source: git/Singular/LIB/crypto.lib @ 8e48d49

///////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Teaching";
info="
LIBRARY:  crypto.lib     Procedures for teaching cryptography
AUTHOR:                  Gerhard Pfister, pfister@mathematik.uni-kl.de

OVERVIEW:
     The library contains procedures to compute the discrete logarithm,
      primality-tests, factorization included elliptic curves.
      The library is intended to be used for teaching purposes but not
      for serious computations. Sufficiently high printlevel allows to
      control each step, thus illustrating the algorithms at work.


PROCEDURES:
 decimal(s);                number corresponding to the hexadecimal number s
 exgcdN(a,n)                compute s,t,d such that d=gcd(a,n)=s*a+t*n
 eexgcdN(L)                 T with sum_i L[i]*T[i]=T[n+1]=gcd(L[1],...,L[n])
 gcdN(a,b)                  compute gcd(a,b)
 lcmN(a,b)                  compute lcm(a,b)
 powerN(m,d,n)              compute m^d mod n
 chineseRem(T,L)            compute x such that x = T[i] mod L[i]
 Jacobi(a,n)                the generalized Legendre symbol of a and n
 primList(n)                the list of all primes <=n
 primL(q)                   first primes p_1,...,p_r such that q<p_1*...*p_r
 intPart(x)                 the integral part of a rational number
 intRoot(m)                 the integral part of the square root of m
 squareRoot(a,p)            the square root of a in Z/p, p prime
 solutionsMod2(M)           basis solutions of Mx=0 over Z/2
 powerX(q,i,I)              q-th power of the i-th variable modulo I
 babyGiant(b,y,p)           discrete logarithm x: b^x=y mod p
 rho(b,y,p)                 discrete logarithm x: b^x=y mod p
 MillerRabin(n,k)           probabilistic primaly-test of Miller-Rabin
 SolowayStrassen(n,k)       probabilistic primaly-test of Soloway-Strassen
 PocklingtonLehmer(N,[])    primaly-test of Pocklington-Lehmer
 PollardRho(n,k,a,[])       Pollard's rho factorization
 pFactor(n,B,P)             Pollard's p-factorization
 quadraticSieve(n,c,B,k)    quadratic sieve factorization
 isOnCurve(N,a,b,P)         P is on the curve y^2z=x^3+a*xz^2+b*z^3  over Z/N
 ellipticAdd(N,a,b,P,Q)     P+Q, addition on elliptic curves
 ellipticMult(N,a,b,P,k)    k*P on elliptic curves
 ellipticRandomCurve(N)     generates y^2z=x^3+a*xz^2+b*z^3  over Z/N randomly
 ellipticRandomPoint(N,a,b) random point on y^2z=x^3+a*xz^2+b*z^3  over Z/N
 countPoints(N,a,b)         number of points of y^2=x^3+a*x+b  over Z/N
 ellipticAllPoints(N,a,b)   points of y^2=x^3+a*x+b  over Z/N
 ShanksMestre(q,a,b,[])     number of points of y^2=x^3+a*x+b  over Z/N
 Schoof(N,a,b)              number of points of y^2=x^3+a*x+b  over Z/N
 generateG(a,b,m)           m-th division polynomial of y^2=x^3+a*x+b  over Z/N
 factorLenstraECM(N,S,B,[]) Lenstra's factorization
 ECPP(N)                    primaly-test of Goldwasser-Kilian

              [parameters in square brackets are optional]
";

LIB "poly.lib";

///////////////////////////////////////////////////////////////////////////////


//=============================================================================
//=========================== basic prozedures ================================
//=============================================================================

proc decimal(string s)
"USAGE:  decimal(s); s = string
RETURN: the (decimal) number corresponding to the hexadecimal number s
EXAMPLE:example decimal; shows an example
"
{
   int n=size(s);
   int i;
   bigint k;
   bigint t=16;
   bigint m=0;
   for(i=1;i<=n;i++)
   {
      if(s[i]=="1"){k=1;}
      if(s[i]=="2"){k=2;}
      if(s[i]=="3"){k=3;}
      if(s[i]=="4"){k=4;}
      if(s[i]=="5"){k=5;}
      if(s[i]=="6"){k=6;}
      if(s[i]=="7"){k=7;}
      if(s[i]=="8"){k=8;}
      if(s[i]=="9"){k=9;}
      if(s[i]=="a"){k=10;}
      if(s[i]=="b"){k=11;}
      if(s[i]=="c"){k=12;}
      if(s[i]=="d"){k=13;}
      if(s[i]=="e"){k=14;}
      if(s[i]=="f"){k=15;}
      m=m*t+k;
   }
   return(m);
}
example
{ "EXAMPLE:"; echo = 2;
   string s  ="8edfe37dae96cfd2466d77d3884d4196";
   decimal(s);
}

proc exgcdN(number a, number n)
"USAGE:  exgcdN(a,n);
RETURN: a list s,t,d of numbers satisfying d=gcd(a,n)=s*a+t*n
EXAMPLE:example exgcdN; shows an example
"
{
  number x=a mod n;
  if(x==0){return(list(0,1,n))}
  if (x<0) { x=x+n;}
  list l=exgcdN(n,x);
  return(list(l[2],l[1]-(a-x)*l[2]/n,l[3]))
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   exgcdN(24,15);
}

proc eexgcdN(list L)
"USAGE:  eexgcdN(L);
RETURN: list T such that sum_i L[i]*T[i]=T[n+1]=gcd(L[1],...,L[n])
EXAMPLE:example eexgcdN; shows an example
"
{
   if(size(L)==2){return(exgcdN(L[1],L[2]));}
   number p=L[size(L)];
   L=delete(L,size(L));
   list T=eexgcdN(L);
   list S=exgcdN(T[size(T)],p);
   int i;
   for(i=1;i<=size(T)-1;i++)
   {
      T[i]=T[i]*S[1];
   }
   p=T[size(T)];
   T[size(T)]=S[2];
   T[size(T)+1]=S[3];
   return(T);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   eexgcdN(list(24,15,21));
}

proc gcdN(number a, number b)
"USAGE:  gcdN(a,b);
RETURN: gcd(a,b)
EXAMPLE:example gcdN; shows an example
"
{
   //if((a mod b)==0){return(b)}
   //return(gcdN(b,a mod b));
   return(gcd(a,b));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   gcdN(24,15);
}

proc lcmN(number a, number b)
"USAGE:  lcmN(a,b);
RETURN: lcm(a,b);
EXAMPLE:example lcmN; shows an example
"
{
   //number d=gcdN(a,b);
   //return(a*b/d);
   return (a*b/gcd(a,b));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   lcmN(24,15);
}

proc powerN(number m, number d, number n)
"USAGE:  powerN(m,d,n);
RETURN: m^d mod n
EXAMPLE:example powerN; shows an example
"
{
   if(d==0){return(1);}
   int i;
   if(n==0)
   {
      for(i=12;i>=2;i--)
      {
         if((d mod i)==0){return(powerN(m,d/i,n)^i);}
      }
      return(m*powerN(m,d-1,n));
   }
   for(i=12;i>=2;i--)
   {
      if((d mod i)==0)
      {
        number rr=powerN(m,d/i,n)^i mod n;
        if (rr<0) { rr=rr+n;}
        return(rr);
      }
   }
   return(m*powerN(m,d-1,n) mod n);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   powerN(24,15,7);
}

proc chineseRem(list T,list L)
"USAGE:  chineseRem(T,L);
RETURN: x such that x = T[i] mod L[i]
NOTE:   chinese remainder theorem
EXAMPLE:example chineseRem; shows an example
"
{
   int i;
   int n=size(L);
   number N=1;
   for(i=1;i<=n;i++)
   {
      N=N*L[i];
   }
   list M;
   for(i=1;i<=n;i++)
   {
      M[i]=N/L[i];
   }
   list S=eexgcdN(M);
   number x;
   for(i=1;i<=n;i++)
   {
      x=x+S[i]*M[i]*T[i];
   }
   x=x mod N;
   return(x);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   chineseRem(list(24,15,7),list(2,3,5));
}

proc Jacobi(number a, number n)
"USAGE:  Jacobi(a,n);
RETURN: the generalized Legendre symbol
NOTE: if n is an odd prime then Jacobi(a,n)=0,1,-1 if n|a, a=x^2 mod n,else
EXAMPLE:example Jacobi; shows an example
"
{
   int i;
   int z=1;
   number t=1;
   number k;

   if((((n-1)/2) mod 2)!=0){z=-1;}
   if(a<0){return(z*Jacobi(-a,n));}
   a=a mod n;
   if(n==1){return(1);}
   if(a==0){return(0);}

   while(a!=0)
   {
      while((a mod 2)==0)
      {
         a=a/2;
         if(((n mod 8)==3)||((n mod 8)==5)){t=-t;}
      }
      k=a;a=n;n=k;
      if(((a mod 4)==3)&&((n mod 4)==3)){t=-t;}
      a=a mod n;
   }
   if (n==1){return(t);}
   return(0);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   Jacobi(13580555397810650806,5792543);
}

proc primList(int n)
"USAGE:  primList(n);
RETURN: the list of all primes <=n
EXAMPLE:example primList; shows an example
"
{
   int i,j;
   list re;
   re[1]=2;
   re[2]=3;
   for(i=5;i<=n;i=i+2)
   {
     j=1;
     while(j<=size(re))
     {
        if((i mod re[j])==0){break;}
        j++;
     }
     if(j==size(re)+1){re[size(re)+1]=i;}
   }
   return(re);
}
example
{ "EXAMPLE:"; echo = 2;
   list L=primList(100);
   size(L);
   L[size(L)];
}

proc primL(number q)
"USAGE:  primL(q);
RETURN: list of the first primes p_1,...,p_r such that q>p_1*...*p_(r-1)
        and q<p_1*...*p_r
EXAMPLE:example primL; shows an example
"
{
   int i,j;
   list re;
   re[1]=2;
   re[2]=3;
   number s=6;
   i=3;
   while(s<=q)
   {
     i=i+2;
     j=1;
     while(j<=size(re))
     {
        if((i mod re[j])==0){break;}
        j++;
     }
     if(j==size(re)+1)
     {
        re[size(re)+1]=i;
        s=s*i;
     }
   }
   return(re);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   primL(20);
}

proc intPart(number x)
"USAGE:  intPart(x);
RETURN: the integral part of a rational number
EXAMPLE:example intPart; shows an example
"
{
  if (x>=0)
  {
    return((numerator(x)-(numerator(x) mod denominator(x)))/denominator(x));
  }
  else
  {
    return((numerator(x)-(numerator(x) mod denominator(x)+denominator(x)))/denominator(x));
  }
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   intPart(7/3);
}

proc intRoot(number m)
"USAGE:  intRoot(m);
RETURN: the integral part of the square root of m
EXAMPLE:example intRoot; shows an example
"
{
   number x=1;
   number t=x^2;
   number s=(x+1)^2;
   while(((m>t)&&(m>s))||((m<t)&&(m<s)))
   {
      if (x==0) { t=0; }
      else
      {
        x=intPart(x/2+m/(2*x));  //Newton step
        t=x^2;
      }
      if(t>m)
      {
         s=(x-1)^2;
      }
      else
      {
         s=(x+1)^2;
      }
   }
   if(t>m){return(x-1);}
   if(s==m){return(x+1);}
   return(x);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   intRoot(20);
}

proc squareRoot(number a, number p)
"USAGE:  squareRoot(a,p);
RETURN: the square root of a in Z/p, p prime
NOTE:   assumes the Jacobi symbol is 1 or p=2.
EXAMPLE:example squareRoot; shows an example
"
{
   if(p==2){return(a);}
   if((a mod p)==0){return(0);}
   if (a<0) { a=a+p; }
   if(powerN(a,p-1,p)!=1)
   {
      "p is not prime";
      return(number(-5));
   }
   number n=random(1,2147483647) mod p;
   if(n==0){n=n+1;}
   number j=Jacobi(n,p);
   if(j==0)
   {
      "p is not prime";
      return(number(-5));
   }
   if(j==1)
   {
      return(squareRoot(a,p));
   }
   number q=p-1;
   number e=0;
   number two=2;
   number z,m,t;
   while((q mod 2)==0)
   {
      e=e+1;
      q=q/2;
   }
   number y=powerN(n,q,p);
   number r=e;
   number x=powerN(a,(q-1)/2,p);
   number b=a*x^2 mod p;
   x=a*x mod p;

   while(((b-1) mod p)!=0)
   {
      m=0;z=b;
      while(((z-1) mod p)!=0)
      {
         z=z^2 mod p;
         m=m+1;
      }
      t=powerN(y,powerN(two,r-m-1,p),p);
      y=t^2 mod p;
      r=m;
      x=x*t mod p;
      b=b*y mod p;
   }
   return(x);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   squareRoot(8315890421938608,32003);
}


proc solutionsMod2(matrix M)
"USAGE:  solutionsMod2(M);
RETURN: an intmat containing a basis of the vector space of solutions of the
        linear system of equations defined by M over the prime field of
        characteristic 2
EXAMPLE:example solutionsMod2; shows an example
"
{
   def R=basering;
   ring Rhelp=2,z,(c,dp);
   matrix M=imap(R,M);
   matrix S=syz(M);
   setring(R);
   matrix S=imap(Rhelp,S);
   int i,j;
   intmat v[nrows(S)][ncols(S)];
   for(i=1;i<=nrows(S);i++)
   {
      for(j=1;j<=ncols(S);j++)
      {
         if(S[i,j]==1){v[i,j]=1;}
      }
   }
   return(v);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   matrix M[3][3]=1,2,3,4,5,6,7,6,5;
   solutionsMod2(M);
}

proc powerX(int q, int i, ideal I)
"USAGE:  powerX(q,i,I);
RETURN: the q-th power of the i-th variable modulo I
ASSUME: I is a standard basis
EXAMPLE:example powerX; shows an example
"
{
   if(q<=181){return(reduce(var(i)^int(q),I));}
   if((q mod 5)==0){return(reduce(powerX(q div 5,i,I)^5,I));}
   if((q mod 4)==0){return(reduce(powerX(q div 4,i,I)^4,I));}
   if((q mod 3)==0){return(reduce(powerX(q div 3,i,I)^3,I));}
   if((q mod 2)==0){return(reduce(powerX(q div 2,i,I)^2,I));}
   return(reduce(var(i)*powerX(q-1,i,I),I));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y),dp;
   powerX(100,2,std(ideal(x3-1,y2-x)));
}

//======================================================================
//=========================== Discrete Logarithm =======================
//======================================================================

//============== Shank's baby step - giant step ========================

proc babyGiant(number b, number y, number p)
"USAGE:  babyGiant(b,y,p);
RETURN: the discrete logarithm x: b^x=y mod p
NOTE:   This procedure works based on Shank's baby step - giant step method.
EXAMPLE:example babyGiant; shows an example
"
{
   int i,j,m;
   list l;
   number h=1;
   number x;

//choose m minimal such that m^2>p
   for(i=1;i<=p;i++){if(i^2>p) break;}
   m=i;

//baby-step:  compute the table y*b^i for 1<=i<=m
   for(i=1;i<=m;i++){l[i]=y*b^i mod p;}

//giant-step: compute b^(m+j), 1<=j<=m and search in the baby-step table
//for an i with y*b^i=b^(m*j). If found then x=m*j-i
   number g=b^m mod p;
   while(j<m)
   {
      j++;
      h=h*g mod p;
      for(i=1;i<=m;i++)
      {
         if(h==l[i])
         {
            x=m*j-i;
            j=m;
            break;
         }
     }
   }
   return(x);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number b=2;
   number y=10;
   number p=101;
   babyGiant(b,y,p);
}

//==============  Pollards rho  =================================

proc rho(number b, number y, number p)
"USAGE:  rho(b,y,p);
RETURN: the discrete logarithm x=log_b(y): b^x=y mod p
NOTE: Pollard's rho:
       choose random f_0 in 0,...,p-2 ,e_0=0, define x_0=b^f_0, define
       x_i=y^e_i*b^f_i as below. For i large enough there is i with
       x_(i/2)=x_i. Let s:=e_(i/2)-e_i mod p-1 and t:=f_i-f_(i/2) mod p-1,
       d=gcd(s,p-1)=u*s+v*(p-1) then x=tu/d +j*(p-1)/d for some j (to be
       found by trying)
EXAMPLE:example rho; shows an example
"
{
   int i=1;
   int j;
   number s,t;
   list e,f,x;

   e[1]=0;
   f[1]=random(0,2147483629) mod (p-1);
   x[1]=powerN(b,f[1],p);
   while(i)
   {
      if((x[i] mod 3)==1)
      {
         x[i+1]=y*x[i] mod p;
         e[i+1]=e[i]+1 mod (p-1);
         f[i+1]=f[i];
      }
      if((x[i] mod 3)==2)
      {
         x[i+1]=x[i]^2 mod p;
         e[i+1]=e[i]*2 mod (p-1);
         f[i+1]=f[i]*2 mod (p-1);
      }
      if((x[i] mod 3)==0)
      {
         x[i+1]=x[i]*b mod p;
         e[i+1]=e[i];
         f[i+1]=f[i]+1 mod (p-1);
      }
      i++;
      for(j=i-1;j>=1;j--)
      {
         if(x[i]==x[j])
         {
            s=(e[j]-e[i]) mod (p-1);
            t=(f[i]-f[j]) mod (p-1);
            if(s!=0)
            {
               i=0;
            }
            else
            {
               e[1]=0;
               f[1]=random(0,2147483629) mod (p-1);
               x[1]=powerN(b,f[1],p);
               i=1;
            }
            break;
         }
      }
   }

   list w=exgcdN(s,p-1);
   number u=w[1];
   number d=w[3];

   number a=(t*u/d) mod (p-1);

   number pn=powerN(b,a,p);
   if (pn<0) { pn=pn+p;}
   while(powerN(b,a,p)!=y)
   {
      a=(a+(p-1)/d) mod (p-1);
      if (a<0) { a=a+p-1; }
   }
   return(a);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x,dp;
   number b=2;
   number y=10;
   number p=101;
   rho(b,y,p);
}
//====================================================================
//====================== Primality Tests =============================
//====================================================================

//================================= Miller-Rabin =====================

proc MillerRabin(number n, int k)
"USAGE:  MillerRabin(n,k);
RETURN: 1 if n is prime, 0 else
NOTE: probabilistic test of Miller-Rabin with k loops to test if n is prime.
       Using the theorem: If n is prime, n-1=2^s*r, r odd, then
       powerN(a,r,n)=1 or powerN(a,r*2^i,n)=-1 for some i
EXAMPLE:example MillerRabin; shows an example
"
{
   if(n<0){n=-n;}
   if((n==2)||(n==3)){return(1);}
   if((n mod 2)==0){return(0);}

   int i;
   number a,b,j,r,s;
   r=n-1;
   s=0;
   while((r mod 2)==0)
   {
      s=s+1;
      r=r/2;
   }
   while(i<k)
   {
      i++;
      a=random(2,2147483629) mod n; if(a==0){a=3;}
      if(exgcdN(a,n)[3]!=1){return(0);}
      b=powerN(a,r,n);
      if(b!=1)
      {
         j=0;
         while(j<s)
         {
           if(((b+1) mod n)==0) break;
           b=powerN(b,2,n);
           j=j+1;
         }
         if(j==s){return(0);}
      }
   }
   return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number x=2;
   x=x^787-1;
   MillerRabin(x,3);
}

//======================= Soloway-Strassen  ==========================

proc SolowayStrassen(number n, int k)
"USAGE:  SolowayStrassen(n,k);
RETURN: 1 if n is prime, 0 else
NOTE: probabilistic test of Soloway-Strassen with k loops to test if n is
       prime using the theorem: If n is prime then
       powerN(a,(n-1)/2,n)=Jacobi(a,n) mod n
EXAMPLE:example SolowayStrassen; shows an example
"
{
   if(n<0){n=-n;}
   if((n==2)||(n==3)){return(1);}
   if((n mod 2)==0){return(0);}

   number a,pn,jn;
   int i;
   while(i<k)
   {
      i++;
      a=random(2,2147483629) mod n; if(a==0){a=3;}
      if(gcd(a,n)!=1){return(0);}
      pn=powerN(a,(n-1)/2,n);
      if (pn<0) { pn=pn+n;}
      jn=Jacobi(a,n) mod n;
      if (jn<0) { jn=jn+n;}
      if(pn!=jn){return(0);}
   }
   return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number h=10;
   number p=h^100+267;
   //p=h^100+43723;
   //p=h^200+632347;
   SolowayStrassen(h,3);
}


/*
ring R=0,z,dp;
number p=398075086424064937397125500550386491199064362342526708406385189575946388957261768583317;
number q=472772146107435302536223071973048224632914695302097116459852171130520711256363590397527;
number n=p*q;
SolowayStrassen(n,3);
*/

//===================== Pocklington-Lehmer ==============================

proc PocklingtonLehmer(number N, list #)
"USAGE: PocklingtonLehmer(N); optional: PocklingtonLehmer(N,L);
        L a list of the first k primes
RETURN:message N is not prime or {A,{p},{a_p}} as certificate for N being prime
NOTE:assumes that it is possible to factorize N-1=A*B such that gcd(A,B)=1,
      the factorization of A is completely known and A^2>N .
      N is prime if and only if for each prime factor p of A we can find
      a_p such that a_p^(N-1)=1 mod N and gcd(a_p^((N-1)/p)-1,N)=1
EXAMPLE:example PocklingtonLehmer; shows an example
"
{
   number m=intRoot(N);
   if(size(#)>0)
   {
      list S=PollardRho(N-1,10000,1,#);
   }
   else
   {
      list S=PollardRho(N-1,10000,1);
   }
   int i,j;
   number A=1;
   number p,a,g;
   list PA;
   list re;

   while(i<size(S))
   {
      p=S[i+1];
      A=A*p;
      PA[i+1]=p;
      if(A>m){break;}

      while(1)
      {
        p=p*S[i+1];
        if(((N-1) mod p)==0)
        {
           A=A*p;
        }
        else
        {
           break;
        }
      }
      i++;
   }
   if(A<=m)
   {
     A=N/A;
     PA=list(S[size(S)]);
   }
   for(i=1;i<=size(PA);i++)
   {
      a=1;
      while(a<N-1)
      {
         a=a+1;
         if(powerN(a,N-1,N)!=1){return("not prime");}
         g=gcd(powerN(a,(N-1)/PA[i],N),N);
         if(g==1)
         {
           re[size(re)+1]=list(PA[i],a);
           break;
         }
         if(g<N){"not prime";return(g);}
      }
   }
   return(list(A,re));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number N=105554676553297;
   PocklingtonLehmer(N);
   list L=primList(1000);
   PocklingtonLehmer(N,L);
}

//=======================================================================
//======================= Factorization =================================
//=======================================================================

//======================= Pollards rho  =================================

proc PollardRho(number n, int k, int allFactors, list #)
"USAGE:  PollardRho(n,k,allFactors); optional: PollardRho(n,k,allFactors,L);
         L a list of the first k primes
RETURN: a list of factors of n (which could be just n),if allFactors=0@*
        a list of all factors of n ,if allFactors=1
NOTE: probabilistic rho-algorithm of Pollard to find a factor of n in k loops.
      Creates a sequence x_i such that (x_i)^2=(x_2i)^2 mod n for some i,
      computes gcd(x_i-x_2i,n) to find a divisor. To define the sequence
      choose x,a and define x_n+1=x_n^2+a mod n, x_1=x.
      If allFactors is 1, it tries to find recursively all prime factors
      using the Soloway-Strassen test.
SEE ALSO: primefactors
EXAMPLE:example PollardRho; shows an example
"
{
    int i,j;
    list L=primList(100);
    list re,se;
    if(n<0){n=-n;}
    if(n==1){return(re);}

//this is optional: test whether a prime of the list # devides n
    if(size(#)>0)
    {
       L=#;
    }
    for(i=1;i<=size(L);i++)
    {
       if((n mod L[i])==0)
       {
          re[size(re)+1]=L[i];
          while((n mod L[i])==0)
          {
             n=n/L[i];
          }
       }
       if(n==1){return(re);}
    }
    int e=size(re);
//here the rho-algorithm starts
    number a,d,x,y;
    while(n>1)
    {
       a=random(2,2147483629);
       x=random(2,2147483629);
       y=x;
       d=1;
       i=0;
       while(i<k)
       {
          i++;
          x=powerN(x,2,n); x=(x+a) mod n;
          y=powerN(y,2,n); y=(y+a) mod n;
          y=powerN(y,2,n); y=(y+a) mod n;
          d=gcd(x-y,n);
          if(d>1)
          {
             re[size(re)+1]=d;
             while((n mod d)==0)
             {
               n=n/d;
             }
             break;
          }
          if(i==k)
          {
             re[size(re)+1]=n;
             n=1;
          }
       }
    }
    if(allFactors)      //want to obtain all prime factors
    {
       i=e;
       while(i<size(re))
       {
          i++;

          if(!SolowayStrassen(re[i],5))
          {
             se=PollardRho(re[i],2*k,1);
             re[i]=se[size(se)];
             for(j=1;j<=size(se)-1;j++)
             {
                re[size(re)+1]=se[j];
             }
             i--;
          }
       }
    }
    return(re);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number h=10;
   number p=h^30+4;
   PollardRho(p,5000,0);
}

//======================== Pollards p-factorization ================
proc pFactor(number n,int B, list P)
"USAGE:  pFactor(n,B,P); n to be factorized, B a bound , P a list of primes
RETURN: a list of factors of n or n if no factor found
NOTE: Pollard's p-factorization
       creates the product k of powers of primes (bounded by B)  from
       the list P with the idea that for a prime divisor p of n we have
       p-1|k, and then p divides gcd(a^k-1,n) for some random a
EXAMPLE:example pFactor; shows an example
"
{
   int i;
   number k=1;
   number w;
   while(i<size(P))
   {
      i++;
      w=P[i];
      if(w>B) break;
      while(w*P[i]<=B)
      {
         w=w*P[i];
      }
      k=k*w;
   }
   number a=random(2,2147483629);
   number d=gcd(powerN(a,k,n)-1,n);
   if((d>1)&&(d<n)){return(d);}
   return(n);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   list L=primList(1000);
   pFactor(1241143,13,L);
   number h=10;
   h=h^30+25;
   pFactor(h,20,L);
}

//==================== quadratic sieve ==============================

proc quadraticSieve(number n, int c, list B, int k)
"USAGE:  quadraticSieve(n,c,B,k); n to be factorized, [-c,c] the
         sieve-intervall, B a list of primes,
         k for using the first k elements in B
RETURN: a list of factors of n or the message: no divisor found
NOTE: The idea being used is to find x,y such that x^2=y^2 mod n then
      gcd(x-y,n) can be a proper divisor of n
EXAMPLE:example quadraticSieve; shows an example
"
{
   number f,d;
   int i,j,l,s,p;
   list S,tmp;
   intvec v;
   v[k]=0;

//compute the integral part of the square root of n
   number m=intRoot(n);

//consider the function f(X)=(X+m)^2-n and compute for s in [-c,c] the values
   while(i<=2*c)
   {
      f=(i-c+m)^2-n;
      tmp[1]=i-c+m;
      tmp[2]=f;
      tmp[3]=v;
      S[i+1]=tmp;
      i++;
   }

//the sieve with p in B
//find all s in [-c,c] such that f(s) has all prime divisors in the first
//k elements of B and the decomposition of f(s). They are characterized
//by 1 or -1 at the second place of S[j]:
//S[j]=j-c+m,f(j-c)/p_1^v_1*...*p_k^v_k, v_1,...,v_k maximal
   for(i=1;i<=k;i++)
   {
      p=B[i];
      if((p>2)&&(Jacobi(n,p)==-1)){i++;continue;}//n is no quadratic rest mod p
      j=1;
      while(j<=p)
      {
         if(j>2*c+1) break;
         f=S[j][2];
         v=S[j][3];
         s=0;
         while((f mod p)==0)
         {
            s++;
            f=f/p;
         }
         if(s)
         {
           S[j][2]=f;
           v[i]=s;
           S[j][3]=v;
           l=j;
           while(l+p<=2*c+1)
           {
              l=l+p;
              f=S[l][2];
              v=S[l][3];
              s=0;
              while((f mod p)==0)
              {
                s++;
                f=f/p;
              }
              S[l][2]=f;
              v[i]=s;
              S[l][3]=v;
           }
         }
         j++;
      }
   }
   list T;
   for(j=1;j<=2*c+1;j++)
   {
      if((S[j][2]==1)||(S[j][2]==-1))
      {
         T[size(T)+1]=S[j];
      }
   }

//the system of equations for the exponents {l_s} for the f(s) such
//product f(s)^l_s is a square (l_s are 1 or 0)
   matrix M[k+1][size(T)];
   for(j=1;j<=size(T);j++)
   {
      if(T[j][2]==-1){M[1,j]=1;}
      for(i=1;i<=k;i++)
      {
         M[i+1,j]=T[j][3][i];
      }
   }
   intmat G=solutionsMod2(M);

//construction of x and y such that x^2=y^2 mod n and d=gcd(x-y,n)
//y=square root of product f(s)^l_s
//x=product s+m
   number x=1;
   number y=1;

   for(i=1;i<=ncols(G);i++)
   {
      kill v;
      intvec v;
      v[k]=0;
      for(j=1;j<=size(T);j++)
      {
         x=x*T[j][1]^G[j,i] mod n;
         if((T[j][2]==-1)&&(G[j,i]==1)){y=-y;}
         v=v+G[j,i]*T[j][3];

      }
      for(l=1;l<=k;l++)
      {
         y=y*B[l]^(v[l] div 2) mod n;
      }
      d=gcd(x-y,n);
      if((d>1)&&(d<n)){return(d);}
   }
   return("no divisor found");
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   list L=primList(5000);
   quadraticSieve(7429,3,L,4);
   quadraticSieve(1241143,100,L,50);
}

//======================================================================
//==================== elliptic curves  ================================
//======================================================================

//================= elementary operations ==============================

proc isOnCurve(number N, number a, number b, list P)
"USAGE:  isOnCurve(N,a,b,P);
RETURN: 1 or 0 (depending on whether P is on the curve or not)
NOTE: checks whether P=(P[1]:P[2]:P[3]) is a point on the elliptic
      curve defined by y^2z=x^3+a*xz^2+b*z^3  over Z/N
EXAMPLE:example isOnCurve; shows an example
"
{
   if(((P[2]^2*P[3]-P[1]^3-a*P[1]*P[3]^2-b*P[3]^3) mod N)!=0){return(0);}
   return(1);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   isOnCurve(32003,5,7,list(10,16,1));
}

proc ellipticAdd(number N, number a, number b, list P, list Q)
"USAGE:  ellipticAdd(N,a,b,P,Q);
RETURN: list L, representing the point P+Q
NOTE: P=(P[1]:P[2]:P[3]), Q=(Q[1]:Q[2]:Q[3]) points on the elliptic curve
      defined by y^2z=x^3+a*xz^2+b*z^3  over Z/N
EXAMPLE:example ellipticAdd; shows an example
"
{
   if(N==2){ERROR("not implemented for 2");}
   int i;
   for(i=1;i<=3;i++)
   {
      P[i]=P[i] mod N; if (P[i]<0) { P[i]=P[i]+N:}
      Q[i]=Q[i] mod N; if (Q[i]<0) { Q[i]=Q[i]+N;}
   }
   list Resu;
   Resu[1]=number(0);
   Resu[2]=number(1);
   Resu[3]=number(0);
   list Error;
   Error[1]=0;
   //test for ellictic curve
   number D=4*a^3+27*b^2;
   number g=gcd(D,N);
   if(g==N){return(Error);}
   if(g!=1)
   {
      P[4]=g;
      return(P);
   }
   if(((P[1]==0)&&(P[2]==0)&&(P[3]==0))||((Q[1]==0)&&(Q[2]==0)&&(Q[3]==0)))
   {
      Error[1]=-2;
      return(Error);
   }
   if(!isOnCurve(N,a,b,P)||!isOnCurve(N,a,b,Q))
   {
      Error[1]=-1;
      return(Error);
   }
   if(P[3]==0){return(Q);}
   if(Q[3]==0){return(P);}
   list I=exgcdN(P[3],N);
   if(I[3]!=1)
   {
      P[4]=I[3];
      return(P);
   }
   P[1]=P[1]*I[1] mod N;
   P[2]=P[2]*I[1] mod N;
   I=exgcdN(Q[3],N);
   if(I[3]!=1)
   {
      P[4]=I[3];
      return(P);
   }
   Q[1]=Q[1]*I[1] mod N;
   Q[2]=Q[2]*I[1] mod N;
   if((P[1]==Q[1])&&(((P[2]+Q[2]) mod N)==0)){return(Resu);}
   number L;
   if((P[1]==Q[1])&&(P[2]==Q[2]))
   {
      I=exgcdN(2*Q[2],N);
      if(I[3]!=1)
      {
         P[4]=I[3];
         return(P);
      }
      L=I[1]*(3*Q[1]^2+a) mod N;
   }
   else
   {
      I=exgcdN(Q[1]-P[1],N);
      if(I[3]!=1)
      {
         P[4]=I[3];
         return(P);
      }
      L=(Q[2]-P[2])*I[1] mod N;
   }
   Resu[1]=(L^2-P[1]-Q[1]) mod N;
   if (Resu[1]<0) { Resu[1]=Resu[1]+N; }
   Resu[2]=(L*(P[1]-Resu[1])-P[2]) mod N;
   if (Resu[2]<0) { Resu[2]=Resu[2]+N; }
   Resu[3]=number(1);
   return(Resu);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number N=11;
   number a=1;
   number b=6;
   list P,Q;
   P[1]=2;
   P[2]=4;
   P[3]=1;
   Q[1]=3;
   Q[2]=5;
   Q[3]=1;
   ellipticAdd(N,a,b,P,Q);
}

proc ellipticMult(number N, number a, number b, list P, number k)
"USAGE:  ellipticMult(N,a,b,P,k);
RETURN: a list L representing the point k*P
NOTE:  P=(P[1]:P[2]:P[3]) a point on the elliptic curve defined by
       y^2z=x^3+a*xz^2+b*z^3  over Z/N
EXAMPLE:example ellipticMult; shows an example
"
{
   if(P[3]==0){return(P);}
   list resu;
   resu[1]=number(0);
   resu[2]=number(1);
   resu[3]=number(0);

   if(k==0){return(resu);}
   if(k==1){return(P);}
   if(k==2){return(ellipticAdd(N,a,b,P,P));}
   if(k==-1)
   {
      resu=P;
      resu[2]=N-P[2];
      return(resu);
   }
   if(k<0)
   {
      resu=ellipticMult(N,a,b,P,-k);
      return(ellipticMult(N,a,b,resu,-1));
   }
   if((k mod 2)==0)
   {
      resu=ellipticMult(N,a,b,P,k/2);
      return(ellipticAdd(N,a,b,resu,resu));
   }
   resu=ellipticMult(N,a,b,P,k-1);
   return(ellipticAdd(N,a,b,resu,P));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number N=11;
   number a=1;
   number b=6;
   list P;
   P[1]=2;
   P[2]=4;
   P[3]=1;
   ellipticMult(N,a,b,P,3);
}

//================== Random for elliptic curves =====================

proc ellipticRandomCurve(number N)
"USAGE:  ellipticRandomCurve(N);
RETURN: a list of two random numbers a,b and 4a^3+27b^2 mod N
NOTE:   y^2z=x^3+a*xz^2+b^2*z^3 defines an elliptic curve over Z/N
EXAMPLE:example ellipticRandomCurve; shows an example
"
{
   int k;
   while(k<=10)
   {
     k++;
     number a=random(1,2147483647) mod N;
     number b=random(1,2147483647) mod N;
     //test for ellictic curve
     number D=4*a^3+27*b^4; //the constant term is b^2
     number g=gcd(D,N);
     if(g<N){return(list(a,b,g));}
   }
   ERROR("no random curve found");
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   ellipticRandomCurve(32003);
}

proc ellipticRandomPoint(number N, number a, number b)
"USAGE:  ellipticRandomPoint(N,a,b);
RETURN: a list representing  a random point (x:y:z) of the elliptic curve
        defined by y^2z=x^3+a*xz^2+b*z^3  over Z/N
EXAMPLE:example ellipticRandomPoint; shows an example
"
{
   number x=random(1,2147483647) mod N;
   number h=x^3+a*x+b;
   h=h mod N;
   list resu;
   resu[1]=x;
   resu[2]=0;
   resu[3]=1;
   if(h==0){return(resu);}

   number n=Jacobi(h,N);
   if(n==0)
   {
      resu=-5;
      "N is not prime";
      return(resu);
   }
   if(n==1)
   {
      resu[2]=squareRoot(h,N);
      return(resu);
   }
   return(ellipticRandomPoint(N,a,b));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   ellipticRandomPoint(32003,3,181);
}



//====================================================================
//======== counting the points of an elliptic curve  =================
//====================================================================

//==================   the trivial approaches  =======================
proc countPoints(number N, number a, number b)
"USAGE:  countPoints(N,a,b);
RETURN: the number of points of the elliptic curve defined by
        y^2=x^3+a*x+b  over Z/N
NOTE: trivial approach
EXAMPLE:example countPoints; shows an example
"
{
  number x;
  number r=N+1;
  while(x<N)
  {
     r=r+Jacobi((x^3+a*x+b) mod N,N);
     x=x+1;
  }
  return(r);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   countPoints(181,71,150);
}

proc ellipticAllPoints(number N, number a, number b)
"USAGE:  ellipticAllPoints(N,a,b);
RETURN: list of points (x:y:z) of the elliptic curve defined by
        y^2z=x^3+a*xz^2+b*z^3  over Z/N
EXAMPLE:example ellipticAllPoints; shows an example
"
{
   list resu,point;
   point[1]=0;
   point[2]=1;
   point[3]=0;
   resu[1]=point;
   point[3]=1;
   number x,h,n;
   while(x<N)
   {
      h=(x^3+a*x+b) mod N;
      if(h==0)
      {
         point[1]=x;
         point[2]=0;
         resu[size(resu)+1]=point;
      }
      else
      {
         n=Jacobi(h,N);
         if(n==1)
         {
            n=squareRoot(h,N);
            point[1]=x;
            point[2]=n;
            resu[size(resu)+1]=point;
            point[2]=N-n;
            resu[size(resu)+1]=point;
         }
      }
      x=x+1;
   }
   return(resu);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   list L=ellipticAllPoints(181,71,150);
   size(L);
   L[size(L)];
}

//================ the algorithm of Shanks and Mestre =================

proc ShanksMestre(number q, number a, number b, list #)
"USAGE:  ShanksMestre(q,a,b); optional: ShanksMestre(q,a,b,s); s the number
         of loops in the algorithm (default s=1)
RETURN: the number of points of the elliptic curve defined by
         y^2=x^3+a*x+b  over Z/N
NOTE: algorithm of Shanks and Mestre (baby-step-giant-step)
EXAMPLE:example ShanksMestre; shows an example
"
{
   number n=intRoot(4*q);
   number m=intRoot(intRoot(16*q))+1;
   number d;
   int i,j,k,s;
   list B,K,T,P,Q,R,mP;
   B[1]=list(0,1,0);
   if(size(#)>0)
   {
      s=#[1];
   }
   else
   {
      s=1;
   }
   while(k<s)
   {
      P =ellipticRandomPoint(q,a,b);
      Q =ellipticMult(q,a,b,P,n+q+1);

      while(j<m)
      {
         j++;
         B[j+1]=ellipticAdd(q,a,b,P,B[j]);  //baby-step list
      }
      mP=ellipticAdd(q,a,b,P,B[j]);
      mP[2]=q-mP[2];
      while(i<m)                            //giant-step
      {
         j=0;
         while(j<m)
         {
            j++;
            if((Q[1]==B[j][1])&&(Q[2]==B[j][2])&&(Q[3]==B[j][3]))
            {

               T[1]=P;
               T[2]=q+1+n-(i*m+j-1);
               K[size(K)+1]=T;
               if(size(K)>1)
               {
                  if(K[size(K)][2]!=K[size(K)-1][2])
                  {
                     d=gcd(K[size(K)][2],K[size(K)-1][2]);
                     if(ellipticMult(q,a,b,K[size(K)],d)[3]==0)
                     {
                       K[size(K)][2]=K[size(K)-1][2];
                     }
                  }
               }
               i=int(m);
               break;
            }
         }
         i=i+1;
         Q=ellipticAdd(q,a,b,mP,Q);
      }
      k++;
   }
   if(size(K)>0)
   {
      int te=1;
      for(i=1;i<=size(K)-1;i++)
      {
         if(K[size(K)][2]!=K[i][2])
         {
           if(ellipticMult(q,a,b,K[i],K[size(K)][2])[3]!=0)
           {
             te=0;
             break;
           }
         }
      }
      if(te)
      {
         return(K[size(K)][2]);
      }
   }
   return(ShanksMestre(q,a,b,s));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   ShanksMestre(32003,71,602);
}

//==================== Schoof's algorithm =============================

proc Schoof(number N,number a, number b)
"USAGE:  Schoof(N,a,b);
RETURN: the number of points of the elliptic curve defined by
        y^2=x^3+a*x+b  over Z/N
NOTE:  algorithm of Schoof
EXAMPLE:example Schoof; shows an example
"
{
   int pr=printlevel;
//test for ellictic curve
   number D=4*a^3+27*b^2;
   number G=gcd(D,N);
   if(G==N){ERROR("not an elliptic curve");}
   if(G!=1){ERROR("not a prime");}

//=== small N
  // if((N<=500)&&(pr<5)){return(countPoints(int(N),a,b));}

//=== the general case
   number q=intRoot(4*N);
   list L=primL(2*q);
   int r=size(L);
   list T;
   int i,j;
   for(j=1;j<=r;j++)
   {
      T[j]=(testElliptic(int(N),a,b,L[j])+int(q)) mod L[j];
   }
   if(pr>=5)
   {
      "===================================================================";
      "Chinese remainder :";
      for(i=1;i<=size(T);i++)
      {
         " x =",T[i]," mod ",L[i];
      }
      "gives t+ integral part of the square root of q (to be positive)";
      chineseRem(T,L);
      "we obtain t = ",chineseRem(T,L)-q;
      "===================================================================";
   }
   number t=chineseRem(T,L)-q;
   return(N+1-t);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   Schoof(32003,71,602);
}

/*
needs 518 sec
Schoof(2147483629,17,3567);
2147168895
*/


proc generateG(number a,number b, int m)
"USAGE:  generateG(a,b,m);
RETURN: m-th division polynomial
NOTE: generate the so-called division polynomials, i.e., the recursively defined
polynomials p_m=generateG(a,b,m) in Z[x, y] such that, for a point (x:y:1) on the
elliptic curve defined by y^2=x^3+a*x+b  over Z/N the point@*
m*P=(x-(p_(m-1)*p_(m+1))/p_m^2 :(p_(m+2)*p_(m-1)^2-p_(m-2)*p_(m+1)^2)/4y*p_m^3 :1)
and m*P=0 if and only if p_m(P)=0
EXAMPLE:example generateG; shows an example
"
{
   if(m==0){return(poly(0));}
   if(m==1){return(poly(1));}
   if(m==2){return(2*var(1));}
   if(m==3){return(3*var(2)^4+6*a*var(2)^2+12*b*var(2)-a^2);}
   if(m==4)
   {
      return(4*var(1)*(var(2)^6+5*a*var(2)^4+20*b*var(2)^3-5*a^2*var(2)^2
        -4*a*b*var(2)-8*b^2-a^3));
   }
   if((m mod 2)==0)
   {
      return((generateG(a,b,m div 2+2)*generateG(a,b,m div 2-1)^2
        -generateG(a,b,m div 2-2)*generateG(a,b,m div 2+1)^2)
        *generateG(a,b,m div 2)/(2*var(1)));
   }
   return(generateG(a,b,(m-1) div 2+2)*generateG(a,b,(m-1) div 2)^3
     -generateG(a,b,(m-1) div 2-1)*generateG(a,b,(m-1) div 2+1)^3);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y),dp;
   generateG(7,15,4);
}


proc testElliptic(int q,number a,number b,int l)
"USAGE:  testElliptic(q,a,b,l);
RETURN: an integer t, the trace of the Frobenius
NOTE: the kernel for the Schoof algorithm: looks for the t such that for all
      points (x:y:1) in C[l]={P in C | l*P=0},C the elliptic curve defined by
      y^2=x^3+a*x+b  over Z/q with group structure induced by 0=(0:1:0),
      (x:y:1)^(q^2)-t*(x:y:1)^q -ql*(x:y:1)=(0:1:0), ql= q mod l, trace of
      Frobenius.
EXAMPLE:example testElliptic; shows an example
"
{
   int pr=printlevel;
   def R=basering;
   ring S=q,(y,x),(L(100000),lp);
   number a=imap(R,a);
   number b=imap(R,b);
   poly F=y2-x3-a*x-b;       // the curve C
   poly G=generateG(a,b,l);
   ideal I=std(ideal(F,G));  // the points C[l]
   poly xq=powerX(q,2,I);
   poly yq=powerX(q,1,I);
   poly xq2=reduce(subst(xq,x,xq,y,yq),I);
   poly yq2=reduce(subst(yq,x,xq,y,yq),I);
   ideal J;
   int ql=q mod l;
   if(ql==0){ERROR("q is not prime");}
   int t;
   poly F1,F2,G1,G2,P1,P2,Q1,Q2,H1,H2,L1,L2;

   if(pr>=5)
   {
      "===================================================================";
      "q=",q;
      "l=",l;
      "q mod l=",ql;
      "the Groebner basis for C[l]:";I;
      "x^q mod I = ",xq;
      "x^(q^2) mod I = ",xq2;
      "y^q mod I = ",yq;
      "y^(q^2) mod I = ",yq2;
      pause();
   }
   //==== l=2 =============================================================
   if(l==2)
   {
      xq=powerX(q,2,std(x3+a*x+b));
      J=std(ideal(xq-x,x3+a*x+b));
      if(deg(J[1])==0){t=1;}
      if(pr>=5)
      {
         "===================================================================";
         "the case l=2";
         "the gcd(x^q-x,x^3+ax+b)=",J[1];
         pause();
      }
      setring R;
      return(t);
   }
   //=== (F1/G1,F2/G2)=[ql](x,y) ==========================================
   if(ql==1)
   {
      F1=x;G1=1;F2=y;G2=1;
   }
   else
   {
      G1=reduce(generateG(a,b,ql)^2,I);
      F1=reduce(x*G1-generateG(a,b,ql-1)*generateG(a,b,ql+1),I);
      G2=reduce(4*y*generateG(a,b,ql)^3,I);
      F2=reduce(generateG(a,b,ql+2)*generateG(a,b,ql-1)^2
               -generateG(a,b,ql-2)*generateG(a,b,ql+1)^2,I);

   }
   if(pr>=5)
   {
      "===================================================================";
      "the point ql*(x,y)=(F1/G1,F2/G2)";
      "F1=",F1;
      "G1=",G1;
      "F2=",F2;
      "G2=",G2;
      pause();
   }
   //==== the case t=0 :  the equations for (x,y)^(q^2)=-[ql](x,y) ===
   J[1]=xq2*G1-F1;
   J[2]=yq2*G2+F2;
   if(pr>=5)
   {
      "===================================================================";
      "the case t=0 mod l";
      "the equations for (x,y)^(q^2)=-[ql](x,y) :";
      J;
      "the test, if they vanish for all points in C[l]:";
      reduce(J,I);
      pause();
   }
   //=== test if all points of C[l] satisfy  (x,y)^(q^2)=-[ql](x,y)
   //=== if so: t mod l =0 is returned
   if(size(reduce(J,I))==0){setring R;return(0);}

   //==== test for (x,y)^(q^2)=[ql](x,y) for some point

   J=xq2*G1-F1,yq2*G2-F2;
   J=std(J+I);
   if(pr>=5)
   {
      "===================================================================";
      "test if (x,y)^(q^2)=[ql](x,y) for one point";
      "if so, the Frobenius has an eigenvalue 2ql/t: (x,y)^q=(2ql/t)*(x,y)";
      "it follows that t^2=4q mod l";
      "if w is one square root of q mod l";
      "t =2w mod l or -2w mod l ";
      "-------------------------------------------------------------------";
      "the equations for (x,y)^(q^2)=[ql](x,y) :";
      xq2*G1-F1,yq2*G2-F2;
      "the test if one point satisfies them";
      J;
      pause();
   }
   if(deg(J[1])>0)
   {
      setring R;
      int w=int(squareRoot(q,l));
      setring S;
      //=== +/-2w mod l zurueckgeben, wenn (x,y)^q=+/-[w](x,y)
      //==== the case t>0 :  (Q1/P1,Q2/P2)=[w](x,y) ==============
      if(w==1)
      {
         Q1=x;P1=1;Q2=y;P2=1;
      }
      else
      {
         P1=reduce(generateG(a,b,w)^2,I);
         Q1=reduce(x*G1-generateG(a,b,w-1)*generateG(a,b,w+1),I);
         P2=reduce(4*y*generateG(a,b,w)^3,I);
         Q2=reduce(generateG(a,b,w+2)*generateG(a,b,w-1)^2
               -generateG(a,b,w-2)*generateG(a,b,w+1)^2,I);
      }
      J=xq*P1-Q1,yq*P2-Q2;
      J=std(I+J);
      if(pr>=5)
      {
      "===================================================================";
      "the Frobenius has an eigenvalue, one of the roots of  w^2=q mod l:";
      "one root is:";w;
      "test, if it is the eigenvalue (if not it must be -w):";
      "the equations for (x,y)^q=w*(x,y)";I;xq*P1-Q1,yq*P2-Q2;
      "the Groebner basis";
       J;
         pause();
      }
      if(deg(J[1])>0){return(2*w mod l);}
      return(-2*w mod l);
   }

   //==== the case t>0 :  (Q1/P1,Q2/P2)=(x,y)^(q^2)+[ql](x,y) =====
   P1=reduce(G1*G2^2*(F1-xq2*G1)^2,I);
   Q1=reduce((F2-yq2*G2)^2*G1^3-F1*G2^2*(F1-xq2*G1)^2-xq2*P1,I);
   P2=reduce(P1*G2*(F1-xq2*G1),I);
   Q2=reduce((xq2*P1-Q1)*(F2-yq2*G2)*G1-yq2*P2,I);

   if(pr>=5)
   {
      "we are in the general case:";
      "(x,y)^(q^2)!=ql*(x,y) and (x,y)^(q^2)!=-ql*(x,y) ";
      "the point (Q1/P1,Q2/P2)=(x,y)^(q^2)+[ql](x,y)";
      "Q1=",Q1;
      "P1=",P1;
      "Q2=",Q2;
      "P2=",P2;
      pause();
   }
   while(t<(l-1) div 2)
   {
      t++;
      //====  (H1/L1,H2/L2)=[t](x,y)^q ===============================
      if(t==1)
      {
         H1=xq;L1=1;
         H2=yq;L2=1;
      }
      else
      {
         H1=x*generateG(a,b,t)^2-generateG(a,b,t-1)*generateG(a,b,t+1);
         H1=subst(H1,x,xq,y,yq);
         H1=reduce(H1,I);
         L1=generateG(a,b,t)^2;
         L1=subst(L1,x,xq,y,yq);
         L1=reduce(L1,I);
         H2=generateG(a,b,t+2)*generateG(a,b,t-1)^2
           -generateG(a,b,t-2)*generateG(a,b,t+1)^2;
         H2=subst(H2,x,xq,y,yq);
         H2=reduce(H2,I);
         L2=4*y*generateG(a,b,t)^3;
         L2=subst(L2,x,xq,y,yq);
         L2=reduce(L2,I);
      }
      J=Q1*L1-P1*H1,Q2*L2-P2*H2;
      if(pr>=5)
      {
      "we test now the different t, 0<t<=(l-1)/2:";
      "the point (H1/L1,H2/L2)=[t](x,y)^q :";
      "H1=",H1;
      "L1=",L1;
      "H2=",H2;
      "L2=",L2;
      "the equations for (x,y)^(q^2)+[ql](x,y)=[t](x,y)^q :";J;
      "the test";reduce(J,I);
      "the test for l-t (the x-cordinate is the same):";
       Q1*L1-P1*H1,Q2*L2+P2*H2;
       reduce(ideal(Q1*L1-P1*H1,Q2*L2+P2*H2),I);
         pause();
      }
      if(size(reduce(J,I))==0){setring R;return(t);}
      J=Q1*L1-P1*H1,Q2*L2+P2*H2;
      if(size(reduce(J,I))==0){setring R;return(l-t);}
   }
   ERROR("something is wrong in testElliptic");
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   testElliptic(1267985441,338474977,64740730,3);
}

//============================================================================
//================== Factorization and Primality Test ========================
//============================================================================

//============= Lenstra's ECM Factorization ==================================

proc factorLenstraECM(number N, list S, int B, list #)
"USAGE:  factorLenstraECM(N,S,B); optional: factorLenstraECM(N,S,B,d);
         d+1 the number of loops in the algorithm (default d=0)
RETURN: a factor of N or the message no factor found
NOTE: - computes a factor of N using Lenstra's ECM factorization@*
      - the idea is that the fact that N is not prime is dedected using
        the operations on the elliptic curve
      - is similarly to Pollard's p-1-factorization
EXAMPLE:example factorLenstraECM; shows an example
"
{
   list L,P;
   number g,M,w;
   int i,j,k,d;
   int l=size(S);
   if(size(#)>0)
   {
      d=#[1];
   }

   while(i<=d)
   {
      L=ellipticRandomCurve(N);
      if(L[3]>1){return(L[3]);} //the discriminant was not invertible
      P=list(0,L[2],1);
      j=0;
      M=1;
      while(j<l)
      {
         j++;
         w=S[j];
         if(w>B) break;
         while(w*S[j]<B)
         {
           w=w*S[j];
         }
         M=M*w;
         P=ellipticMult(N,L[1],L[2]^2,P,w);
         if(size(P)==4){return(P[4]);}  //some inverse did not exsist
         if(P[3]==0){break;}            //the case M*P=0
      }
      i++;
   }
   return("no factor found");
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   list L=primList(1000);
   factorLenstraECM(181*32003,L,10,5);
   number h=10;
   h=h^30+25;
   factorLenstraECM(h,L,4,3);
}

//================= ECPP (Goldwasser-Kilian) a primaly-test =============

proc ECPP(number N)
"USAGE:  ECPP(N);
RETURN: message:N is not prime or {L,P,m,q} as certificate for N being prime@*
         L a list (y^2=x^3+L[1]*x+L[2] defines an elliptic curve C)@*
         P a list ((P[1]:P[2]:P[3]) is a point of C)@*
         m,q integers
ASSUME: gcd(N,6)=1
NOTE:   The basis of the algorithm is the following theorem:
         Given C, an elliptic curve over Z/N, P a point of C(Z/N),
         m an integer, q a prime with the following properties:
         - q|m
         - q>(4-th root(N) +1)^2
         - m*P=0=(0:1:0)
         - (m/q)*P=(x:y:z) and z a unit in Z/N
         Then N is prime.
EXAMPLE:example ECPP; shows an example
"
{
   list L,S,P;
   number m,q;
   int i;

   number n=intRoot(intRoot(N));
   n=(n+1)^2;                         //lower bound for q
   while(1)
   {
      L=ellipticRandomCurve(N);       //a random elliptic curve C
      m=ShanksMestre(N,L[1],L[2],3);  //number of points of the curve C
      S=PollardRho(m,10000,1);        //factorization of m
      for(i=1;i<=size(S);i++)         //search for q between the primes
      {
         q=S[i];
         if(n<q){break;}
      }
      if(n<q){break;}
   }
   number u=m/q;
   while(1)
   {
      P=ellipticRandomPoint(N,L[1],L[2]);  //a random point on C
      "P=",P;
      if(ellipticMult(N,L[1],L[2],P,m)[3]!=0){"N is not prime";return(-5);}
      if(ellipticMult(N,L[1],L[2],P,u)[3]!=0)
      {
         L=delete(L,3);
         return(list(L,P,m,q));
      }
   }
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,z,dp;
   number N=1267985441;
   ECPP(N);
}

static proc wordToNumber(string s)
{
   int i;
   intvec v;
   number n;
   number t=27;
   for(i=size(s);i>0;i--)
   {
      if(s[i]=="a"){v[i]=0;}
      if(s[i]=="b"){v[i]=1;}
      if(s[i]=="c"){v[i]=2;}
      if(s[i]=="d"){v[i]=3;}
      if(s[i]=="e"){v[i]=4;}
      if(s[i]=="f"){v[i]=5;}
      if(s[i]=="g"){v[i]=6;}
      if(s[i]=="h"){v[i]=7;}
      if(s[i]=="i"){v[i]=8;}
      if(s[i]=="j"){v[i]=9;}
      if(s[i]=="k"){v[i]=10;}
      if(s[i]=="l"){v[i]=11;}
      if(s[i]=="m"){v[i]=12;}
      if(s[i]=="n"){v[i]=13;}
      if(s[i]=="o"){v[i]=14;}
      if(s[i]=="p"){v[i]=15;}
      if(s[i]=="q"){v[i]=16;}
      if(s[i]=="r"){v[i]=17;}
      if(s[i]=="s"){v[i]=18;}
      if(s[i]=="t"){v[i]=19;}
      if(s[i]=="u"){v[i]=20;}
      if(s[i]=="v"){v[i]=21;}
      if(s[i]=="w"){v[i]=22;}
      if(s[i]=="x"){v[i]=23;}
      if(s[i]=="y"){v[i]=24;}
      if(s[i]=="z"){v[i]=25;}
      if(s[i]==" "){v[i]=26;}
   }
   for(i=1;i<=size(s);i++)
   {
      n=n+v[i]*t^(i-1);
   }
   return(n);
}

static proc numberToWord(number n)
{
   int i,j;
   string v;
   list s;
   number t=27;
   number mm;
   number nn=n;
   while(nn>t)
   {
      j++;
      mm=nn mod t;
      s[j]=mm;
      nn=(nn-mm)/t;
   }
   j++;
   s[j]=nn;
   for(i=1;i<=j;i++)
   {
      if(s[i]==0){v=v+"a";}
      if(s[i]==1){v=v+"b";}
      if(s[i]==2){v=v+"c";}
      if(s[i]==3){v=v+"d";}
      if(s[i]==4){v=v+"e";}
      if(s[i]==5){v=v+"f";}
      if(s[i]==6){v=v+"g";}
      if(s[i]==7){v=v+"h";}
      if(s[i]==8){v=v+"i";}
      if(s[i]==9){v=v+"j";}
      if(s[i]==10){v=v+"k";}
      if(s[i]==11){v=v+"l";}
      if(s[i]==12){v=v+"m";}
      if(s[i]==13){v=v+"n";}
      if(s[i]==14){v=v+"o";}
      if(s[i]==15){v=v+"p";}
      if(s[i]==16){v=v+"q";}
      if(s[i]==17){v=v+"r";}
      if(s[i]==18){v=v+"s";}
      if(s[i]==19){v=v+"t";}
      if(s[i]==20){v=v+"u";}
      if(s[i]==21){v=v+"v";}
      if(s[i]==22){v=v+"w";}
      if(s[i]==23){v=v+"x";}
      if(s[i]==24){v=v+"y";}
      if(s[i]==25){v=v+"z";}
      if(s[i]==26){v=v+" ";}
   }
   return(v);
}

proc code(string s)
"USAGE:  code(s); s a string
ASSUME:  s contains only small letters and space
COMPUTE: a number, RSA-coding of the string s
RETURN:  return RSA-coding of the string s as string
EXAMPLE: code;  shows an example
"
{
   ring r=0,x,dp;
   number
p=398075086424064937397125500550386491199064362342526708406385189575946388957261768583317;
   number
q=472772146107435302536223071973048224632914695302097116459852171130520711256363590397527;
   number n=p*q;
   number phi=(p-1)*(q-1);
   number e=1234567891;
   list L=exgcdN(e,phi);
   number d=L[1];
   number m=wordToNumber(s);
   number c=powerN(m,e,n);
   string cc=string(c);
   return(cc);
}
example
{"EXAMPLE:";  echo = 2;
  string s="i go to school";
  code(s);
}

proc decodeString(string g)
"USAGE:  decodeString(s); s a string
ASSUME:  s is a string of a number, the output of code
COMPUTE: a string, RSA-decoding of the string s
RETURN:  return RSA-decoding of the string s as string
EXAMPLE: decodeString;  shows an example
"
{
   ring r=0,x,dp;
   number
p=398075086424064937397125500550386491199064362342526708406385189575946388957261768583317;
   number
q=472772146107435302536223071973048224632914695302097116459852171130520711256363590397527;
   number n=p*q;
   number phi=(p-1)*(q-1);
   number e=1234567891;
   list L=exgcdN(e,phi);
   number d=L[1];
   execute("number c="+g+";");
   number f=powerN(c,d,n);
   string s=numberToWord(f);
   return(s);
}
example
{"EXAMPLE:";  echo = 2;
  string
s="78638618599886548153321853785991541374544958648147340831959482696082179852616053583234149080198937632782579537867262780982185252913122030800897193851413140758915381848932565";
  string t=decodeString(s);
  t;
}

/*
//===============================================================
//=======  Example for DSA  =====================================
//===============================================================
Suppose a file test is given.It contains "Oscar".

//Hash-function MD5 under Linux

md5sum test           8edfe37dae96cfd2466d77d3884d4196

//================================================================

ring R=0,x,dp;

number q=2^19+21;          //524309
number o=2*3*23*number(7883)*number(16170811);

number p=o*q+1;            //9223372036869000547
number b=2;
number g=power(2,o,p);     //8308467587808723131

number a=111111;
number A=power(g,a,p);     //8566038811843553785

number h =decimal("8edfe37dae96cfd2466d77d3884d4196");

                           //189912871665444375716340628395668619670
h= h mod q;                //259847

number k=123456;

number ki=exgcd(k,q)[1];    //50804
                            //inverse von k mod q

number r= power(g,k,p) mod q;    //76646

number s=ki*(h+a*r) mod q;       //2065

//========== signatur is (r,s)=(76646,2065) =====================
//==================== verification  ============================

number si=exgcd(s,q)[1];  //inverse von s mod q
number e1=si*h mod q;
number e2=si*r mod q;
number rr=((power(g,e1,p)*power(A,e2,p)) mod p) mod q;    //76646

//===============================================================
//=======  Example for knapsack  ================================
//===============================================================
ring R=(5^5,t),x,dp;
R;
//   # ground field : 3125
//   primitive element : t
//   minpoly        : 1*t^5+4*t^1+2*t^0
//   number of vars : 1
//        block   1 : ordering dp
//                  : names    x
//        block   2 : ordering C

proc findEx(number n, number g)
{
   int i;
   for(i=0;i<=size(basering)-1;i++)
   {
      if(g^i==n){return(i);}
   }
}

number g=t^3;                       //choice of the primitive root

findEx(t+1,g);
//2091
findEx(t+2,g);
//2291
findEx(t+3,g);
//1043

intvec b=1,2091,2291,1043;          // k=4
int z=199;
intvec v=1043+z,1+z,2091+z,2291+z;  //permutation   pi=(0123)
v;
1242,200,2290,2490

//(1101)=(e_3,e_2,e_1,e_0)
//encoding 2490+2290+1242=6022 und 1+1+0+1=3

//(6022,3) decoding: c-z*c'=6022-199*3=5425

ring S=5,x,dp;
poly F=x5+4x+2;
poly G=reduce((x^3)^5425,std(F));
G;
//x3+x2+x+1

factorize(G);
//[1]:
//   _[1]=1
//   _[2]=x+1
//   _[3]=x-2
//   _[4]=x+2
//[2]:
//   1,1,1,1

//factors x+1,x+2,x+3, i.e.  (1110)=(e_pi(3),e_pi(2),e_pi(1),e_pi(0))

//pi(0)=1,pi(1)=2,pi(2)=3,pi(3)=0 gives:  (1101)

*/