[449fbf] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[341696] | 2 | version="$Id$"; |
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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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[1a3911] | 6 | AUTHOR: Stefan Steidel, steidel@mathematik.uni-kl.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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[d317a9] | 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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[449fbf] | 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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[1a3911] | 22 | sqr(w,k) computes the square root of w w.r.t. accuracy k |
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[3eadab] | 23 | expo(z,k) computes exp(z) |
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| 24 | jOft(t,k) computes the j-invariant of t |
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| 25 | round(r) rounds r to the nearest number out of Z |
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| 26 | HilbertClassPoly(D,k) computes the Hilbert Class Polynomial |
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| 27 | rootsModp(p,P) computes roots of the polynomial P modulo p |
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| 28 | wUnit(D) computes the number of units in Q(sqr(D)) |
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| 29 | Atkin(N,K,B) tries to prove that N is prime |
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[d1b0065] | 30 | |
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[449fbf] | 31 | "; |
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| 32 | |
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[abb4919] | 33 | LIB "crypto.lib"; |
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[449fbf] | 34 | LIB "general.lib"; |
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| 35 | LIB "ntsolve.lib"; |
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| 36 | LIB "inout.lib"; |
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| 37 | |
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| 38 | /////////////////////////////////////////////////////////////////////////////// |
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| 39 | |
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[3eadab] | 40 | proc newTest(list L, number D) |
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| 41 | "USAGE: newTest(L,D); |
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[53e03a6] | 42 | RETURN: 1, if D does not already exist in L, |
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| 43 | -1, if D does already exist in L |
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| 44 | EXAMPLE:example new; shows an example |
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| 45 | " |
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| 46 | { |
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[d317a9] | 47 | number a=1; // a=1 means: D does not already exist in L |
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[ea9f7aa] | 48 | int i; |
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| 49 | for(i=1;i<=size(L);i++) |
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| 50 | { |
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| 51 | if(D==L[i]) |
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| 52 | { |
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[d317a9] | 53 | a=-1; // a=-1 means: D does already exist in L |
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[ea9f7aa] | 54 | break; |
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| 55 | } |
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| 56 | } |
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| 57 | return(a); |
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[53e03a6] | 58 | } |
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[449fbf] | 59 | example |
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[53e03a6] | 60 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 61 | ring r = 0,x,dp; |
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| 62 | list L=8976,-223456,556,-778,3,-55603,45,766677; |
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| 63 | number D=-55603; |
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[3eadab] | 64 | newTest(L,D); |
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[53e03a6] | 65 | } |
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[449fbf] | 66 | |
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[d317a9] | 67 | |
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[449fbf] | 68 | proc bubblesort(list L) |
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[53e03a6] | 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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[ea9f7aa] | 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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[53e03a6] | 92 | } |
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[449fbf] | 93 | example |
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[53e03a6] | 94 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 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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[53e03a6] | 98 | } |
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[449fbf] | 99 | |
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[d317a9] | 100 | |
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[449fbf] | 101 | proc disc(number N, int k) |
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[53e03a6] | 102 | "USAGE: disc(N,k); |
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[1a3911] | 103 | RETURN: list L of negative discriminants D, sorted in decreasing order |
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[53e03a6] | 104 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4N |
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| 105 | 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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| 106 | EXAMPLE:example disc; shows an example |
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| 107 | " |
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| 108 | { |
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[ea9f7aa] | 109 | list L=-3,-4,-7; |
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| 110 | number D; |
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| 111 | number B; |
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| 112 | int a,b; |
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| 113 | for(b=0;b<=k;b++) |
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| 114 | { |
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| 115 | B=b^2; |
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| 116 | for(a=int(intPart(B/4))+1;a<=k;a++) |
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| 117 | { |
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| 118 | D=-4*a+B; |
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[3eadab] | 119 | if((D<0)&&((D mod 4)!=2)&&((D mod 4)!=3)&&(absValue(D)<4*N)&&(newTest(L,D)==1)) |
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[ea9f7aa] | 120 | { |
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| 121 | L[size(L)+1]=D; |
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| 122 | } |
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| 123 | } |
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| 124 | } |
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| 125 | L=bubblesort(L); |
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| 126 | return(L); |
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[53e03a6] | 127 | } |
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[449fbf] | 128 | example |
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[53e03a6] | 129 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 130 | ring R = 0,x,dp; |
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| 131 | disc(2003,50); |
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[53e03a6] | 132 | } |
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[449fbf] | 133 | |
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| 134 | |
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| 135 | proc Cornacchia(number d, number p) |
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[53e03a6] | 136 | "USAGE: Cornacchia(d,p); |
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| 137 | RETURN: x,y such that x^2+d*y^2=p with p prime, |
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| 138 | -1, if the Diophantine equation has no solution, |
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| 139 | 0, if the parameters are wrong selected |
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| 140 | ASSUME: 0<d<p |
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| 141 | EXAMPLE:example Cornacchia; shows an example |
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| 142 | " |
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| 143 | { |
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[d317a9] | 144 | if((d<0)||(p<d)) // (0)[Test if assumptions well-defined] |
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[ea9f7aa] | 145 | { |
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| 146 | return(0); |
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| 147 | // ERROR("Parameters wrong selected! It has to be 0<d<p!"); |
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| 148 | } |
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| 149 | else |
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| 150 | { |
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[19ffafb] | 151 | number k,x(0),a,b,l,r,c,i; |
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[ea9f7aa] | 152 | |
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[d317a9] | 153 | k=Jacobi(-d,p); // (1)[Test if residue] |
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[ea9f7aa] | 154 | if(k==-1) |
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| 155 | { |
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| 156 | return(-1); |
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| 157 | // ERROR("The Diophantine equation has no solution!"); |
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| 158 | } |
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| 159 | else |
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| 160 | { |
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[19ffafb] | 161 | x(0)=squareRoot(-d,p); // (2)[Compute square root] |
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| 162 | if((p/2>=x(0))||(p<=x(0))) |
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| 163 | { |
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| 164 | x(0)=-x(0) mod p; |
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| 165 | } |
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| 166 | a=p; |
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| 167 | b=x(0); |
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| 168 | l=intRoot(p); |
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[d317a9] | 169 | while(b>l) // (3)[Euclidean algorithm] |
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[19ffafb] | 170 | { |
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[ea9f7aa] | 171 | r=a mod b; |
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| 172 | a=b; |
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| 173 | b=r; |
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[19ffafb] | 174 | } |
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[d317a9] | 175 | c=(p-b^2)/d; // (4)[Test solution] |
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[19ffafb] | 176 | i=intRoot(c); |
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| 177 | if((((p-b^2) mod d)!=0)||(c!=i^2)) |
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| 178 | { |
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| 179 | return(-1); |
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| 180 | } |
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| 181 | else |
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| 182 | { |
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[ea9f7aa] | 183 | list L=b,i; |
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| 184 | return(L); |
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[19ffafb] | 185 | } |
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[ea9f7aa] | 186 | } |
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| 187 | } |
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[2472afe] | 188 | } |
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[449fbf] | 189 | example |
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[53e03a6] | 190 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 191 | ring R = 0,x,dp; |
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| 192 | Cornacchia(55,9551); |
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[53e03a6] | 193 | } |
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[449fbf] | 194 | |
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[d317a9] | 195 | |
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[449fbf] | 196 | proc CornacchiaModified(number D, number p) |
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[53e03a6] | 197 | "USAGE: CornacchiaModified(D,p); |
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| 198 | RETURN: x,y such that x^2+|D|*y^2=p with p prime, |
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| 199 | -1, if the Diophantine equation has no solution, |
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| 200 | 0, if the parameters are wrong selected |
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| 201 | ASSUME: D<0, D kongruent 0 or 1 modulo 4 and |D|<4p |
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| 202 | EXAMPLE:example CornacchiaModified; shows an example |
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| 203 | " |
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| 204 | { |
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[26508d] | 205 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)||(absValue(D)>=4*p)) |
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[d317a9] | 206 | { // (0)[Test if assumptions well-defined] |
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[26508d] | 207 | return(0); |
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| 208 | // ERROR("Parameters wrong selected!"); |
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| 209 | } |
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| 210 | else |
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| 211 | { |
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[d317a9] | 212 | if(p==2) // (1)[Case p=2] |
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[26508d] | 213 | { |
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| 214 | if((D+8)==intRoot(D+8)^2) |
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[19ffafb] | 215 | { |
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[26508d] | 216 | return(intRoot(D+8),1); |
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[19ffafb] | 217 | } |
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[449fbf] | 218 | else |
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[19ffafb] | 219 | { |
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[26508d] | 220 | return(-1); |
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| 221 | // ERROR("The Diophantine equation has no solution!"); |
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| 222 | } |
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| 223 | } |
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| 224 | else |
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| 225 | { |
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| 226 | number k,x(0),a,b,l,r,c; |
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[d317a9] | 227 | k=Jacobi(D,p); // (2)[Test if residue] |
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[26508d] | 228 | if(k==-1) |
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| 229 | { |
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| 230 | return(-1); |
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| 231 | // ERROR("The Diophantine equation has no solution!"); |
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[19ffafb] | 232 | } |
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[26508d] | 233 | else |
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| 234 | { |
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[d317a9] | 235 | x(0)=squareRoot(D,p); // (3)[Compute square root] |
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[26508d] | 236 | if((x(0) mod 2)!=(D mod 2)) |
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| 237 | { |
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| 238 | x(0)=p-x(0); |
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| 239 | } |
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| 240 | a=2*p; |
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| 241 | b=x(0); |
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| 242 | l=intRoot(4*p); |
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[d317a9] | 243 | while(b>l) // (4)[Euclidean algorithm] |
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[26508d] | 244 | { |
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| 245 | r=a mod b; |
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| 246 | a=b; |
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| 247 | b=r; |
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| 248 | } |
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[d317a9] | 249 | c=(4*p-b^2)/absValue(D); // (5)[Test solution] |
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[26508d] | 250 | if((((4*p-b^2) mod absValue(D))!=0)||(c!=intRoot(c)^2)) |
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| 251 | { |
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| 252 | return(-1); |
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| 253 | // ERROR("The Diophantine equation has no solution!"); |
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| 254 | } |
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| 255 | else |
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| 256 | { |
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| 257 | list L=b,intRoot(c); |
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| 258 | return(L); |
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| 259 | } |
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| 260 | } |
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| 261 | } |
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| 262 | } |
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[53e03a6] | 263 | } |
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[449fbf] | 264 | example |
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[53e03a6] | 265 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 266 | ring R = 0,x,dp; |
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| 267 | CornacchiaModified(-107,1319); |
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[53e03a6] | 268 | } |
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[449fbf] | 269 | |
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[d317a9] | 270 | |
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[449fbf] | 271 | proc pFactor1(number n,int B, list P) |
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[53e03a6] | 272 | "USAGE: pFactor1(n,B,P); n to be factorized, B a bound , P a list of primes |
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| 273 | RETURN: a list of factors of n or the message: no factor found |
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| 274 | NOTE: Pollard's p-factorization |
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| 275 | creates the product k of powers of primes (bounded by B) from |
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| 276 | the list P with the idea that for a prime divisor p of n p-1|k |
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| 277 | then p devides gcd(a^k-1,n) for some random a |
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| 278 | EXAMPLE:example pFactor1; shows an example |
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| 279 | " |
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| 280 | { |
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[26508d] | 281 | int i; |
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| 282 | number k=1; |
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| 283 | number w; |
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| 284 | while(i<size(P)) |
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| 285 | { |
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| 286 | i++; |
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| 287 | w=P[i]; |
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| 288 | if(w>B) {break;} |
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| 289 | while(w*P[i]<=B) |
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| 290 | { |
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| 291 | w=w*P[i]; |
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| 292 | } |
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| 293 | k=k*w; |
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| 294 | } |
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| 295 | number a=random(2,2147483629); |
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| 296 | number d=gcdN(powerN(a,k,n)-1,n); |
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| 297 | if((d>1)&&(d<n)){return(d);} |
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| 298 | return(n); |
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[53e03a6] | 299 | } |
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[449fbf] | 300 | example |
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| 301 | { "EXAMPLE:"; echo = 2; |
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| 302 | ring R = 0,z,dp; |
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| 303 | list L=primList(1000); |
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| 304 | pFactor1(1241143,13,L); |
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| 305 | number h=10; |
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| 306 | h=h^30+25; |
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| 307 | pFactor1(h,20,L); |
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| 308 | } |
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| 309 | |
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[d317a9] | 310 | |
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[449fbf] | 311 | proc maximum(list L) |
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[53e03a6] | 312 | "USAGE: maximum(list L); |
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| 313 | RETURN: the maximal number contained in list L |
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| 314 | EXAMPLE:example maximum; shows an example |
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| 315 | " |
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| 316 | { |
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[26508d] | 317 | number max=L[1]; |
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| 318 | int i; |
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| 319 | for(i=2;i<=size(L);i++) |
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| 320 | { |
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| 321 | if(L[i]>max) |
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| 322 | { |
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| 323 | max=L[i]; |
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| 324 | } |
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| 325 | } |
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| 326 | return(max); |
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[53e03a6] | 327 | } |
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[449fbf] | 328 | example |
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[53e03a6] | 329 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 330 | ring r = 0,x,dp; |
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| 331 | list L=465,867,1233,4567,776544,233445,2334,556; |
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| 332 | maximum(L); |
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[53e03a6] | 333 | } |
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[449fbf] | 334 | |
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[d317a9] | 335 | |
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[3eadab] | 336 | static proc cmod(number x, number y) |
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[53e03a6] | 337 | "USAGE: cmod(x,y); |
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| 338 | RETURN: x mod y |
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| 339 | ASSUME: x,y out of Z and x,y<=2147483647 |
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| 340 | NOTE: this algorithm is a helping procedure to be able to calculate |
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| 341 | x mod y with x,y out of Z while working in the complex field |
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| 342 | EXAMPLE:example cmod; shows an example |
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| 343 | " |
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| 344 | { |
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[26508d] | 345 | int rest=int(x-y*int(x/y)); |
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| 346 | if(rest<0) |
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| 347 | { |
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| 348 | rest=rest+int(y); |
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| 349 | } |
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| 350 | return(rest); |
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[53e03a6] | 351 | } |
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[449fbf] | 352 | example |
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[53e03a6] | 353 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 354 | ring r = (complex,30,i),x,dp; |
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| 355 | number x=-1004456; |
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| 356 | number y=1233; |
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| 357 | cmod(x,y); |
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[53e03a6] | 358 | } |
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[449fbf] | 359 | |
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[d317a9] | 360 | |
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[26508d] | 361 | proc sqr(number w, int k) |
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[53e03a6] | 362 | "USAGE: sqr(w,k); |
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| 363 | RETURN: the square root of w |
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| 364 | ASSUME: w>=0 |
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| 365 | NOTE: k describes the number of decimals being calculated in the real numbers, |
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| 366 | k, intPart(k/5) are inputs for the procedure "nt_solve" |
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| 367 | EXAMPLE:example sqr; shows an example |
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| 368 | " |
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| 369 | { |
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| 370 | poly f=var(1)^2-w; |
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| 371 | def S=basering; |
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| 372 | ring R=(real,k),var(1),dp; |
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| 373 | poly f=imap(S,f); |
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| 374 | ideal I=nt_solve(f,1.1,list(k,int(intPart(k/5)))); |
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| 375 | number c=leadcoef(I[1]); |
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| 376 | setring S; |
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| 377 | number c=imap(R,c); |
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| 378 | return(c); |
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| 379 | } |
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[449fbf] | 380 | example |
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[53e03a6] | 381 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 382 | ring R = (real,60),x,dp; |
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[0c0b9f1] | 383 | number ww=288469650108669535726081; |
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| 384 | sqr(ww,60); |
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[53e03a6] | 385 | } |
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[449fbf] | 386 | |
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| 387 | |
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[3eadab] | 388 | proc expo(number z, int k) |
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| 389 | "USAGE: expo(z,k); |
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[53e03a6] | 390 | RETURN: e^z to the order k |
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| 391 | NOTE: k describes the number of summands being calculated in the exponential power series |
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[1a3911] | 392 | EXAMPLE:example expo; shows an example |
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[53e03a6] | 393 | " |
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| 394 | { |
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| 395 | number q=1; |
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| 396 | number e=1; |
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| 397 | int n; |
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| 398 | for(n=1;n<=k;n++) |
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| 399 | { |
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| 400 | q=q*z/n; |
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| 401 | e=e+q; |
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| 402 | } |
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| 403 | return(e); |
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| 404 | } |
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[449fbf] | 405 | example |
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[53e03a6] | 406 | { "EXAMPLE:"; echo = 2; |
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[449fbf] | 407 | ring r = (real,30),x,dp; |
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| 408 | number z=40.35; |
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[3eadab] | 409 | expo(z,1000); |
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[53e03a6] | 410 | } |
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[449fbf] | 411 | |
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| 412 | |
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[3eadab] | 413 | proc jOft(number t, int k) |
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| 414 | "USAGE: jOft(t,k); |
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[53e03a6] | 415 | RETURN: the j-invariant of t |
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| 416 | ASSUME: t is a complex number with positive imaginary part |
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| 417 | NOTE: k describes the number of summands being calculated in the power series, |
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[1a3911] | 418 | 10*k is input for the procedure @code{expo} |
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| 419 | EXAMPLE:example jOft; shows an example |
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[53e03a6] | 420 | " |
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| 421 | { |
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[26508d] | 422 | number q1,q2,qr1,qi1,tr,ti,m1,m2,f,j; |
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| 423 | |
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| 424 | number pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989; |
---|
| 425 | |
---|
| 426 | tr=repart(t); |
---|
| 427 | ti=impart(t); |
---|
| 428 | if(tr==-1/2){qr1=-1;} |
---|
| 429 | if(tr==0){qr1=1;} |
---|
| 430 | if((tr!=-1/2)&&(tr!=0)) |
---|
| 431 | { |
---|
| 432 | tr=tr-round(tr); |
---|
| 433 | qr1=expo(2*i*pi*tr,10*k); |
---|
| 434 | } |
---|
| 435 | |
---|
| 436 | qi1=expo(-pi*ti,10*k); |
---|
| 437 | q1=qr1*qi1^2; |
---|
| 438 | q2=q1^2; |
---|
| 439 | |
---|
| 440 | int n=1; |
---|
| 441 | while(n<=k) |
---|
| 442 | { |
---|
| 443 | m1=m1+(-1)^n*(q1^(n*(3*n-1)/2)+q1^(n*(3*n+1)/2)); |
---|
| 444 | m2=m2+(-1)^n*(q2^(n*(3*n-1)/2)+q2^(n*(3*n+1)/2)); |
---|
| 445 | n++; |
---|
| 446 | } |
---|
| 447 | |
---|
| 448 | f=q1*((1+m2)/(1+m1))^24; |
---|
[449fbf] | 449 | |
---|
[26508d] | 450 | j=(256*f+1)^3/f; |
---|
| 451 | return(j); |
---|
| 452 | } |
---|
[449fbf] | 453 | example |
---|
[53e03a6] | 454 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 455 | ring r = (complex,30,i),x,dp; |
---|
| 456 | number t=(-7+i*sqr(7,250))/2; |
---|
[3eadab] | 457 | jOft(t,50); |
---|
[53e03a6] | 458 | } |
---|
[449fbf] | 459 | |
---|
[d317a9] | 460 | |
---|
[449fbf] | 461 | proc round(number r) |
---|
[53e03a6] | 462 | "USAGE: round(r); |
---|
| 463 | RETURN: the nearest number to r out of Z |
---|
| 464 | ASSUME: r should be a rational or a real number |
---|
| 465 | EXAMPLE:example round; shows an example |
---|
| 466 | " |
---|
| 467 | { |
---|
| 468 | number a=absValue(r); |
---|
| 469 | number v=r/a; |
---|
| 470 | |
---|
| 471 | number d=10; |
---|
| 472 | int e; |
---|
| 473 | while(1) |
---|
| 474 | { |
---|
| 475 | e=e+1; |
---|
| 476 | if(a-d^e<0) |
---|
[449fbf] | 477 | { |
---|
[53e03a6] | 478 | e=e-1; |
---|
| 479 | break; |
---|
[449fbf] | 480 | } |
---|
[53e03a6] | 481 | } |
---|
[449fbf] | 482 | |
---|
[53e03a6] | 483 | number b=a; |
---|
| 484 | int k; |
---|
| 485 | for(k=0;k<=e;k++) |
---|
| 486 | { |
---|
| 487 | while(1) |
---|
| 488 | { |
---|
| 489 | b=b-d^(e-k); |
---|
| 490 | if(b<0) |
---|
| 491 | { |
---|
| 492 | b=b+d^(e-k); |
---|
| 493 | break; |
---|
| 494 | } |
---|
| 495 | } |
---|
| 496 | } |
---|
| 497 | |
---|
| 498 | if(b<1/2) |
---|
| 499 | { |
---|
| 500 | return(v*(a-b)); |
---|
| 501 | } |
---|
| 502 | else |
---|
| 503 | { |
---|
| 504 | return(v*(a+1-b)); |
---|
| 505 | } |
---|
| 506 | } |
---|
[449fbf] | 507 | example |
---|
[53e03a6] | 508 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 509 | ring R = (real,50),x,dp; |
---|
| 510 | number r=7357683445788723456321.6788643224; |
---|
| 511 | round(r); |
---|
[53e03a6] | 512 | } |
---|
[449fbf] | 513 | |
---|
| 514 | |
---|
[3eadab] | 515 | proc HilbertClassPoly(number D, int k) |
---|
| 516 | "USAGE: HilbertClassPoly(D,k); |
---|
| 517 | RETURN: the monic polynomial of degree h(D) in Z[X] of which jOft((D+sqr(D))/2) is a root |
---|
[53e03a6] | 518 | ASSUME: D is a negative discriminant |
---|
[1a3911] | 519 | NOTE: k is input for the procedure "jOft", |
---|
[53e03a6] | 520 | 5*k is input for the procedure "sqr", |
---|
| 521 | 10*k describes the number of decimals being calculated in the complex numbers |
---|
[1a3911] | 522 | EXAMPLE:example HilbertClassPoly; shows an example |
---|
[53e03a6] | 523 | " |
---|
| 524 | { |
---|
[d317a9] | 525 | if(D>=0) // (0)[Test if assumptions well-defined] |
---|
[26508d] | 526 | { |
---|
| 527 | ERROR("Parameter wrong selected!"); |
---|
| 528 | } |
---|
| 529 | else |
---|
| 530 | { |
---|
| 531 | def S=basering; |
---|
| 532 | ring R=0,x,dp; |
---|
[449fbf] | 533 | |
---|
[26508d] | 534 | string s1,s2,s3; |
---|
| 535 | number a1,b1,t1,g1; |
---|
| 536 | number D=imap(S,D); |
---|
| 537 | number B=intRoot(absValue(D)/3); |
---|
| 538 | |
---|
| 539 | ring C=(complex,10*k,i),x,dp; |
---|
| 540 | number D=imap(S,D); |
---|
| 541 | |
---|
[d317a9] | 542 | poly P=1; // (1)[Initialize] |
---|
[26508d] | 543 | number b=cmod(D,2); |
---|
| 544 | number B=imap(R,B); |
---|
| 545 | |
---|
| 546 | number t,a,g,tau,j; |
---|
| 547 | list L; |
---|
| 548 | |
---|
| 549 | int step=2; |
---|
| 550 | while(1) |
---|
| 551 | { |
---|
| 552 | if(step==2) // (2)[Initialize a] |
---|
| 553 | { |
---|
| 554 | t=(b^2-D)/4; |
---|
| 555 | L=b,1; |
---|
| 556 | a=maximum(L); |
---|
| 557 | step=3; |
---|
| 558 | } |
---|
| 559 | |
---|
[d317a9] | 560 | if(step==3) // (3)[Test] |
---|
[26508d] | 561 | { |
---|
| 562 | if((cmod(t,a)!=0)) |
---|
| 563 | { |
---|
| 564 | step=4; |
---|
| 565 | } |
---|
| 566 | else |
---|
| 567 | { |
---|
| 568 | s1=string(a); |
---|
| 569 | s2=string(b); |
---|
| 570 | s3=string(t); |
---|
| 571 | |
---|
| 572 | setring R; |
---|
| 573 | execute("a1="+s1+";"); |
---|
| 574 | execute("b1="+s2+";"); |
---|
| 575 | execute("t1="+s3+";"); |
---|
| 576 | g1=gcd(gcd(a1,b1),t1/a1); |
---|
| 577 | setring C; |
---|
| 578 | g=imap(R,g1); |
---|
| 579 | |
---|
| 580 | if(g!=1) |
---|
| 581 | { |
---|
| 582 | step=4; |
---|
| 583 | } |
---|
| 584 | else |
---|
| 585 | { |
---|
| 586 | tau=(-b+i*sqr(absValue(D),5*k))/(2*a); |
---|
| 587 | j=jOft(tau,k); |
---|
| 588 | if((a==b)||(a^2==t)||(b==0)) |
---|
| 589 | { |
---|
| 590 | P=P*(var(1)-repart(j)); |
---|
| 591 | step=4; |
---|
| 592 | } |
---|
| 593 | else |
---|
| 594 | { |
---|
| 595 | P=P*(var(1)^2-2*repart(j)*var(1)+repart(j)^2+impart(j)^2); |
---|
| 596 | step=4; |
---|
| 597 | } |
---|
| 598 | } |
---|
| 599 | } |
---|
| 600 | } |
---|
| 601 | |
---|
[d317a9] | 602 | if(step==4) // (4)[Loop on a] |
---|
[26508d] | 603 | { |
---|
| 604 | a=a+1; |
---|
| 605 | if(a^2<=t) |
---|
| 606 | { |
---|
| 607 | step=3; |
---|
| 608 | continue; |
---|
| 609 | } |
---|
| 610 | else |
---|
| 611 | { |
---|
| 612 | step=5; |
---|
| 613 | } |
---|
| 614 | } |
---|
| 615 | |
---|
[d317a9] | 616 | if(step==5) // (5)[Loop on b] |
---|
[26508d] | 617 | { |
---|
| 618 | b=b+2; |
---|
| 619 | if(b<=B) |
---|
| 620 | { |
---|
| 621 | step=2; |
---|
| 622 | } |
---|
| 623 | else |
---|
| 624 | { |
---|
| 625 | break; |
---|
| 626 | } |
---|
| 627 | } |
---|
| 628 | } |
---|
| 629 | |
---|
| 630 | matrix M=coeffs(P,var(1)); |
---|
| 631 | |
---|
| 632 | list liste; |
---|
| 633 | int n; |
---|
| 634 | for(n=1;n<=nrows(M);n++) |
---|
| 635 | { |
---|
| 636 | liste[n]=round(repart(number(M[n,1]))); |
---|
| 637 | } |
---|
| 638 | |
---|
| 639 | poly Q; |
---|
| 640 | int m; |
---|
| 641 | for(m=1;m<=size(liste);m++) |
---|
| 642 | { |
---|
| 643 | Q=Q+liste[m]*var(1)^(m-1); |
---|
| 644 | } |
---|
| 645 | |
---|
| 646 | string s=string(Q); |
---|
| 647 | setring S; |
---|
| 648 | execute("poly Q="+s+";"); |
---|
| 649 | return(Q); |
---|
| 650 | } |
---|
[53e03a6] | 651 | } |
---|
[449fbf] | 652 | example |
---|
[53e03a6] | 653 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 654 | ring r = 0,x,dp; |
---|
| 655 | number D=-23; |
---|
[3eadab] | 656 | HilbertClassPoly(D,50); |
---|
[53e03a6] | 657 | } |
---|
[449fbf] | 658 | |
---|
[d317a9] | 659 | |
---|
[3eadab] | 660 | proc rootsModp(int p, poly P) |
---|
| 661 | "USAGE: rootsModp(p,P); |
---|
[53e03a6] | 662 | RETURN: list of roots of the polynomial P modulo p with p prime |
---|
| 663 | ASSUME: p>=3 |
---|
| 664 | NOTE: this algorithm will be called recursively, and it is understood |
---|
[0dd77c2] | 665 | that all the operations are done in Z/pZ (excepting squareRoot(d,p)) |
---|
[3eadab] | 666 | EXAMPLE:example rootsModp; shows an example |
---|
[53e03a6] | 667 | " |
---|
| 668 | { |
---|
[d317a9] | 669 | if(p<3) // (0)[Test if assumptions well-defined] |
---|
[26508d] | 670 | { |
---|
| 671 | ERROR("Parameter wrong selected, since p<3!"); |
---|
| 672 | } |
---|
| 673 | else |
---|
| 674 | { |
---|
| 675 | def S=basering; |
---|
| 676 | ring R=p,var(1),dp; |
---|
[449fbf] | 677 | |
---|
[26508d] | 678 | poly P=imap(S,P); |
---|
| 679 | number d; |
---|
| 680 | int a; |
---|
| 681 | list L; |
---|
| 682 | |
---|
[d317a9] | 683 | poly A=gcd(var(1)^p-var(1),P); // (1)[Isolate roots in Z/pZ] |
---|
[26508d] | 684 | if(subst(A,var(1),0)==0) |
---|
| 685 | { |
---|
| 686 | L[1]=0; |
---|
| 687 | A=A/var(1); |
---|
| 688 | } |
---|
| 689 | |
---|
[d317a9] | 690 | if(deg(A)==0) // (2)[Small degree?] |
---|
[26508d] | 691 | { |
---|
| 692 | return(L); |
---|
| 693 | } |
---|
| 694 | |
---|
| 695 | if(deg(A)==1) |
---|
| 696 | { |
---|
| 697 | matrix M=coeffs(A,var(1)); |
---|
| 698 | L[size(L)+1]=-leadcoef(M[1,1])/leadcoef(M[2,1]); |
---|
| 699 | setring S; |
---|
| 700 | list L=imap(R,L); |
---|
| 701 | return(L); |
---|
| 702 | } |
---|
| 703 | |
---|
| 704 | if(deg(A)==2) |
---|
| 705 | { |
---|
| 706 | matrix M=coeffs(A,var(1)); |
---|
| 707 | d=leadcoef(M[2,1])^2-4*leadcoef(M[1,1])*leadcoef(M[3,1]); |
---|
| 708 | |
---|
| 709 | ring T=0,var(1),dp; |
---|
| 710 | number d=imap(R,d); |
---|
| 711 | number e=squareRoot(d,p); |
---|
| 712 | setring R; |
---|
| 713 | number e=imap(T,e); |
---|
| 714 | |
---|
| 715 | L[size(L)+1]=(-leadcoef(M[2,1])+e)/(2*leadcoef(M[3,1])); |
---|
| 716 | L[size(L)+1]=(-leadcoef(M[2,1])-e)/(2*leadcoef(M[3,1])); |
---|
| 717 | setring S; |
---|
| 718 | list L=imap(R,L); |
---|
| 719 | return(L); |
---|
| 720 | } |
---|
| 721 | |
---|
[d317a9] | 722 | poly B=1; // (3)[Random splitting] |
---|
[26508d] | 723 | poly C; |
---|
| 724 | while((deg(B)==0)||(deg(B)==deg(A))) |
---|
| 725 | { |
---|
| 726 | a=random(0,p-1); |
---|
| 727 | B=gcd((var(1)+a)^((p-1)/2)-1,A); |
---|
| 728 | C=A/B; |
---|
| 729 | } |
---|
| 730 | |
---|
[d317a9] | 731 | setring S; // (4)[Recurse] |
---|
[26508d] | 732 | poly B=imap(R,B); |
---|
| 733 | poly C=imap(R,C); |
---|
| 734 | list l=L+rootsModp(p,B)+rootsModp(p,C); |
---|
| 735 | return(l); |
---|
| 736 | } |
---|
[53e03a6] | 737 | } |
---|
[449fbf] | 738 | example |
---|
[53e03a6] | 739 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 740 | ring r = 0,x,dp; |
---|
| 741 | poly f=x4+2x3-5x2+x; |
---|
[3eadab] | 742 | rootsModp(7,f); |
---|
[449fbf] | 743 | poly g=x5+112x4+655x3+551x2+1129x+831; |
---|
[3eadab] | 744 | rootsModp(1223,g); |
---|
[53e03a6] | 745 | } |
---|
[449fbf] | 746 | |
---|
[d317a9] | 747 | |
---|
[3eadab] | 748 | proc wUnit(number D) |
---|
| 749 | "USAGE: wUnit(D); |
---|
[53e03a6] | 750 | RETURN: the number of roots of unity in the quadratic order of discriminant D |
---|
| 751 | ASSUME: D<0 a discriminant kongruent to 0 or 1 modulo 4 |
---|
| 752 | EXAMPLE:example w; shows an example |
---|
| 753 | " |
---|
| 754 | { |
---|
| 755 | if((D>=0)||((D mod 4)==2)||((D mod 4)==3)) |
---|
| 756 | { |
---|
| 757 | ERROR("Parameter wrong selected!"); |
---|
| 758 | } |
---|
| 759 | else |
---|
| 760 | { |
---|
| 761 | if(D<-4) {return(2);} |
---|
| 762 | if(D==-4){return(4);} |
---|
| 763 | if(D==-3){return(6);} |
---|
| 764 | } |
---|
| 765 | } |
---|
[449fbf] | 766 | example |
---|
[53e03a6] | 767 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 768 | ring r = 0,x,dp; |
---|
| 769 | number D=-3; |
---|
[3eadab] | 770 | wUnit(D); |
---|
[53e03a6] | 771 | } |
---|
[449fbf] | 772 | |
---|
| 773 | |
---|
| 774 | proc Atkin(number N, int K, int B) |
---|
[53e03a6] | 775 | "USAGE: Atkin(N,K,B); |
---|
| 776 | RETURN: 1, if N is prime, |
---|
| 777 | -1, if N is not prime, |
---|
[d317a9] | 778 | 0, if the algorithm is not applicable, since there are too few discriminants |
---|
[53e03a6] | 779 | ASSUME: N is coprime to 6 and different from 1 |
---|
[1a3911] | 780 | NOTE: K/2 is input for the procedure "disc",@* |
---|
[d317a9] | 781 | K is input for the procedure "HilbertClassPoly",@* |
---|
[1a3911] | 782 | B describes the number of recursions being calculated.@* |
---|
| 783 | The basis of the algorithm is the following theorem: |
---|
[d1b0065] | 784 | Let N be an integer coprime to 6 and different from 1 and E be an |
---|
[d317a9] | 785 | ellipic curve modulo N.@* Assume that we know an integer m and a |
---|
[1a3911] | 786 | point P of E(Z/NZ) satisfying the following conditions.@* |
---|
[d317a9] | 787 | (1) There exists a prime divisor q of m such that q > (4-th root(N)+1)^2.@* |
---|
| 788 | (2) m*P = O(E) = (0:1:0).@* |
---|
| 789 | (3) (m/q)*P = (x:y:t) with t element of (Z/NZ)*.@* |
---|
[53e03a6] | 790 | Then N is prime. |
---|
| 791 | EXAMPLE:example Atkin; shows an example |
---|
| 792 | " |
---|
| 793 | { |
---|
[d1b0065] | 794 | if(N==1) {return(-1);} // (0)[Test if assumptions well-defined] |
---|
| 795 | if((N==2)||(N==3)) {return(1);} |
---|
| 796 | if(gcdN(N,6)!=1) |
---|
| 797 | { |
---|
[d317a9] | 798 | if(printlevel>=1) {"gcd(N,6) = "+string(gcdN(N,6));pause();"";} |
---|
[d1b0065] | 799 | return(-1); |
---|
| 800 | } |
---|
| 801 | else |
---|
| 802 | { |
---|
| 803 | int i; // (1)[Initialize] |
---|
| 804 | int n(i); |
---|
| 805 | number N(i)=N; |
---|
[d317a9] | 806 | if(printlevel>=1) {"Set i = 0, n = 0 and N(i) = N(0)= "+string(N(i))+".";pause();"";} |
---|
[d1b0065] | 807 | |
---|
| 808 | // declarations: |
---|
| 809 | int j(0),j(1),j(2),j(3),j(4),k; // running indices |
---|
| 810 | list L; // all primes smaller than 1000 |
---|
| 811 | list H; // sequence of negative discriminants |
---|
| 812 | number D; // discriminant out of H |
---|
| 813 | list L1,L2,S,S1,S2,R; // lists of relevant elements |
---|
| 814 | list P,P1,P2; // elliptic points on E(Z/N(i)Z) |
---|
| 815 | number m,q; // m=|E(Z/N(i)Z)| and q|m |
---|
| 816 | number a,b,j,c; // characterize E(Z/N(i)Z) |
---|
| 817 | number g,u; // g out of Z/N(i)Z, u=Jacobi(g,N(i)) |
---|
[3eadab] | 818 | poly T; // T=HilbertClassPoly(D,K) |
---|
[d1b0065] | 819 | matrix M; // M contains the coefficients of T |
---|
| 820 | |
---|
[d317a9] | 821 | if(printlevel>=1) {"List H of possibly suitable discriminants will be calculated.";} |
---|
[d1b0065] | 822 | H=disc(N,K/2); |
---|
[d317a9] | 823 | if(printlevel>=1) {"H = "+string(H);pause();"";} |
---|
[d1b0065] | 824 | |
---|
| 825 | int step=2; |
---|
| 826 | while(1) |
---|
| 827 | { |
---|
| 828 | if(step==2) // (2)[Is N(i) small??] |
---|
[370344] | 829 | { |
---|
[d1b0065] | 830 | 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] | 831 | for(j(0)=1;j(0)<=size(L);j(0)++) |
---|
[d1b0065] | 832 | { |
---|
[3d4a5a] | 833 | if(N(i)==L[j(0)]){return(1);} |
---|
[d1b0065] | 834 | if(((N(i) mod L[j(0)])==0)&&(N(i)!=L[j(0)])) |
---|
| 835 | { |
---|
[d317a9] | 836 | if(printlevel>=1) {"N("+string(i)+") = "+string(N(i))+" is divisible by "+string(L[j(0)])+".";pause();"";} |
---|
[d1b0065] | 837 | step=14; |
---|
| 838 | break; |
---|
| 839 | } |
---|
| 840 | } |
---|
| 841 | if(step==2) |
---|
[370344] | 842 | { |
---|
[d1b0065] | 843 | step=3; |
---|
[370344] | 844 | } |
---|
| 845 | } |
---|
[449fbf] | 846 | |
---|
[d1b0065] | 847 | if(step==3) // (3)[Choose next discriminant] |
---|
| 848 | { |
---|
| 849 | n(i)=n(i)+1; |
---|
| 850 | if(n(i)==size(H)+1) |
---|
| 851 | { |
---|
[d317a9] | 852 | if(printlevel>=1) {"Algorithm is not applicable, since there are not enough suitable discriminants."; |
---|
| 853 | "Increase the parameter of accuracy K and start the algorithm again.";pause();"";} |
---|
[d1b0065] | 854 | return(0); |
---|
| 855 | } |
---|
| 856 | D=H[n(i)]; |
---|
[d317a9] | 857 | if(printlevel>=1) {"Next discriminant D will be chosen. D = "+string(D)+".";pause();"";} |
---|
[d1b0065] | 858 | if(Jacobi(D,N(i))!=1) |
---|
| 859 | { |
---|
[d317a9] | 860 | if(printlevel>=1) {"Jacobi(D,N("+string(i)+")) = "+string(Jacobi(D,N(i)));pause();"";} |
---|
[d1b0065] | 861 | continue; |
---|
| 862 | } |
---|
| 863 | else |
---|
| 864 | { |
---|
| 865 | L1=CornacchiaModified(D,N(i)); |
---|
| 866 | if(size(L1)>1) |
---|
| 867 | { |
---|
[d317a9] | 868 | if(printlevel>=1) {"The solution (x,y) of the equation x^2+|D|y^2 = 4N("+string(i)+") is";L1;pause();"";} |
---|
[d1b0065] | 869 | step=4; |
---|
| 870 | } |
---|
| 871 | else |
---|
| 872 | { |
---|
| 873 | if(L1[1]==-1) |
---|
[370344] | 874 | { |
---|
[d317a9] | 875 | if(printlevel>=1) {"The equation x^2+|D|y^2 = 4N("+string(i)+") has no solution.";pause();"";} |
---|
[d1b0065] | 876 | continue; |
---|
| 877 | } |
---|
| 878 | if(L1[1]==0) |
---|
| 879 | { |
---|
[d317a9] | 880 | if(printLevel>=1) {"Algorithm is not applicable for N("+string(i)+") = "+string(N(i))+","; |
---|
| 881 | "since there are not enough suitable discriminants.";pause();"";} |
---|
[d1b0065] | 882 | step=14; |
---|
| 883 | } |
---|
| 884 | } |
---|
| 885 | } |
---|
| 886 | } |
---|
[449fbf] | 887 | |
---|
[d1b0065] | 888 | if(step==4) // (4)[Factor m] |
---|
| 889 | { |
---|
[d317a9] | 890 | if(printlevel>=1) {"List L2 of possible m = |E(Z/N("+string(i)+")Z)| will be calculated.";} |
---|
[d1b0065] | 891 | if(absValue(L1[1])^2<=4*N(i)) {L2=N(i)+1+L1[1],N(i)+1-L1[1];} |
---|
| 892 | if(D==-4) |
---|
| 893 | { |
---|
| 894 | if(absValue(2*L1[2])^2<=4*N(i)) {L2[size(L2)+1]=N(i)+1+2*L1[2]; |
---|
| 895 | L2[size(L2)+1]=N(i)+1-2*L1[2];} |
---|
| 896 | } |
---|
[d317a9] | 897 | // At this point "<=4*N(i)" has been replaced by "<=16*N(i)". |
---|
[d1b0065] | 898 | if(D==-3) |
---|
| 899 | { |
---|
| 900 | if(absValue(L1[1]+3*L1[2])^2<=16*N(i)) {L2[size(L2)+1]=N(i)+1+(L1[1]+3*L1[2])/2; |
---|
| 901 | L2[size(L2)+1]=N(i)+1-(L1[1]+3*L1[2])/2;} |
---|
| 902 | if(absValue(L1[1]-3*L1[2])^2<=16*N(i)) {L2[size(L2)+1]=N(i)+1+(L1[1]-3*L1[2])/2; |
---|
| 903 | L2[size(L2)+1]=N(i)+1-(L1[1]-3*L1[2])/2;} |
---|
| 904 | } |
---|
| 905 | /////////////////////////////////////////////////////////////// |
---|
| 906 | if(size(L2)==0) |
---|
| 907 | { |
---|
[d317a9] | 908 | if(printlevel>=1) {"Due to the theorem of Hasse there were no possible m = |E(Z/N("+string(i)+")Z)|"; |
---|
| 909 | "found for D = "+string(D)+".";} |
---|
[d1b0065] | 910 | step=3; |
---|
| 911 | continue; |
---|
| 912 | } |
---|
| 913 | else |
---|
| 914 | { |
---|
[d317a9] | 915 | if(printlevel>=1) {"L2 = ";L2;pause();"";} |
---|
[d1b0065] | 916 | } |
---|
[449fbf] | 917 | |
---|
[d317a9] | 918 | if(printlevel>=1) {"List S of factors of all possible m will be calculated.";} |
---|
[d1b0065] | 919 | S=list(); |
---|
| 920 | for(j(1)=1;j(1)<=size(L2);j(1)++) |
---|
| 921 | { |
---|
| 922 | m=L2[j(1)]; |
---|
| 923 | if(m!=0) |
---|
| 924 | { |
---|
| 925 | S1=PollardRho(m,10000,1,L); |
---|
| 926 | S2=pFactor(m,100,L); |
---|
| 927 | S[size(S)+1]=list(m,S1+S2); |
---|
| 928 | } |
---|
| 929 | } |
---|
[d317a9] | 930 | if(printlevel>=1) {"S=";S;pause();"";} |
---|
[d1b0065] | 931 | step=5; |
---|
| 932 | } |
---|
[449fbf] | 933 | |
---|
[d1b0065] | 934 | if(step==5) // (5)[Does a suitable m exist??] |
---|
| 935 | { |
---|
| 936 | for(j(2)=1;j(2)<=size(S);j(2)++) |
---|
| 937 | { |
---|
| 938 | m=L2[j(2)]; |
---|
| 939 | for(j(3)=1;j(3)<=size(S[j(2)][2]);j(3)++) |
---|
| 940 | { |
---|
| 941 | q=S[j(2)][2][j(3)]; |
---|
[d317a9] | 942 | // sqr(sqr(N(i),50),50) replaces intRoot(intRoot(N(i))) |
---|
[d1b0065] | 943 | if((q>(sqr(sqr(N(i),50),50)+1)^2) && (MillerRabin(q,5)==1)) |
---|
| 944 | { |
---|
| 945 | step=6; |
---|
| 946 | break; |
---|
| 947 | } |
---|
| 948 | ////////////////////////////////////////////////////// |
---|
| 949 | } |
---|
| 950 | if(step==6) |
---|
| 951 | { |
---|
[d317a9] | 952 | if(printlevel>=1) {"Suitable pair (m,q) has been found such that q|m,"; |
---|
| 953 | "q > (4-th root(N("+string(i)+"))+1)^2 and q passes the Miller-Rabin-Test."; |
---|
| 954 | "m = "+string(m)+",";"q = "+string(q);pause();"";} |
---|
[d1b0065] | 955 | break; |
---|
| 956 | } |
---|
| 957 | else |
---|
| 958 | { |
---|
| 959 | step=3; |
---|
| 960 | } |
---|
| 961 | } |
---|
| 962 | if(step==3) |
---|
| 963 | { |
---|
[d317a9] | 964 | if(printlevel>=1) {"No suitable pair (m,q) has been found such that q|m,"; |
---|
| 965 | "q > (4-th root(N("+string(i)+"))+1)^2 and q passes the Miller-Rabin-Test."; |
---|
| 966 | pause();"";} |
---|
[d1b0065] | 967 | continue; |
---|
| 968 | } |
---|
| 969 | } |
---|
[449fbf] | 970 | |
---|
[d1b0065] | 971 | if(step==6) // (6)[Compute elliptic curve] |
---|
| 972 | { |
---|
| 973 | if(D==-4) |
---|
| 974 | { |
---|
| 975 | a=-1; |
---|
| 976 | b=0; |
---|
[d317a9] | 977 | if(printlevel>=1) {"Since D = -4, set a = -1 and b = 0.";pause();"";} |
---|
[d1b0065] | 978 | } |
---|
| 979 | if(D==-3) |
---|
| 980 | { |
---|
| 981 | a=0; |
---|
| 982 | b=-1; |
---|
[d317a9] | 983 | if(printlevel>=1) {"Since D = -3, set a = 0 and b = -1.";pause();"";} |
---|
[d1b0065] | 984 | } |
---|
| 985 | if(D<-4) |
---|
| 986 | { |
---|
[d317a9] | 987 | if(printlevel>=1) {"The minimal polynomial T of j((D+sqr(D))/2) in Z[X] will be calculated for D="+string(D)+".";} |
---|
[3eadab] | 988 | T=HilbertClassPoly(D,K); |
---|
[d317a9] | 989 | if(printlevel>=1) {"T = "+string(T);pause();"";} |
---|
[d1b0065] | 990 | |
---|
| 991 | M=coeffs(T,var(1)); |
---|
| 992 | T=0; |
---|
| 993 | |
---|
| 994 | for(j(4)=1;j(4)<=nrows(M);j(4)++) |
---|
| 995 | { |
---|
| 996 | M[j(4),1]=leadcoef(M[j(4),1]) mod N(i); |
---|
| 997 | T=T+M[j(4),1]*var(1)^(j(4)-1); |
---|
| 998 | } |
---|
[d317a9] | 999 | if(printlevel>=1) {"Set T = T mod N("+string(i)+").";"T = "+string(T);pause();"";} |
---|
[d1b0065] | 1000 | |
---|
[3eadab] | 1001 | R=rootsModp(int(N(i)),T); |
---|
[d1b0065] | 1002 | if(deg(T)>size(R)) |
---|
| 1003 | { |
---|
[d317a9] | 1004 | ERROR("The polynomial T does not completely split into linear factors modulo N("+string(i)+")." |
---|
| 1005 | "Increase the parameter of accuracy K and start the algorithm again."); |
---|
[2472afe] | 1006 | } |
---|
[d317a9] | 1007 | if(printlevel>=1) {if(deg(T)>1) {"The "+string(deg(T))+" zeroes of T modulo N("+string(i)+") are"; |
---|
| 1008 | R;pause();"";} |
---|
| 1009 | if(deg(T)==1){"The zero of T modulo N("+string(i)+") is";R;pause();"";}} |
---|
[d1b0065] | 1010 | |
---|
| 1011 | j=R[1]; |
---|
| 1012 | c=j*exgcdN(j-1728,N(i))[1]; |
---|
| 1013 | a=-3*c mod N(i); |
---|
| 1014 | b=2*c mod N(i); |
---|
[d317a9] | 1015 | if(printlevel>=1) {"Choose the zero j = "+string(j)+" and set"; "c = j/(j-1728) mod N("+string(i)+"), a = -3c mod N("+string(i)+"), b = 2c mod N("+string(i)+")."; |
---|
| 1016 | "a = "+string(a)+",";"b = "+string(b);pause();"";} |
---|
[d1b0065] | 1017 | } |
---|
| 1018 | step=7; |
---|
| 1019 | } |
---|
[449fbf] | 1020 | |
---|
[d1b0065] | 1021 | if(step==7) // (7)[Find g] |
---|
| 1022 | { |
---|
| 1023 | if(D==-3) |
---|
| 1024 | { |
---|
| 1025 | while(1) |
---|
| 1026 | { |
---|
| 1027 | g=random(1,2147483647) mod N(i); |
---|
| 1028 | u=Jacobi(g,N(i)); |
---|
| 1029 | if((u==-1)&&(powerN(g,(N(i)-1)/3,N(i))!=1)) |
---|
| 1030 | { |
---|
[d317a9] | 1031 | if(printlevel>=1) {"g = "+string(g);pause();"";} |
---|
[d1b0065] | 1032 | break; |
---|
| 1033 | } |
---|
| 1034 | } |
---|
| 1035 | } |
---|
| 1036 | else |
---|
| 1037 | { |
---|
| 1038 | while(1) |
---|
| 1039 | { |
---|
| 1040 | g=random(1,2147483647) mod N(i); |
---|
| 1041 | u=Jacobi(g,N(i)); |
---|
| 1042 | if(u==-1) |
---|
| 1043 | { |
---|
[d317a9] | 1044 | if(printlevel>=1) {"g = "+string(g);pause();"";} |
---|
[d1b0065] | 1045 | break; |
---|
| 1046 | } |
---|
| 1047 | } |
---|
| 1048 | } |
---|
| 1049 | step=8; |
---|
| 1050 | } |
---|
[449fbf] | 1051 | |
---|
[d1b0065] | 1052 | if(step==8) // (8)[Find P] |
---|
| 1053 | { |
---|
[d317a9] | 1054 | if(printlevel>=1) {"A random point P on the elliptic curve corresponding"; |
---|
| 1055 | "to the equation y^2 = x^3+ax+b for";"N("+string(i)+") = "+string(N(i))+","; |
---|
| 1056 | " a = "+string(a)+",";" b = "+string(b);"will be chosen.";} |
---|
[d1b0065] | 1057 | P=ellipticRandomPoint(N(i),a,b); |
---|
[d317a9] | 1058 | if(printlevel>=1) {"P = ("+string(P)+")";pause();"";} |
---|
[449fbf] | 1059 | |
---|
[d1b0065] | 1060 | if(size(P)==1) |
---|
| 1061 | { |
---|
| 1062 | step=14; |
---|
| 1063 | } |
---|
| 1064 | else |
---|
| 1065 | { |
---|
| 1066 | step=9; |
---|
| 1067 | } |
---|
| 1068 | } |
---|
[449fbf] | 1069 | |
---|
[d1b0065] | 1070 | if(step==9) // (9)[Find right curve] |
---|
| 1071 | { |
---|
[d317a9] | 1072 | if(printlevel>=1) {"The points P2 = (m/q)*P and P1 = q*P2 on the curve will be calculated.";} |
---|
[d1b0065] | 1073 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
| 1074 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
[d317a9] | 1075 | if(printlevel>=1) {"P1 = ("+string(P1)+"),";"P2 = ("+string(P2)+")";pause();"";} |
---|
[449fbf] | 1076 | |
---|
[d1b0065] | 1077 | if((P1[1]==0)&&(P1[2]==1)&&(P1[3]==0)) |
---|
| 1078 | { |
---|
| 1079 | step=12; |
---|
| 1080 | } |
---|
| 1081 | else |
---|
| 1082 | { |
---|
[d317a9] | 1083 | if(printlevel>=1) {"Since P1 != (0:1:0), it holds m != |E(Z/N("+string(i)+")Z)| for the coefficients a = "+string(a)+" and b = "+string(b)+"."; |
---|
| 1084 | "Therefore choose new coefficients a and b.";pause();"";} |
---|
[d1b0065] | 1085 | step=10; |
---|
| 1086 | } |
---|
| 1087 | } |
---|
[449fbf] | 1088 | |
---|
[2472afe] | 1089 | if(step==10) // (10)[Change coefficients] |
---|
[d1b0065] | 1090 | { |
---|
| 1091 | k=k+1; |
---|
[3eadab] | 1092 | if(k>=wUnit(D)) |
---|
[d1b0065] | 1093 | { |
---|
[d317a9] | 1094 | if(printlevel>=1) {"Since k = wUnit(D) = "+string(k)+", it holds that N("+string(i)+") = "+string(N(i))+" is not prime.";pause();"";} |
---|
[d1b0065] | 1095 | step=14; |
---|
| 1096 | } |
---|
| 1097 | else |
---|
| 1098 | { |
---|
| 1099 | if(D<-4) {a=a*g^2 mod N(i); b=b*g^3 mod N(i); |
---|
[d317a9] | 1100 | if(printlevel>=1) {"Since D < -4, set a = a*g^2 mod N("+string(i)+") and b = b*g^3 mod N("+string(i)+")."; |
---|
| 1101 | "a = "+string(a)+",";"b = "+string(b)+",";"k = "+string(k);pause();"";}} |
---|
[d1b0065] | 1102 | if(D==-4){a=a*g mod N(i); |
---|
[d317a9] | 1103 | if(printlevel>=1) {"Since D = -4, set a = a*g mod N("+string(i)+").";"a = "+string(a)+","; |
---|
| 1104 | "b = "+string(b)+",";"k = "+string(k);pause();"";}} |
---|
[d1b0065] | 1105 | if(D==-3){b=b*g mod N(i); |
---|
[d317a9] | 1106 | if(printlevel>=1) {"Since D = -3, set b = b*g mod N("+string(i)+").";"a = "+string(a)+","; |
---|
| 1107 | "b = "+string(b)+",";"k = "+string(k);pause();"";}} |
---|
[d1b0065] | 1108 | step=8; |
---|
| 1109 | continue; |
---|
| 1110 | } |
---|
| 1111 | } |
---|
[449fbf] | 1112 | |
---|
[d1b0065] | 1113 | if(step==11) // (11)[Find a new P] |
---|
| 1114 | { |
---|
[d317a9] | 1115 | if(printlevel>=1) {"A new random point P on the elliptic curve will be chosen,"; |
---|
| 1116 | "since also P2 = (0:1:0).";} |
---|
[d1b0065] | 1117 | P=ellipticRandomPoint(N(i),a,b); |
---|
[d317a9] | 1118 | if(printlevel>=1) {"P = ("+string(P)+")";pause();"";} |
---|
[449fbf] | 1119 | |
---|
[d1b0065] | 1120 | if(size(P)==1) |
---|
| 1121 | { |
---|
| 1122 | step=14; |
---|
| 1123 | } |
---|
| 1124 | else |
---|
| 1125 | { |
---|
[d317a9] | 1126 | if(printlevel>=1) {"The points P2 = (m/q)*P and P1 = q*P2 on the curve will be calculated.";} |
---|
[d1b0065] | 1127 | P2=ellipticMult(N(i),a,b,P,m/q); |
---|
| 1128 | P1=ellipticMult(N(i),a,b,P2,q); |
---|
[d317a9] | 1129 | if(printlevel>=1) {"P1 = ("+string(P1)+"),";"P2 = ("+string(P2)+")";pause();"";} |
---|
[d1b0065] | 1130 | |
---|
| 1131 | if((P1[1]!=0)||(P1[2]!=1)||(P1[3]!=0)) |
---|
| 1132 | { |
---|
[d317a9] | 1133 | if(printlevel>=1) {"Since P1 != (0:1:0), it holds m != |E(Z/N("+string(i)+")Z)| for the coefficients a = "+string(a)+" and b = "+string(b)+"."; |
---|
| 1134 | "Therefore choose new coefficients a and b.";pause();"";} |
---|
[d1b0065] | 1135 | step=10; |
---|
| 1136 | continue; |
---|
| 1137 | } |
---|
| 1138 | else |
---|
| 1139 | { |
---|
| 1140 | step=12; |
---|
| 1141 | } |
---|
| 1142 | } |
---|
| 1143 | } |
---|
[449fbf] | 1144 | |
---|
[d1b0065] | 1145 | if(step==12) // (12)[Check P] |
---|
| 1146 | { |
---|
| 1147 | if((P2[1]==0)&&(P2[2]==1)&&(P2[3]==0)) |
---|
| 1148 | { |
---|
| 1149 | step=11; |
---|
| 1150 | continue; |
---|
| 1151 | } |
---|
| 1152 | else |
---|
| 1153 | { |
---|
| 1154 | step=13; |
---|
| 1155 | } |
---|
| 1156 | } |
---|
[449fbf] | 1157 | |
---|
[d1b0065] | 1158 | if(step==13) // (13)[Recurse] |
---|
| 1159 | { |
---|
| 1160 | if(i<B) |
---|
| 1161 | { |
---|
[d317a9] | 1162 | if(printlevel>=1) {string(i+1)+". Recursion:";""; |
---|
| 1163 | "N("+string(i)+") = "+string(N(i))+" suffices the conditions of the underlying theorem,"; |
---|
| 1164 | "since P1 = (0:1:0) and P2[3] in (Z/N("+string(i)+")Z)*.";""; |
---|
| 1165 | "Now check if also the found factor q="+string(q)+" suffices these assumptions."; |
---|
| 1166 | "Therefore set i = i+1, N("+string(i+1)+") = q = "+string(q)+" and restart the algorithm.";pause();"";} |
---|
[d1b0065] | 1167 | i=i+1; |
---|
| 1168 | int n(i); |
---|
| 1169 | number N(i)=q; |
---|
| 1170 | k=0; |
---|
| 1171 | step=2; |
---|
| 1172 | continue; |
---|
| 1173 | } |
---|
| 1174 | else |
---|
| 1175 | { |
---|
[d317a9] | 1176 | if(printlevel>=1) {"N(B) = N("+string(i)+") = "+string(N(i))+" suffices the conditions of the underlying theorem,"; |
---|
| 1177 | "since P1 = (0:1:0) and P2[3] in (Z/N("+string(i)+")Z)*."; |
---|
| 1178 | "In particular N = "+string(N)+" is prime.";pause();"";} |
---|
[d1b0065] | 1179 | return(1); |
---|
| 1180 | } |
---|
| 1181 | } |
---|
[370344] | 1182 | |
---|
[d1b0065] | 1183 | if(step==14) // (14)[Backtrack] |
---|
| 1184 | { |
---|
| 1185 | if(i>0) |
---|
| 1186 | { |
---|
[d317a9] | 1187 | if(printlevel>=1) {"Set i = i-1 and restart the algorithm for N("+string(i-1)+") = "+string(N(i-1))+" with"; |
---|
| 1188 | "a new discriminant.";pause();"";} |
---|
[d1b0065] | 1189 | i=i-1; |
---|
| 1190 | k=0; |
---|
| 1191 | step=3; |
---|
| 1192 | } |
---|
| 1193 | else |
---|
| 1194 | { |
---|
[d317a9] | 1195 | if(printlevel>=1) {"N(0) = N = "+string(N)+" and therefore N is not prime.";pause();"";} |
---|
[d1b0065] | 1196 | return(-1); |
---|
| 1197 | } |
---|
| 1198 | } |
---|
| 1199 | } |
---|
| 1200 | } |
---|
[53e03a6] | 1201 | } |
---|
[449fbf] | 1202 | example |
---|
[53e03a6] | 1203 | { "EXAMPLE:"; echo = 2; |
---|
[449fbf] | 1204 | ring R = 0,x,dp; |
---|
| 1205 | Atkin(7691,100,5); |
---|
[d317a9] | 1206 | Atkin(3473,10,2); |
---|
| 1207 | printlevel=1; |
---|
[449fbf] | 1208 | Atkin(10000079,100,2); |
---|
[53e03a6] | 1209 | } |
---|
[3eadab] | 1210 | |
---|