |
---|
1277 | |
---|
1278 | if((P1[1]!=0)||(P1[2]!=1)||(P1[3]!=0)) |
---|
1279 | { |
---|
1280 | if(printlevel>=1) |
---|
1281 | { |
---|
1282 | "Da P1!=(0:1:0), ist, fuer die Koeffizienten a="+string(a)+" und b="+string(b)+", m!=|E(Z/N("+string(i)+")Z)|."; |
---|
1283 | "Waehle daher neue Koeffizienten a und b."; |
---|
1284 | pause(); |
---|
1285 | } |
---|
1286 | step=10; |
---|
1287 | continue; |
---|
1288 | } |
---|
1289 | else |
---|
1290 | { |
---|
1291 | step=12; |
---|
1292 | } |
---|
1293 | } |
---|
1294 | } |
---|
1295 | |
---|
1296 | if(step==12) // (12)[Check P] |
---|
1297 | { |
---|
1298 | if((P2[1]==0)&&(P2[2]==1)&&(P2[3]==0)) |
---|
1299 | { |
---|
1300 | step=11; |
---|
1301 | continue; |
---|
1302 | } |
---|
1303 | else |
---|
1304 | { |
---|
1305 | step=13; |
---|
1306 | } |
---|
1307 | } |
---|
1308 | |
---|
1309 | if(step==13) // (13)[Recurse] |
---|
1310 | { |
---|
1311 | if(i<B) |
---|
1312 | { |
---|
1313 | if(printlevel>=1) |
---|
1314 | { |
---|
1315 | string(i+1)+". Rekursion:";""; |
---|
1316 | "N("+string(i)+")="+string(N(i))+" erfuellt die Bedingungen des zugrunde liegenden Satzes,"; |
---|
1317 | "da P1=(0:1:0) und P2[3] aus (Z/N("+string(i)+")Z)*.";""; |
---|
1318 | "Untersuche nun, ob auch der gefundene Faktor q="+string(q)+" diese Bedingungen erfuellt."; |
---|
1319 | "Setze dazu i=i+1, N("+string(i+1)+")=q="+string(q)+" und beginne den Algorithmus von vorne."; |
---|
1320 | pause(); |
---|
1321 | } |
---|
1322 | i=i+1; |
---|
1323 | int n(i); |
---|
1324 | number N(i)=q; |
---|
1325 | k=0; |
---|
1326 | step=2; |
---|
1327 | continue; |
---|
1328 | } |
---|
1329 | else |
---|
1330 | { |
---|
1331 | if(printlevel>=1) |
---|
1332 | { |
---|
1333 | "N(B)=N("+string(i)+")="+string(N(i))+" erfuellt die Bedingungen des zugrunde liegenden Satzes,"; |
---|
1334 | "da P1=(0:1:0) und P2[3] aus (Z/N("+string(i)+")Z)*."; |
---|
1335 | "Insbesondere ist N="+string(N)+" prim."; |
---|
1336 | pause(); |
---|
1337 | } |
---|
1338 | return(1); |
---|
1339 | } |
---|
1340 | } |
---|
1341 | |
---|
1342 | if(step==14) // (14)[Backtrack] |
---|
1343 | { |
---|
1344 | if(i>0) |
---|
1345 | { |
---|
1346 | if(printlevel>=1) |
---|
1347 | { |
---|
1348 | "Setze i=i-1 und starte den Algorithmus fuer N("+string(i-1)+")="+string(N(i-1))+" mit neuer Diskriminanten von vorne."; |
---|
1349 | pause(); |
---|
1350 | } |
---|
1351 | i=i-1; |
---|
1352 | k=0; |
---|
1353 | step=3; |
---|
1354 | } |
---|
1355 | else |
---|
1356 | { |
---|
1357 | if(printlevel>=1) |
---|
1358 | { |
---|
1359 | "N(0)=N="+string(N)+" und daher ist N nicht prim."; |
---|
1360 | pause(); |
---|
1361 | } |
---|
1362 | return(-1); |
---|
1363 | } |
---|
1364 | } |
---|
1365 | } |
---|
1366 | } |
---|
1367 | } |
---|
1368 | example |
---|
1369 | { "EXAMPLE:"; echo = 2; |
---|
1370 | ring R = 0,x,dp; |
---|
1371 | printlevel=1; |
---|
1372 | Atkin(7691,100,5); |
---|
1373 | Atkin(8543,100,4); |
---|
1374 | Atkin(100019,100,5); |
---|
1375 | Atkin(10000079,100,2); |
---|
1376 | } |
---|