[449fbf] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[7d56875] | 2 | version="$Id: atkins.lib,v 1.5 2008-10-09 09:31:57 Singular Exp $"; |
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[449fbf] | 3 | category="Teaching"; |
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| 4 | info=" |
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[abb4919] | 5 | LIBRARY: atkins.lib Procedures for teaching cryptography |
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| 6 | AUTHOR: Stefan Steidel, Stefan.Steidel@gmx.de |
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[449fbf] | 7 | |
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| 8 | NOTE: The library contains auxiliary procedures to compute the elliptic |
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[0c0b9f1] | 9 | curve primality test of Atkin and the Atkin's Test itself. |
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[449fbf] | 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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[3eadab] | 16 | newTest(L,D) checks if number D already exists in list L |
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| 17 | bubblesort(L) sorts elements of the list L |
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| 18 | disc(N,k) generates a list of negative discriminants |
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| 19 | Cornacchia(d,p) computes solution (x,y) for x^2+d*y^2=p |
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| 20 | CornacchiaModified(D,p) computes solution (x,y) for x^2+|D|*y^2=4p |
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| 21 | maximum(L) computes the maximal number contained in L |
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| 22 | expo(z,k) computes exp(z) |
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| 23 | jOft(t,k) computes the j-invariant of t |
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| 24 | round(r) rounds r to the nearest number out of Z |
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| 25 | HilbertClassPoly(D,k) computes the Hilbert Class Polynomial |
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| 26 | rootsModp(p,P) computes roots of the polynomial P modulo p |
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| 27 | wUnit(D) computes the number of units in Q(sqr(D)) |
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| 28 | Atkin(N,K,B) tries to prove that N is prime |
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[d1b0065] | 29 | |
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[449fbf] | 30 | "; |
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| 31 | |
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[abb4919] | 32 | LIB "crypto.lib"; |
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[449fbf] | 33 | LIB "general.lib"; |
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| 34 | LIB "ntsolve.lib"; |
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| 35 | LIB "inout.lib"; |
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| 36 | |
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| 37 | /////////////////////////////////////////////////////////////////////////////// |
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| 38 | |
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[3eadab] | 39 | proc newTest(list L, number D) |
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| 40 | "USAGE: newTest(L,D); |
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[53e03a6] | 41 | RETURN: 1, if D does not already exist in L, |
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| 42 | -1, if D does already exist in L |
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| 43 | EXAMPLE:example new; shows an example |
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| 44 | " |
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| 45 | { |
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[ea9f7aa] | 46 | number a=1; // a=1 bedeutet: D noch nicht in L vorhanden |
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| 47 | int i; |
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| 48 | for(i=1;i<=size(L);i++) |
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| 49 | { |
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| 50 | if(D==L[i]) |
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| 51 | { |
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| 52 | a=-1; // a=-1 bedeutet: D bereits in L vorhanden |
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| 53 | break; |
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| 54 | } |
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| 55 | } |
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| 56 | return(a); |
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[53e03a6] | 57 | } |
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[449fbf] | 58 | example |
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[53e03a6] | 59 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 60 | ring r = 0,x,dp; |
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| 61 | list L=8976,-223456,556,-778,3,-55603,45,766677; |
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| 62 | number D=-55603; |
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[3eadab] | 63 | newTest(L,D); |
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[53e03a6] | 64 | } |
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[449fbf] | 65 | |
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| 66 | proc bubblesort(list L) |
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[53e03a6] | 67 | "USAGE: bubblesort(L); |
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| 68 | RETURN: list L, sort in decreasing order |
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| 69 | EXAMPLE:example bubblesort; shows an example |
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| 70 | " |
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| 71 | { |
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[ea9f7aa] | 72 | number b; |
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| 73 | int n,i,j; |
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| 74 | while(j==0) |
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| 75 | { |
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| 76 | i=i+1; |
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| 77 | j=1; |
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| 78 | for(n=1;n<=size(L)-i;n++) |
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| 79 | { |
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| 80 | if(L[n]<L[n+1]) |
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| 81 | { |
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| 82 | b=L[n]; |
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| 83 | L[n]=L[n+1]; |
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| 84 | L[n+1]=b; |
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| 85 | j=0; |
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| 86 | } |
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| 87 | } |
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| 88 | } |
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| 89 | return(L); |
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[53e03a6] | 90 | } |
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[449fbf] | 91 | example |
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[53e03a6] | 92 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 93 | ring r = 0,x,dp; |
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| 94 | list L=-567,-233,446,12,-34,8907; |
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| 95 | bubblesort(L); |
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[53e03a6] | 96 | } |
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[449fbf] | 97 | |
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| 98 | proc disc(number N, int k) |
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[53e03a6] | 99 | "USAGE: disc(N,k); |
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| 100 | RETURN: list L of negative discriminants D, sort in decreasing order |
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| 101 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4N |
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| 102 | 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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| 103 | EXAMPLE:example disc; shows an example |
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| 104 | " |
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| 105 | { |
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[ea9f7aa] | 106 | list L=-3,-4,-7; |
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| 107 | number D; |
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| 108 | number B; |
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| 109 | int a,b; |
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| 110 | for(b=0;b<=k;b++) |
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| 111 | { |
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| 112 | B=b^2; |
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| 113 | for(a=int(intPart(B/4))+1;a<=k;a++) |
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| 114 | { |
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| 115 | D=-4*a+B; |
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[3eadab] | 116 | if((D<0)&&((D mod 4)!=2)&&((D mod 4)!=3)&&(absValue(D)<4*N)&&(newTest(L,D)==1)) |
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[ea9f7aa] | 117 | { |
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| 118 | L[size(L)+1]=D; |
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| 119 | } |
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| 120 | } |
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| 121 | } |
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| 122 | L=bubblesort(L); |
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| 123 | return(L); |
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[53e03a6] | 124 | } |
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[449fbf] | 125 | example |
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[53e03a6] | 126 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 127 | ring R = 0,x,dp; |
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| 128 | disc(2003,50); |
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[53e03a6] | 129 | } |
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[449fbf] | 130 | |
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| 131 | |
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| 132 | |
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| 133 | proc Cornacchia(number d, number p) |
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[53e03a6] | 134 | "USAGE: Cornacchia(d,p); |
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| 135 | RETURN: x,y such that x^2+d*y^2=p with p prime, |
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| 136 | -1, if the Diophantine equation has no solution, |
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| 137 | 0, if the parameters are wrong selected |
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| 138 | ASSUME: 0<d<p |
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| 139 | EXAMPLE:example Cornacchia; shows an example |
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| 140 | " |
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| 141 | { |
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[19ffafb] | 142 | if((d<0)||(p<d)) // (0)[Test if assumptions well-defined] |
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[ea9f7aa] | 143 | { |
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| 144 | return(0); |
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| 145 | // ERROR("Parameters wrong selected! It has to be 0<d<p!"); |
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| 146 | } |
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| 147 | else |
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| 148 | { |
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[19ffafb] | 149 | number k,x(0),a,b,l,r,c,i; |
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[ea9f7aa] | 150 | |
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[19ffafb] | 151 | k=Jacobi(-d,p); // (1)[Test if residue] |
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[ea9f7aa] | 152 | if(k==-1) |
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| 153 | { |
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| 154 | return(-1); |
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| 155 | // ERROR("The Diophantine equation has no solution!"); |
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| 156 | } |
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| 157 | else |
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| 158 | { |
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[19ffafb] | 159 | x(0)=squareRoot(-d,p); // (2)[Compute square root] |
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| 160 | if((p/2>=x(0))||(p<=x(0))) |
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| 161 | { |
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| 162 | x(0)=-x(0) mod p; |
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| 163 | } |
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| 164 | a=p; |
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| 165 | b=x(0); |
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| 166 | l=intRoot(p); |
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| 167 | while(b>l) // (3)[Euclidean algorithm] |
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| 168 | { |
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[ea9f7aa] | 169 | r=a mod b; |
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| 170 | a=b; |
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| 171 | b=r; |
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[19ffafb] | 172 | } |
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| 173 | c=(p-b^2)/d; // (4)[Test solution] |
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| 174 | i=intRoot(c); |
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| 175 | if((((p-b^2) mod d)!=0)||(c!=i^2)) |
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| 176 | { |
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| 177 | return(-1); |
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| 178 | } |
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| 179 | else |
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| 180 | { |
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[ea9f7aa] | 181 | list L=b,i; |
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| 182 | return(L); |
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[19ffafb] | 183 | } |
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[ea9f7aa] | 184 | } |
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| 185 | } |
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[2472afe] | 186 | } |
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[449fbf] | 187 | example |
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[53e03a6] | 188 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 189 | ring R = 0,x,dp; |
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| 190 | Cornacchia(55,9551); |
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[53e03a6] | 191 | } |
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[449fbf] | 192 | |
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| 193 | proc CornacchiaModified(number D, number p) |
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[53e03a6] | 194 | "USAGE: CornacchiaModified(D,p); |
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| 195 | RETURN: x,y such that x^2+|D|*y^2=p with p prime, |
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| 196 | -1, if the Diophantine equation has no solution, |
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| 197 | 0, if the parameters are wrong selected |
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| 198 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4p |
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| 199 | EXAMPLE:example CornacchiaModified; shows an example |
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| 200 | " |
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| 201 | { |
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[26508d] | 202 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)||(absValue(D)>=4*p)) |
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| 203 | {// (0)[Test if assumptions well-defined] |
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| 204 | return(0); |
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| 205 | // ERROR("Parameters wrong selected!"); |
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| 206 | } |
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| 207 | else |
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| 208 | { |
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| 209 | if(p==2) // (1)[Case p=2] |
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| 210 | { |
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| 211 | if((D+8)==intRoot(D+8)^2) |
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[19ffafb] | 212 | { |
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[26508d] | 213 | return(intRoot(D+8),1); |
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[19ffafb] | 214 | } |
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[449fbf] | 215 | else |
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[19ffafb] | 216 | { |
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[26508d] | 217 | return(-1); |
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| 218 | // ERROR("The Diophantine equation has no solution!"); |
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| 219 | } |
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| 220 | } |
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| 221 | else |
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| 222 | { |
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| 223 | number k,x(0),a,b,l,r,c; |
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| 224 | k=Jacobi(D,p); // (2)[Test if residue] |
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| 225 | if(k==-1) |
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| 226 | { |
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| 227 | return(-1); |
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| 228 | // ERROR("The Diophantine equation has no solution!"); |
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[19ffafb] | 229 | } |
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[26508d] | 230 | else |
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| 231 | { |
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| 232 | x(0)=squareRoot(D,p); // (3)[Compute square root] |
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| 233 | if((x(0) mod 2)!=(D mod 2)) |
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| 234 | { |
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| 235 | x(0)=p-x(0); |
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| 236 | } |
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| 237 | a=2*p; |
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| 238 | b=x(0); |
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| 239 | l=intRoot(4*p); |
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| 240 | while(b>l) // (4)[Euclidean algorithm] |
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| 241 | { |
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| 242 | r=a mod b; |
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| 243 | a=b; |
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| 244 | b=r; |
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| 245 | } |
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| 246 | c=(4*p-b^2)/absValue(D);// (5)[Test solution] |
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| 247 | if((((4*p-b^2) mod absValue(D))!=0)||(c!=intRoot(c)^2)) |
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| 248 | { |
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| 249 | return(-1); |
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| 250 | // ERROR("The Diophantine equation has no solution!"); |
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| 251 | } |
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| 252 | else |
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| 253 | { |
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| 254 | list L=b,intRoot(c); |
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| 255 | return(L); |
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| 256 | } |
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| 257 | } |
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| 258 | } |
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| 259 | } |
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[53e03a6] | 260 | } |
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[449fbf] | 261 | example |
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[53e03a6] | 262 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 263 | ring R = 0,x,dp; |
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| 264 | CornacchiaModified(-107,1319); |
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[53e03a6] | 265 | } |
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[449fbf] | 266 | |
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| 267 | proc pFactor1(number n,int B, list P) |
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[53e03a6] | 268 | "USAGE: pFactor1(n,B,P); n to be factorized, B a bound , P a list of primes |
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| 269 | RETURN: a list of factors of n or the message: no factor found |
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| 270 | NOTE: Pollard's p-factorization |
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| 271 | creates the product k of powers of primes (bounded by B) from |
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| 272 | the list P with the idea that for a prime divisor p of n p-1|k |
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| 273 | then p devides gcd(a^k-1,n) for some random a |
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| 274 | EXAMPLE:example pFactor1; shows an example |
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| 275 | " |
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| 276 | { |
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[26508d] | 277 | int i; |
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| 278 | number k=1; |
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| 279 | number w; |
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| 280 | while(i<size(P)) |
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| 281 | { |
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| 282 | i++; |
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| 283 | w=P[i]; |
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| 284 | if(w>B) {break;} |
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| 285 | while(w*P[i]<=B) |
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| 286 | { |
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| 287 | w=w*P[i]; |
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| 288 | } |
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| 289 | k=k*w; |
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| 290 | } |
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| 291 | number a=random(2,2147483629); |
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| 292 | number d=gcdN(powerN(a,k,n)-1,n); |
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| 293 | if((d>1)&&(d<n)){return(d);} |
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| 294 | return(n); |
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[53e03a6] | 295 | } |
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[449fbf] | 296 | example |
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| 297 | { "EXAMPLE:"; echo = 2; |
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| 298 | ring R = 0,z,dp; |
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| 299 | list L=primList(1000); |
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| 300 | pFactor1(1241143,13,L); |
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| 301 | number h=10; |
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| 302 | h=h^30+25; |
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| 303 | pFactor1(h,20,L); |
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| 304 | } |
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| 305 | |
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| 306 | proc maximum(list L) |
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[53e03a6] | 307 | "USAGE: maximum(list L); |
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| 308 | RETURN: the maximal number contained in list L |
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| 309 | EXAMPLE:example maximum; shows an example |
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| 310 | " |
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| 311 | { |
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[26508d] | 312 | number max=L[1]; |
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| 313 | int i; |
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| 314 | for(i=2;i<=size(L);i++) |
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| 315 | { |
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| 316 | if(L[i]>max) |
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| 317 | { |
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| 318 | max=L[i]; |
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| 319 | } |
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| 320 | } |
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| 321 | return(max); |
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[53e03a6] | 322 | } |
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[449fbf] | 323 | example |
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[53e03a6] | 324 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 325 | ring r = 0,x,dp; |
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| 326 | list L=465,867,1233,4567,776544,233445,2334,556; |
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| 327 | maximum(L); |
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[53e03a6] | 328 | } |
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[449fbf] | 329 | |
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[3eadab] | 330 | static proc cmod(number x, number y) |
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[53e03a6] | 331 | "USAGE: cmod(x,y); |
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| 332 | RETURN: x mod y |
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| 333 | ASSUME: x,y out of Z and x,y<=2147483647 |
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| 334 | NOTE: this algorithm is a helping procedure to be able to calculate |
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| 335 | x mod y with x,y out of Z while working in the complex field |
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| 336 | EXAMPLE:example cmod; shows an example |
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| 337 | " |
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| 338 | { |
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[26508d] | 339 | int rest=int(x-y*int(x/y)); |
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| 340 | if(rest<0) |
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| 341 | { |
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| 342 | rest=rest+int(y); |
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| 343 | } |
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| 344 | return(rest); |
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[53e03a6] | 345 | } |
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[449fbf] | 346 | example |
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[53e03a6] | 347 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 348 | ring r = (complex,30,i),x,dp; |
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| 349 | number x=-1004456; |
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| 350 | number y=1233; |
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| 351 | cmod(x,y); |
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[53e03a6] | 352 | } |
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[449fbf] | 353 | |
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[26508d] | 354 | proc sqr(number w, int k) |
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[53e03a6] | 355 | "USAGE: sqr(w,k); |
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| 356 | RETURN: the square root of w |
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| 357 | ASSUME: w>=0 |
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| 358 | NOTE: k describes the number of decimals being calculated in the real numbers, |
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| 359 | k, intPart(k/5) are inputs for the procedure "nt_solve" |
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| 360 | EXAMPLE:example sqr; shows an example |
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| 361 | " |
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| 362 | { |
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| 363 | poly f=var(1)^2-w; |
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| 364 | def S=basering; |
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| 365 | ring R=(real,k),var(1),dp; |
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| 366 | poly f=imap(S,f); |
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| 367 | ideal I=nt_solve(f,1.1,list(k,int(intPart(k/5)))); |
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| 368 | number c=leadcoef(I[1]); |
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| 369 | setring S; |
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| 370 | number c=imap(R,c); |
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| 371 | return(c); |
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| 372 | } |
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[449fbf] | 373 | example |
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[53e03a6] | 374 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 375 | ring R = (real,60),x,dp; |
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[0c0b9f1] | 376 | number ww=288469650108669535726081; |
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| 377 | sqr(ww,60); |
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[53e03a6] | 378 | } |
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[449fbf] | 379 | |
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| 380 | |
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| 381 | |
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[3eadab] | 382 | proc expo(number z, int k) |
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| 383 | "USAGE: expo(z,k); |
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[53e03a6] | 384 | RETURN: e^z to the order k |
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| 385 | NOTE: k describes the number of summands being calculated in the exponential power series |
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| 386 | EXAMPLE:example e; shows an example |
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| 387 | " |
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| 388 | { |
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| 389 | number q=1; |
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| 390 | number e=1; |
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| 391 | int n; |
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| 392 | for(n=1;n<=k;n++) |
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| 393 | { |
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| 394 | q=q*z/n; |
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| 395 | e=e+q; |
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| 396 | } |
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| 397 | return(e); |
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| 398 | } |
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[449fbf] | 399 | |
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| 400 | example |
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[53e03a6] | 401 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 402 | ring r = (real,30),x,dp; |
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| 403 | number z=40.35; |
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[3eadab] | 404 | expo(z,1000); |
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[53e03a6] | 405 | } |
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[449fbf] | 406 | |
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| 407 | |
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| 408 | |
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[3eadab] | 409 | proc jOft(number t, int k) |
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| 410 | "USAGE: jOft(t,k); |
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[53e03a6] | 411 | RETURN: the j-invariant of t |
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| 412 | ASSUME: t is a complex number with positive imaginary part |
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| 413 | NOTE: k describes the number of summands being calculated in the power series, |
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| 414 | 10*k is input for the procedure "e" |
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| 415 | EXAMPLE:example jot; shows an example |
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| 416 | " |
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| 417 | { |
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[26508d] | 418 | number q1,q2,qr1,qi1,tr,ti,m1,m2,f,j; |
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| 419 | |
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| 420 | number pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989; |
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| 421 | |
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| 422 | tr=repart(t); |
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| 423 | ti=impart(t); |
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| 424 | if(tr==-1/2){qr1=-1;} |
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| 425 | if(tr==0){qr1=1;} |
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| 426 | if((tr!=-1/2)&&(tr!=0)) |
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| 427 | { |
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| 428 | tr=tr-round(tr); |
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| 429 | qr1=expo(2*i*pi*tr,10*k); |
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| 430 | } |
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| 431 | |
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| 432 | qi1=expo(-pi*ti,10*k); |
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| 433 | q1=qr1*qi1^2; |
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| 434 | q2=q1^2; |
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| 435 | |
---|
| 436 | int n=1; |
---|
| 437 | while(n<=k) |
---|
| 438 | { |
---|
| 439 | m1=m1+(-1)^n*(q1^(n*(3*n-1)/2)+q1^(n*(3*n+1)/2)); |
---|
| 440 | m2=m2+(-1)^n*(q2^(n*(3*n-1)/2)+q2^(n*(3*n+1)/2)); |
---|
| 441 | n++; |
---|
| 442 | } |
---|
| 443 | |
---|
| 444 | f=q1*((1+m2)/(1+m1))^24; |
---|
[449fbf] | 445 | |
---|
[26508d] | 446 | j=(256*f+1)^3/f; |
---|
| 447 | return(j); |
---|
| 448 | } |
---|
[449fbf] | 449 | example |
---|
[53e03a6] | 450 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 451 | ring r = (complex,30,i),x,dp; |
---|
| 452 | number t=(-7+i*sqr(7,250))/2; |
---|
[3eadab] | 453 | jOft(t,50); |
---|
[53e03a6] | 454 | } |
---|
[449fbf] | 455 | |
---|
| 456 | proc round(number r) |
---|
[53e03a6] | 457 | "USAGE: round(r); |
---|
| 458 | RETURN: the nearest number to r out of Z |
---|
| 459 | ASSUME: r should be a rational or a real number |
---|
| 460 | EXAMPLE:example round; shows an example |
---|
| 461 | " |
---|
| 462 | { |
---|
| 463 | number a=absValue(r); |
---|
| 464 | number v=r/a; |
---|
| 465 | |
---|
| 466 | number d=10; |
---|
| 467 | int e; |
---|
| 468 | while(1) |
---|
| 469 | { |
---|
| 470 | e=e+1; |
---|
| 471 | if(a-d^e<0) |
---|
[449fbf] | 472 | { |
---|
[53e03a6] | 473 | e=e-1; |
---|
| 474 | break; |
---|
[449fbf] | 475 | } |
---|
[53e03a6] | 476 | } |
---|
[449fbf] | 477 | |
---|
[53e03a6] | 478 | number b=a; |
---|
| 479 | int k; |
---|
| 480 | for(k=0;k<=e;k++) |
---|
| 481 | { |
---|
| 482 | while(1) |
---|
| 483 | { |
---|
| 484 | b=b-d^(e-k); |
---|
| 485 | if(b<0) |
---|
| 486 | { |
---|
| 487 | b=b+d^(e-k); |
---|
| 488 | break; |
---|
| 489 | } |
---|
| 490 | } |
---|
| 491 | } |
---|
| 492 | |
---|
| 493 | if(b<1/2) |
---|
| 494 | { |
---|
| 495 | return(v*(a-b)); |
---|
| 496 | } |
---|
| 497 | else |
---|
| 498 | { |
---|
| 499 | return(v*(a+1-b)); |
---|
| 500 | } |
---|
| 501 | } |
---|
[449fbf] | 502 | example |
---|
[53e03a6] | 503 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 504 | ring R = (real,50),x,dp; |
---|
| 505 | number r=7357683445788723456321.6788643224; |
---|
| 506 | round(r); |
---|
[53e03a6] | 507 | } |
---|
[449fbf] | 508 | |
---|
| 509 | |
---|
| 510 | |
---|
[3eadab] | 511 | proc HilbertClassPoly(number D, int k) |
---|
| 512 | "USAGE: HilbertClassPoly(D,k); |
---|
| 513 | RETURN: the monic polynomial of degree h(D) in Z[X] of which jOft((D+sqr(D))/2) is a root |
---|
[53e03a6] | 514 | ASSUME: D is a negative discriminant |
---|
| 515 | NOTE: k is input for the procedure "jot", |
---|
| 516 | 5*k is input for the procedure "sqr", |
---|
| 517 | 10*k describes the number of decimals being calculated in the complex numbers |
---|
| 518 | EXAMPLE:example HilbertClassPolynomial; shows an example |
---|
| 519 | " |
---|
| 520 | { |
---|
[26508d] | 521 | if(D>=0) // (0)[Test if assumptions well-defined] |
---|
| 522 | { |
---|
| 523 | ERROR("Parameter wrong selected!"); |
---|
| 524 | } |
---|
| 525 | else |
---|
| 526 | { |
---|
| 527 | def S=basering; |
---|
| 528 | ring R=0,x,dp; |
---|
[449fbf] | 529 | |
---|
[26508d] | 530 | string s1,s2,s3; |
---|
| 531 | number a1,b1,t1,g1; |
---|
| 532 | number D=imap(S,D); |
---|
| 533 | number B=intRoot(absValue(D)/3); |
---|
| 534 | |
---|
| 535 | ring C=(complex,10*k,i),x,dp; |
---|
| 536 | number D=imap(S,D); |
---|
| 537 | |
---|
| 538 | poly P=1; // (1)[Initialize] |
---|
| 539 | number b=cmod(D,2); |
---|
| 540 | number B=imap(R,B); |
---|
| 541 | |
---|
| 542 | number t,a,g,tau,j; |
---|
| 543 | list L; |
---|
| 544 | |
---|
| 545 | int step=2; |
---|
| 546 | while(1) |
---|
| 547 | { |
---|
| 548 | if(step==2) // (2)[Initialize a] |
---|
| 549 | { |
---|
| 550 | t=(b^2-D)/4; |
---|
| 551 | L=b,1; |
---|
| 552 | a=maximum(L); |
---|
| 553 | step=3; |
---|
| 554 | } |
---|
| 555 | |
---|
| 556 | if(step==3) // (3)[Test] |
---|
| 557 | { |
---|
| 558 | if((cmod(t,a)!=0)) |
---|
| 559 | { |
---|
| 560 | step=4; |
---|
| 561 | } |
---|
| 562 | else |
---|
| 563 | { |
---|
| 564 | s1=string(a); |
---|
| 565 | s2=string(b); |
---|
| 566 | s3=string(t); |
---|
| 567 | |
---|
| 568 | setring R; |
---|
| 569 | execute("a1="+s1+";"); |
---|
| 570 | execute("b1="+s2+";"); |
---|
| 571 | execute("t1="+s3+";"); |
---|
| 572 | g1=gcd(gcd(a1,b1),t1/a1); |
---|
| 573 | setring C; |
---|
| 574 | g=imap(R,g1); |
---|
| 575 | |
---|
| 576 | if(g!=1) |
---|
| 577 | { |
---|
| 578 | step=4; |
---|
| 579 | } |
---|
| 580 | else |
---|
| 581 | { |
---|
| 582 | tau=(-b+i*sqr(absValue(D),5*k))/(2*a); |
---|
| 583 | j=jOft(tau,k); |
---|
| 584 | if((a==b)||(a^2==t)||(b==0)) |
---|
| 585 | { |
---|
| 586 | P=P*(var(1)-repart(j)); |
---|
| 587 | step=4; |
---|
| 588 | } |
---|
| 589 | else |
---|
| 590 | { |
---|
| 591 | P=P*(var(1)^2-2*repart(j)*var(1)+repart(j)^2+impart(j)^2); |
---|
| 592 | step=4; |
---|
| 593 | } |
---|
| 594 | } |
---|
| 595 | } |
---|
| 596 | } |
---|
| 597 | |
---|
| 598 | if(step==4) // (4)[Loop on a] |
---|
| 599 | { |
---|
| 600 | a=a+1; |
---|
| 601 | if(a^2<=t) |
---|
| 602 | { |
---|
| 603 | step=3; |
---|
| 604 | continue; |
---|
| 605 | } |
---|
| 606 | else |
---|
| 607 | { |
---|
| 608 | step=5; |
---|
| 609 | } |
---|
| 610 | } |
---|
| 611 | |
---|
| 612 | if(step==5) // (5)[Loop on b] |
---|
| 613 | { |
---|
| 614 | b=b+2; |
---|
| 615 | if(b<=B) |
---|
| 616 | { |
---|
| 617 | step=2; |
---|
| 618 | } |
---|
| 619 | else |
---|
| 620 | { |
---|
| 621 | break; |
---|
| 622 | } |
---|
| 623 | } |
---|
| 624 | } |
---|
| 625 | |
---|
| 626 | matrix M=coeffs(P,var(1)); |
---|
| 627 | |
---|
| 628 | list liste; |
---|
| 629 | int n; |
---|
| 630 | for(n=1;n<=nrows(M);n++) |
---|
| 631 | { |
---|
| 632 | liste[n]=round(repart(number(M[n,1]))); |
---|
| 633 | } |
---|
| 634 | |
---|
| 635 | poly Q; |
---|
| 636 | int m; |
---|
| 637 | for(m=1;m<=size(liste);m++) |
---|
| 638 | { |
---|
| 639 | Q=Q+liste[m]*var(1)^(m-1); |
---|
| 640 | } |
---|
| 641 | |
---|
| 642 | string s=string(Q); |
---|
| 643 | setring S; |
---|
| 644 | execute("poly Q="+s+";"); |
---|
| 645 | return(Q); |
---|
| 646 | } |
---|
[53e03a6] | 647 | } |
---|
[449fbf] | 648 | example |
---|
[53e03a6] | 649 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 650 | ring r = 0,x,dp; |
---|
| 651 | number D=-23; |
---|
[3eadab] | 652 | HilbertClassPoly(D,50); |
---|
[53e03a6] | 653 | } |
---|
[449fbf] | 654 | |
---|
[3eadab] | 655 | proc rootsModp(int p, poly P) |
---|
| 656 | "USAGE: rootsModp(p,P); |
---|
[53e03a6] | 657 | RETURN: list of roots of the polynomial P modulo p with p prime |
---|
| 658 | ASSUME: p>=3 |
---|
| 659 | NOTE: this algorithm will be called recursively, and it is understood |
---|
| 660 | that all the operations are done in Z/pZ (excepting sqareRoot(d,p)) |
---|
[3eadab] | 661 | EXAMPLE:example rootsModp; shows an example |
---|
[53e03a6] | 662 | " |
---|
| 663 | { |
---|
[26508d] | 664 | if(p<3) // (0)[Test if assumptions well-defined] |
---|
| 665 | { |
---|
| 666 | ERROR("Parameter wrong selected, since p<3!"); |
---|
| 667 | } |
---|
| 668 | else |
---|
| 669 | { |
---|
| 670 | def S=basering; |
---|
| 671 | ring R=p,var(1),dp; |
---|
[449fbf] | 672 | |
---|
[26508d] | 673 | poly P=imap(S,P); |
---|
| 674 | number d; |
---|
| 675 | int a; |
---|
| 676 | list L; |
---|
| 677 | |
---|
| 678 | poly A=gcd(var(1)^p-var(1),P); // (1)[Isolate roots in Z/pZ] |
---|
| 679 | if(subst(A,var(1),0)==0) |
---|
| 680 | { |
---|
| 681 | L[1]=0; |
---|
| 682 | A=A/var(1); |
---|
| 683 | } |
---|
| 684 | |
---|
| 685 | if(deg(A)==0) // (2)[Small degree?] |
---|
| 686 | { |
---|
| 687 | return(L); |
---|
| 688 | } |
---|
| 689 | |
---|
| 690 | if(deg(A)==1) |
---|
| 691 | { |
---|
| 692 | matrix M=coeffs(A,var(1)); |
---|
| 693 | L[size(L)+1]=-leadcoef(M[1,1])/leadcoef(M[2,1]); |
---|
| 694 | setring S; |
---|
| 695 | list L=imap(R,L); |
---|
| 696 | return(L); |
---|
| 697 | } |
---|
| 698 | |
---|
| 699 | if(deg(A)==2) |
---|
| 700 | { |
---|
| 701 | matrix M=coeffs(A,var(1)); |
---|
| 702 | d=leadcoef(M[2,1])^2-4*leadcoef(M[1,1])*leadcoef(M[3,1]); |
---|
| 703 | |
---|
| 704 | ring T=0,var(1),dp; |
---|
| 705 | number d=imap(R,d); |
---|
| 706 | number e=squareRoot(d,p); |
---|
| 707 | setring R; |
---|
| 708 | number e=imap(T,e); |
---|
| 709 | |
---|
| 710 | L[size(L)+1]=(-leadcoef(M[2,1])+e)/(2*leadcoef(M[3,1])); |
---|
| 711 | L[size(L)+1]=(-leadcoef(M[2,1])-e)/(2*leadcoef(M[3,1])); |
---|
| 712 | setring S; |
---|
| 713 | list L=imap(R,L); |
---|
| 714 | return(L); |
---|
| 715 | } |
---|
| 716 | |
---|
| 717 | poly B=1; // (3)[Random splitting] |
---|
| 718 | poly C; |
---|
| 719 | while((deg(B)==0)||(deg(B)==deg(A))) |
---|
| 720 | { |
---|
| 721 | a=random(0,p-1); |
---|
| 722 | B=gcd((var(1)+a)^((p-1)/2)-1,A); |
---|
| 723 | C=A/B; |
---|
| 724 | } |
---|
| 725 | |
---|
| 726 | setring S; // (4)[Recurse] |
---|
| 727 | poly B=imap(R,B); |
---|
| 728 | poly C=imap(R,C); |
---|
| 729 | list l=L+rootsModp(p,B)+rootsModp(p,C); |
---|
| 730 | return(l); |
---|
| 731 | } |
---|
[53e03a6] | 732 | } |
---|
[449fbf] | 733 | example |
---|
[53e03a6] | 734 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 735 | ring r = 0,x,dp; |
---|
| 736 | poly f=x4+2x3-5x2+x; |
---|
[3eadab] | 737 | rootsModp(7,f); |
---|
[449fbf] | 738 | poly g=x5+112x4+655x3+551x2+1129x+831; |
---|
[3eadab] | 739 | rootsModp(1223,g); |
---|
[53e03a6] | 740 | } |
---|
[449fbf] | 741 | |
---|
[3eadab] | 742 | proc wUnit(number D) |
---|
| 743 | "USAGE: wUnit(D); |
---|
[53e03a6] | 744 | RETURN: the number of roots of unity in the quadratic order of discriminant D |
---|
| 745 | ASSUME: D<0 a discriminant kongruent to 0 or 1 modulo 4 |
---|
| 746 | EXAMPLE:example w; shows an example |
---|
| 747 | " |
---|
| 748 | { |
---|
| 749 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)) |
---|
| 750 | { |
---|
| 751 | ERROR("Parameter wrong selected!"); |
---|
| 752 | } |
---|
| 753 | else |
---|
| 754 | { |
---|
| 755 | if(D<-4) {return(2);} |
---|
| 756 | if(D==-4){return(4);} |
---|
| 757 | if(D==-3){return(6);} |
---|
| 758 | } |
---|
| 759 | } |
---|
[449fbf] | 760 | example |
---|
[53e03a6] | 761 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 762 | ring r = 0,x,dp; |
---|
| 763 | number D=-3; |
---|
[3eadab] | 764 | wUnit(D); |
---|
[53e03a6] | 765 | } |
---|
[449fbf] | 766 | |
---|
| 767 | |
---|
| 768 | |
---|
| 769 | proc Atkin(number N, int K, int B) |
---|
[53e03a6] | 770 | "USAGE: Atkin(N,K,B); |
---|
| 771 | RETURN: 1, if N is prime, |
---|
| 772 | -1, if N is not prime, |
---|
| 773 | 0, if the algorithm is not applicable, since there are too little discriminants |
---|
| 774 | ASSUME: N is coprime to 6 and different from 1 |
---|
| 775 | NOTE: - K/2 is input for the procedure "disc", |
---|
| 776 | K is input for the procedure "HilbertClassPolynomial", |
---|
| 777 | B describes the number of recursions being calculated |
---|
[7d56875] | 778 | - The basis of the algorithm is the following theorem: |
---|
[d1b0065] | 779 | Let N be an integer coprime to 6 and different from 1 and E be an |
---|
[2472afe] | 780 | ellipic curve modulo N. Assume that we know an integer m and a |
---|
[d1b0065] | 781 | point P of E(Z/NZ) satisfying the following conditions. |
---|
[53e03a6] | 782 | (1) There exists a prime divisor q of m such that q>(4-th root(N)+1)^2. |
---|
| 783 | (2) m*P=O(E)=(0:1:0). |
---|
| 784 | (3) (m/q)*P=(x:y:t) with t element of (Z/NZ)*. |
---|
| 785 | Then N is prime. |
---|
| 786 | EXAMPLE:example Atkin; shows an example |
---|
| 787 | " |
---|
| 788 | { |
---|
[d1b0065] | 789 | if(N==1) {return(-1);} // (0)[Test if assumptions well-defined] |
---|
| 790 | if((N==2)||(N==3)) {return(1);} |
---|
| 791 | if(gcdN(N,6)!=1) |
---|
| 792 | { |
---|
| 793 | if(printlevel>=1) {"ggT(N,6)="+string(gcdN(N,6));pause();} |
---|
| 794 | return(-1); |
---|
| 795 | } |
---|
| 796 | else |
---|
| 797 | { |
---|
| 798 | int i; // (1)[Initialize] |
---|
| 799 | int n(i); |
---|
| 800 | number N(i)=N; |
---|
| 801 | if(printlevel>=1) {"Setze i=0, n=0 und N(i)=N(0)="+string(N(i))+".";pause();} |
---|
| 802 | |
---|
| 803 | // declarations: |
---|
| 804 | int j(0),j(1),j(2),j(3),j(4),k; // running indices |
---|
| 805 | list L; // all primes smaller than 1000 |
---|
| 806 | list H; // sequence of negative discriminants |
---|
| 807 | number D; // discriminant out of H |
---|
| 808 | list L1,L2,S,S1,S2,R; // lists of relevant elements |
---|
| 809 | list P,P1,P2; // elliptic points on E(Z/N(i)Z) |
---|
| 810 | number m,q; // m=|E(Z/N(i)Z)| and q|m |
---|
| 811 | number a,b,j,c; // characterize E(Z/N(i)Z) |
---|
| 812 | number g,u; // g out of Z/N(i)Z, u=Jacobi(g,N(i)) |
---|
[3eadab] | 813 | poly T; // T=HilbertClassPoly(D,K) |
---|
[d1b0065] | 814 | matrix M; // M contains the coefficients of T |
---|
| 815 | |
---|
| 816 | if(printlevel>=1) {"Liste H der moeglichen geeigneten Diskriminanten wird berechnet.";} |
---|
| 817 | H=disc(N,K/2); |
---|
| 818 | if(printlevel>=1) {"H="+string(H);pause();} |
---|
| 819 | |
---|
| 820 | int step=2; |
---|
| 821 | while(1) |
---|
| 822 | { |
---|
| 823 | if(step==2) // (2)[Is N(i) small??] |
---|
[370344] | 824 | { |
---|
[d1b0065] | 825 | 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; |
---|
[2472afe] | 826 | for(j(0)=1;j(0)<=size(L);j(0)++) |
---|
[d1b0065] | 827 | { |
---|
[3d4a5a] | 828 | if(N(i)==L[j(0)]){return(1);} |
---|
[d1b0065] | 829 | if(((N(i) mod L[j(0)])==0)&&(N(i)!=L[j(0)])) |
---|
| 830 | { |
---|
| 831 | if(printlevel>=1) {"N("+string(i)+")="+string(N(i))+" ist durch "+string(L[j(0)])+" teilbar.";pause();} |
---|
| 832 | step=14; |
---|
| 833 | break; |
---|
| 834 | } |
---|
| 835 | } |
---|
| 836 | if(step==2) |
---|
[370344] | 837 | { |
---|
[d1b0065] | 838 | step=3; |
---|
[370344] | 839 | } |
---|
| 840 | } |
---|
[449fbf] | 841 | |
---|
[d1b0065] | 842 | if(step==3) // (3)[Choose next discriminant] |
---|
| 843 | { |
---|
| 844 | n(i)=n(i)+1; |
---|
| 845 | if(n(i)==size(H)+1) |
---|
| 846 | { |
---|
| 847 | if(printlevel>=1) {"Algorithmus nicht anwendbar, da zu wenige geeignete Diskriminanten existieren."; |
---|
| 848 | "Erhoehe den Genauigkeitsparameter K und starte den Algorithmus erneut.";pause();} |
---|
| 849 | return(0); |
---|
| 850 | } |
---|
| 851 | D=H[n(i)]; |
---|
| 852 | if(printlevel>=1) {"Naechste Diskriminante D wird gewaehlt. D="+string(D)+".";pause();} |
---|
| 853 | if(Jacobi(D,N(i))!=1) |
---|
| 854 | { |
---|
| 855 | if(printlevel>=1) {"Jacobi(D,N("+string(i)+"))="+string(Jacobi(D,N(i)));pause();} |
---|
| 856 | continue; |
---|
| 857 | } |
---|
| 858 | else |
---|
| 859 | { |
---|
| 860 | L1=CornacchiaModified(D,N(i)); |
---|
| 861 | if(size(L1)>1) |
---|
| 862 | { |
---|
| 863 | if(printlevel>=1) {"Die Loesung (x,y) der Gleichung x^2+|D|y^2=4N("+string(i)+") lautet";L1;pause();} |
---|
| 864 | step=4; |
---|
| 865 | } |
---|
| 866 | else |
---|
| 867 | { |
---|
| 868 | if(L1[1]==-1) |
---|
[370344] | 869 | { |
---|
[d1b0065] | 870 | if(printlevel>=1) {"Die Gleichung x^2+|D|y^2=4N("+string(i)+") hat keine Loesung.";pause();} |
---|
| 871 | continue; |
---|
| 872 | } |
---|
| 873 | if(L1[1]==0) |
---|
| 874 | { |
---|
[2472afe] | 875 | if(printLevel>=1) {"Algorithmus fuer N("+string(i)+")="+string(N(i))+" nicht anwendbar,"; |
---|
[d1b0065] | 876 | "da zu wenige geeignete Diskriminanten existieren.";pause();} |
---|
| 877 | step=14; |
---|
| 878 | } |
---|
| 879 | } |
---|
| 880 | } |
---|
| 881 | } |
---|
[449fbf] | 882 | |
---|
[d1b0065] | 883 | if(step==4) // (4)[Factor m] |
---|
| 884 | { |
---|
| 885 | if(printlevel>=1) {"Die Liste L2 der moeglichen m=|E(Z/N("+string(i)+")Z)| wird berechnet.";} |
---|
| 886 | if(absValue(L1[1])^2<=4*N(i)) {L2=N(i)+1+L1[1],N(i)+1-L1[1];} |
---|
| 887 | if(D==-4) |
---|
| 888 | { |
---|
| 889 | if(absValue(2*L1[2])^2<=4*N(i)) {L2[size(L2)+1]=N(i)+1+2*L1[2]; |
---|
| 890 | L2[size(L2)+1]=N(i)+1-2*L1[2];} |
---|
| 891 | } |
---|
| 892 | // An dieser Stelle wurde "<=4*N(i)" durch "<=16*N(i)" ersetzt. |
---|
| 893 | if(D==-3) |
---|
| 894 | { |
---|
| 895 | if(absValue(L1[1]+3*L1[2])^2<=16*N(i)) {L2[size(L2)+1]=N(i)+1+(L1[1]+3*L1[2])/2; |
---|
| 896 | L2[size(L2)+1]=N(i)+1-(L1[1]+3*L1[2])/2;} |
---|
| 897 | if(absValue(L1[1]-3*L1[2])^2<=16*N(i)) {L2[size(L2)+1]=N(i)+1+(L1[1]-3*L1[2])/2; |
---|
| 898 | L2[size(L2)+1]=N(i)+1-(L1[1]-3*L1[2])/2;} |
---|
| 899 | } |
---|
| 900 | /////////////////////////////////////////////////////////////// |
---|
| 901 | if(size(L2)==0) |
---|
| 902 | { |
---|
| 903 | if(printlevel>=1) {"Nach dem Satz von Hasse wurden keine moeglichen m=|E(Z/N("+string(i)+")Z)|"; |
---|
| 904 | "fuer D="+string(D)+" gefunden.";} |
---|
| 905 | step=3; |
---|
| 906 | continue; |
---|
| 907 | } |
---|
| 908 | else |
---|
| 909 | { |
---|
| 910 | if(printlevel>=1) {"L2=";L2;pause();} |
---|
| 911 | } |
---|
[449fbf] | 912 | |
---|
[d1b0065] | 913 | if(printlevel>=1) {"Die Liste S der Faktoren aller moeglichen m wird berechnet.";} |
---|
| 914 | S=list(); |
---|
| 915 | for(j(1)=1;j(1)<=size(L2);j(1)++) |
---|
| 916 | { |
---|
| 917 | m=L2[j(1)]; |
---|
| 918 | if(m!=0) |
---|
| 919 | { |
---|
| 920 | S1=PollardRho(m,10000,1,L); |
---|
| 921 | S2=pFactor(m,100,L); |
---|
| 922 | S[size(S)+1]=list(m,S1+S2); |
---|
| 923 | } |
---|
| 924 | } |
---|
| 925 | if(printlevel>=1) {"S=";S;pause();} |
---|
| 926 | step=5; |
---|
| 927 | } |
---|
[449fbf] | 928 | |
---|
[d1b0065] | 929 | if(step==5) // (5)[Does a suitable m exist??] |
---|
| 930 | { |
---|
| 931 | for(j(2)=1;j(2)<=size(S);j(2)++) |
---|
| 932 | { |
---|
| 933 | m=L2[j(2)]; |
---|
| 934 | for(j(3)=1;j(3)<=size(S[j(2)][2]);j(3)++) |
---|
| 935 | { |
---|
| 936 | q=S[j(2)][2][j(3)]; |
---|
| 937 | // sqr(sqr(N(i),50),50) ersetzt intRoot(intRoot(N(i))) |
---|
| 938 | if((q>(sqr(sqr(N(i),50),50)+1)^2) && (MillerRabin(q,5)==1)) |
---|
| 939 | { |
---|
| 940 | step=6; |
---|
| 941 | break; |
---|
| 942 | } |
---|
| 943 | ////////////////////////////////////////////////////// |
---|
| 944 | } |
---|
| 945 | if(step==6) |
---|
| 946 | { |
---|
| 947 | if(printlevel>=1) {"Geeignetes Paar (m,q) gefunden, so dass q|m,"; |
---|
| 948 | "q>(4-th root(N("+string(i)+"))+1)^2 und q den Miller-Rabin-Test passiert."; |
---|
| 949 | "m="+string(m)+",";"q="+string(q);pause();} |
---|
| 950 | break; |
---|
| 951 | } |
---|
| 952 | else |
---|
| 953 | { |
---|
| 954 | step=3; |
---|
| 955 | } |
---|
| 956 | } |
---|
| 957 | if(step==3) |
---|
| 958 | { |
---|
| 959 | if(printlevel>=1) {"Kein geeignetes Paar (m,q), so dass q|m,"; |
---|
| 960 | "q>(4-th root(N("+string(i)+"))+1)^2 und q den Miller-Rabin-Test passiert, gefunden."; |
---|
| 961 | pause();} |
---|
| 962 | continue; |
---|
| 963 | } |
---|
| 964 | } |
---|
[449fbf] | 965 | |
---|
[d1b0065] | 966 | if(step==6) // (6)[Compute elliptic curve] |
---|
| 967 | { |
---|
| 968 | if(D==-4) |
---|
| 969 | { |
---|
| 970 | a=-1; |
---|
| 971 | b=0; |
---|
| 972 | if(printlevel>=1) {"Da D=-4, setze a=-1 und b=0.";pause();} |
---|
| 973 | } |
---|
| 974 | if(D==-3) |
---|
| 975 | { |
---|
| 976 | a=0; |
---|
| 977 | b=-1; |
---|
| 978 | if(printlevel>=1) {"Da D=-3, setze a=0 und b=-1.";pause();} |
---|
| 979 | } |
---|
| 980 | if(D<-4) |
---|
| 981 | { |
---|
| 982 | if(printlevel>=1) {"Das Minimalpolynom T von j((D+sqr(D))/2) aus Z[X] fuer D="+string(D)+" wird berechnet.";} |
---|
[3eadab] | 983 | T=HilbertClassPoly(D,K); |
---|
[d1b0065] | 984 | if(printlevel>=1) {"T="+string(T);pause();} |
---|
| 985 | |
---|
| 986 | M=coeffs(T,var(1)); |
---|
| 987 | T=0; |
---|
| 988 | |
---|
| 989 | for(j(4)=1;j(4)<=nrows(M);j(4)++) |
---|
| 990 | { |
---|
| 991 | M[j(4),1]=leadcoef(M[j(4),1]) mod N(i); |
---|
| 992 | T=T+M[j(4),1]*var(1)^(j(4)-1); |
---|
| 993 | } |
---|
| 994 | if(printlevel>=1) {"Setze T=T mod N("+string(i)+").";"T="+string(T);pause();} |
---|
| 995 | |
---|
[3eadab] | 996 | R=rootsModp(int(N(i)),T); |
---|
[d1b0065] | 997 | if(deg(T)>size(R)) |
---|
| 998 | { |
---|
| 999 | ERROR("Das Polynom T zerfaellt modulo N("+string(i)+") nicht vollstaendig in Linearfaktoren." |
---|
| 1000 | "Erhoehe den Genauigkeitsparameter K und starte den Algorithmus erneut."); |
---|
[2472afe] | 1001 | } |
---|
[d1b0065] | 1002 | if(printlevel>=1) {if(deg(T)>1) {"Die "+string(deg(T))+" Nullstellen von T modulo N("+string(i)+") sind"; |
---|
| 1003 | R;pause();} |
---|
| 1004 | if(deg(T)==1){"Die Nullstelle von T modulo N("+string(i)+") ist";R;pause();}} |
---|
| 1005 | |
---|
| 1006 | j=R[1]; |
---|
| 1007 | c=j*exgcdN(j-1728,N(i))[1]; |
---|
| 1008 | a=-3*c mod N(i); |
---|
| 1009 | b=2*c mod N(i); |
---|
| 1010 | if(printlevel>=1) {"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)+")."; |
---|
| 1011 | "a="+string(a)+",";"b="+string(b);pause();} |
---|
| 1012 | } |
---|
| 1013 | step=7; |
---|
| 1014 | } |
---|
[449fbf] | 1015 | |
---|
[d1b0065] | 1016 | if(step==7) // (7)[Find g] |
---|
| 1017 | { |
---|
| 1018 | if(D==-3) |
---|
| 1019 | { |
---|
| 1020 | while(1) |
---|
| 1021 | { |
---|
| 1022 | g=random(1,2147483647) mod N(i); |
---|
| 1023 | u=Jacobi(g,N(i)); |
---|
| 1024 | if((u==-1)&&(powerN(g,(N(i)-1)/3,N(i))!=1)) |
---|
| 1025 | { |
---|
| 1026 | if(printlevel>=1) {"g="+string(g);pause();} |
---|
| 1027 | break; |
---|
| 1028 | } |
---|
| 1029 | } |
---|
| 1030 | } |
---|
| 1031 | else |
---|
| 1032 | { |
---|
| 1033 | while(1) |
---|
| 1034 | { |
---|
| 1035 | g=random(1,2147483647) mod N(i); |
---|
| 1036 | u=Jacobi(g,N(i)); |
---|
| 1037 | if(u==-1) |
---|
| 1038 | { |
---|
| 1039 | if(printlevel>=1) {"g="+string(g);pause();} |
---|
| 1040 | break; |
---|
| 1041 | } |
---|
| 1042 | } |
---|
| 1043 | } |
---|
| 1044 | step=8; |
---|
| 1045 | } |
---|
[449fbf] | 1046 | |
---|
[d1b0065] | 1047 | if(step==8) // (8)[Find P] |
---|
| 1048 | { |
---|
| 1049 | if(printlevel>=1) {"Ein zufaelliger Punkt P auf der Elliptischen Kurve"; |
---|
| 1050 | "mit der Gleichung y^2=x^3+ax+b fuer";"N("+string(i)+")="+string(N(i))+","; |
---|
| 1051 | " a="+string(a)+",";" b="+string(b);"wird gewaehlt.";} |
---|
| 1052 | P=ellipticRandomPoint(N(i),a,b); |
---|
| 1053 | if(printlevel>=1) {"P=("+string(P)+")";pause();} |
---|
[449fbf] | 1054 | |
---|
[d1b0065] | 1055 | if(size(P)==1) |
---|
| 1056 | { |
---|
| 1057 | step=14; |
---|
| 1058 | } |
---|
| 1059 | else |
---|
| 1060 | { |
---|
| 1061 | step=9; |
---|
| 1062 | } |
---|
| 1063 | } |
---|
[449fbf] | 1064 | |
---|
[d1b0065] | 1065 | if(step==9) // (9)[Find right curve] |
---|
| 1066 | { |
---|
| 1067 | if(printlevel>=1) {"Die Punkte P2=(m/q)*P und P1=q*P2 auf der Kurve werden berechnet.";} |
---|
| 1068 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
| 1069 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
| 1070 | if(printlevel>=1) {"P1=("+string(P1)+"),";"P2=("+string(P2)+")";pause();} |
---|
[449fbf] | 1071 | |
---|
[d1b0065] | 1072 | if((P1[1]==0)&&(P1[2]==1)&&(P1[3]==0)) |
---|
| 1073 | { |
---|
| 1074 | step=12; |
---|
| 1075 | } |
---|
| 1076 | else |
---|
| 1077 | { |
---|
| 1078 | 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)|."; |
---|
| 1079 | "Waehle daher neue Koeffizienten a und b.";pause();} |
---|
| 1080 | step=10; |
---|
| 1081 | } |
---|
| 1082 | } |
---|
[449fbf] | 1083 | |
---|
[2472afe] | 1084 | if(step==10) // (10)[Change coefficients] |
---|
[d1b0065] | 1085 | { |
---|
| 1086 | k=k+1; |
---|
[3eadab] | 1087 | if(k>=wUnit(D)) |
---|
[d1b0065] | 1088 | { |
---|
[3eadab] | 1089 | if(printlevel>=1) {"Da k=wUnit(D)="+string(k)+", ist N("+string(i)+")="+string(N(i))+" nicht prim.";pause();} |
---|
[d1b0065] | 1090 | step=14; |
---|
| 1091 | } |
---|
| 1092 | else |
---|
| 1093 | { |
---|
| 1094 | if(D<-4) {a=a*g^2 mod N(i); b=b*g^3 mod N(i); |
---|
| 1095 | if(printlevel>=1) {"Da D<-4, setze a=a*g^2 mod N("+string(i)+") und b=b*g^3 mod N("+string(i)+")."; |
---|
| 1096 | "a="+string(a)+",";"b="+string(b)+",";"k="+string(k);pause();}} |
---|
| 1097 | if(D==-4){a=a*g mod N(i); |
---|
| 1098 | if(printlevel>=1) {"Da D=-4, setze a=a*g mod N("+string(i)+").";"a="+string(a)+","; |
---|
| 1099 | "b="+string(b)+",";"k="+string(k);pause();}} |
---|
| 1100 | if(D==-3){b=b*g mod N(i); |
---|
| 1101 | if(printlevel>=1) {"Da D=-3, setze b=b*g mod N("+string(i)+").";"a="+string(a)+","; |
---|
| 1102 | "b="+string(b)+",";"k="+string(k);pause();}} |
---|
| 1103 | step=8; |
---|
| 1104 | continue; |
---|
| 1105 | } |
---|
| 1106 | } |
---|
[449fbf] | 1107 | |
---|
[d1b0065] | 1108 | if(step==11) // (11)[Find a new P] |
---|
| 1109 | { |
---|
| 1110 | if(printlevel>=1) {"Ein neuer zufaelliger Punkt P auf der Elliptischen Kurve wird gewaehlt,"; |
---|
| 1111 | "da auch P2=(0:1:0).";} |
---|
| 1112 | P=ellipticRandomPoint(N(i),a,b); |
---|
| 1113 | if(printlevel>=1) {"P=("+string(P)+")";pause();} |
---|
[449fbf] | 1114 | |
---|
[d1b0065] | 1115 | if(size(P)==1) |
---|
| 1116 | { |
---|
| 1117 | step=14; |
---|
| 1118 | } |
---|
| 1119 | else |
---|
| 1120 | { |
---|
| 1121 | if(printlevel>=1) {"Die Punkte P2=(m/q)*P und P1=q*P2 auf der Kurve werden berechnet.";} |
---|
| 1122 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
| 1123 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
| 1124 | if(printlevel>=1) {"P1=("+string(P1)+"),";"P2=("+string(P2)+")";pause();} |
---|
| 1125 | |
---|
| 1126 | if((P1[1]!=0)||(P1[2]!=1)||(P1[3]!=0)) |
---|
| 1127 | { |
---|
| 1128 | 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)|."; |
---|
| 1129 | "Waehle daher neue Koeffizienten a und b.";pause();} |
---|
| 1130 | step=10; |
---|
| 1131 | continue; |
---|
| 1132 | } |
---|
| 1133 | else |
---|
| 1134 | { |
---|
| 1135 | step=12; |
---|
| 1136 | } |
---|
| 1137 | } |
---|
| 1138 | } |
---|
[449fbf] | 1139 | |
---|
[d1b0065] | 1140 | if(step==12) // (12)[Check P] |
---|
| 1141 | { |
---|
| 1142 | if((P2[1]==0)&&(P2[2]==1)&&(P2[3]==0)) |
---|
| 1143 | { |
---|
| 1144 | step=11; |
---|
| 1145 | continue; |
---|
| 1146 | } |
---|
| 1147 | else |
---|
| 1148 | { |
---|
| 1149 | step=13; |
---|
| 1150 | } |
---|
| 1151 | } |
---|
[449fbf] | 1152 | |
---|
[d1b0065] | 1153 | if(step==13) // (13)[Recurse] |
---|
| 1154 | { |
---|
| 1155 | if(i<B) |
---|
| 1156 | { |
---|
| 1157 | if(printlevel>=1) {string(i+1)+". Rekursion:";""; |
---|
| 1158 | "N("+string(i)+")="+string(N(i))+" erfuellt die Bedingungen des zugrunde liegenden Satzes,"; |
---|
| 1159 | "da P1=(0:1:0) und P2[3] aus (Z/N("+string(i)+")Z)*.";""; |
---|
| 1160 | "Untersuche nun, ob auch der gefundene Faktor q="+string(q)+" diese Bedingungen erfuellt."; |
---|
| 1161 | "Setze dazu i=i+1, N("+string(i+1)+")=q="+string(q)+" und beginne den Algorithmus von vorne.";pause();} |
---|
| 1162 | i=i+1; |
---|
| 1163 | int n(i); |
---|
| 1164 | number N(i)=q; |
---|
| 1165 | k=0; |
---|
| 1166 | step=2; |
---|
| 1167 | continue; |
---|
| 1168 | } |
---|
| 1169 | else |
---|
| 1170 | { |
---|
| 1171 | if(printlevel>=1) {"N(B)=N("+string(i)+")="+string(N(i))+" erfuellt die Bedingungen des zugrunde liegenden Satzes,"; |
---|
| 1172 | "da P1=(0:1:0) und P2[3] aus (Z/N("+string(i)+")Z)*."; |
---|
| 1173 | "Insbesondere ist N="+string(N)+" prim.";pause();} |
---|
| 1174 | return(1); |
---|
| 1175 | } |
---|
| 1176 | } |
---|
[370344] | 1177 | |
---|
[d1b0065] | 1178 | if(step==14) // (14)[Backtrack] |
---|
| 1179 | { |
---|
| 1180 | if(i>0) |
---|
| 1181 | { |
---|
[2472afe] | 1182 | if(printlevel>=1) {"Setze i=i-1 und starte den Algorithmus fuer N("+string(i-1)+")="+string(N(i-1))+" mit"; |
---|
[d1b0065] | 1183 | "neuer Diskriminanten von vorne.";pause();} |
---|
| 1184 | i=i-1; |
---|
| 1185 | k=0; |
---|
| 1186 | step=3; |
---|
| 1187 | } |
---|
| 1188 | else |
---|
| 1189 | { |
---|
[3eadab] | 1190 | if(printlevel>=1) {"N(0)=N="+string(N)+" und daher ist N nicht prim.";pause(n);} |
---|
[d1b0065] | 1191 | return(-1); |
---|
| 1192 | } |
---|
| 1193 | } |
---|
| 1194 | } |
---|
| 1195 | } |
---|
[53e03a6] | 1196 | } |
---|
[449fbf] | 1197 | example |
---|
[53e03a6] | 1198 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 1199 | ring R = 0,x,dp; |
---|
| 1200 | printlevel=1; |
---|
| 1201 | Atkin(7691,100,5); |
---|
| 1202 | Atkin(8543,100,4); |
---|
| 1203 | Atkin(100019,100,5); |
---|
| 1204 | Atkin(10000079,100,2); |
---|
[53e03a6] | 1205 | } |
---|
[3eadab] | 1206 | |
---|