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// curve.cc: implementations of elliptic curve class Curve
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2023 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
// originally adapted from Elliptic.cc by Oisin McGuiness
#include <eclib/curve.h>
//Kraus' conditions:
int valid_invariants(const bigint& c4, const bigint& c6)
{
bigint disc = c4*c4*c4 - c6*c6;
if (sign(disc)==0) return 0;
if (ndiv(1728,disc)) return 0; // need c4^3-c6^2=1728D, with D|=0
long x6= mod(c6,27);
if((x6==9)||(x6==-9)) return 0; // need c6 != +-9 (mod 27)
x6 = mod(c6,4);
if(x6==-1) return 1; // OK if c6 = -1 (mod 4)
if(ndiv(16,c4)) return 0; // else need c4=0 (mod 16)
x6=mod(c6,32); // and
return ((x6==0) || (x6==8)); // c6 = 0,8 (mod 32).
}
void c4c6_to_ai(const bigint& c4, const bigint& c6,
bigint& a1, bigint& a2, bigint& a3, bigint& a4,
bigint& a6,
bigint& b2, bigint& b4, bigint& b6, bigint& b8)
{
// cout<<"In c4c6_to_ai() with c4="<<c4<<" and c6="<<c6<<endl;
bigint I12; I12=12;
b2 = mod(-c6,I12); // cout<<"...b2="<<b2<<endl;
const bigint& b22 = b2*b2;
b4 = (b22-c4)/24; // cout<<"...b4="<<b4<<endl;
b6 = (-b2*b22+36*b2*b4-c6)/216; // cout<<"...b6="<<b6<<endl;
b8 = (b2*b6 - b4*b4) / 4; // cout<<"...b8="<<b8<<endl;
a1 = (odd(b2) ? 1 : 0);
a3 = (odd(b6) ? 1 : 0);
a2 = (b2-a1)/4; // N.B. a1 == a1*a1 (= 0,1)
a4 = (b4-a1*a3)/2;
a6 = (b6-a3)/4; // N.B. a3 == a3*a3 (= 0,1)
// cout<<"...returning a1="<<a1<<", a2="<<a2<<", a3="<<a3
// <<", a4="<<a4<<", a6="<<a6<<endl;
}
void c4c6_to_ai(const bigint& c4, const bigint& c6,
bigint& a1, bigint& a2, bigint& a3, bigint& a4,
bigint& a6)
{
bigint b2, b4, b6, b8;
c4c6_to_ai(c4,c6,a1,a2,a3,a4,a6,b2,b4,b6,b8);
}
void minimise_c4c6(const bigint& c4, const bigint& c6, const bigint& discr,
bigint& newc4, bigint& newc6, bigint& newdiscr, bigint& u)
{
long a,b;
u = 1; int u_is_1 = 1;
newc4=c4; newc6=c6;
const bigint& c62 = sqr(c6);
newdiscr = (sqr(c4)*c4-c62)/1728; // this must be set before returning
bigint g=gcd(c4,c6); if(is_one(g)) return;
g = gcd( c62, newdiscr ); if(is_one(g)) return;
const vector<bigint>& p_list = pdivs(g);
// cout<<"g = "<<g<<endl;
for (const auto& p : p_list)
{
long d = (long)floor(val(p,g)/12.0);
// cout<<"With p="<<p<<", initial d="<<d<<endl;
if (p==2)
{
a = mod(c4 >> (4*d) , 16);
b = mod(c6 >> (6*d) , 32); if(b<0) b+=32;
// cout<<"a="<<a<<", b="<<b<<endl;
if (( (b%4)!=3) && !( (a==0) && (( b==0) || (b==8) )))
{
d--;
}
}
else if (p==3) if (val(3,c6)==(6*d + 2)) d--;
if(d>0) {u *= pow(p,d); u_is_1 = 0;}
// cout<<"With p="<<p<<", final d = "<<d<<", u="<<u<<endl;
}
if(u_is_1) return;
bigint u2, u4, u6, u12;
mulx(u,u,u2); mulx(u2,u2,u4); mulx(u2,u4,u6); mulx(u6,u6,u12);
newc4 = c4 / u4;
newc6 = c6 / u6;
newdiscr /= u12;
}
//constructor for curve with invariants as argument
Curve::Curve(const bigint& c4, const bigint& c6)
{
if (valid_invariants(c4, c6))
{
c4c6_to_ai(c4,c6,a1,a2,a3,a4,a6);
}
else
{
// cout << " ## attempt to call Curve constructor"
// << " with invalid invariants c4 = "<<c4<<", c6 = "<<c6
// << ": reading as null curve\n";
a1=0; a2=0; a3=0; a4=0; a6=0;
}
}
void Curve::input(istream& is)
{
char c; // to eat commas and detect [ from {;
// `{' flags curve input by invariants, eg {1,3}
// `[' by coeffs a1--a6, eg [1,2,3,4,6]
// seperators and terminators must then be present
// (any nonnumeric will do after the first { or [ )
// else assumes a1 a2 a3 a4 a6 separated by whitespace
is>>skipws;
is>>c;
// cout<<"First char read = "<<c<<"\n";
switch (c) {
case '{':
{
// cout<<"Reading {c4,c6}...\n";
bigint c4, c6;
is >> c4 >> c;
if(c!=',')
{
cout << "syntax error on curve input" << endl;
return;
}
is >> c6 >> c;
if(c!='}')
{
cout << "syntax error on curve input" << endl;
return;
}
if (valid_invariants(c4, c6))
{
const bigint& b2 = bigint(mod(-c6,12));
const bigint& b22 = b2*b2;
const bigint& b4 = (b22-c4)/24;
const bigint& b6 = (-b2*b22+36*b2*b4-c6)/216;
a1 = (odd(b2) ? 1 : 0);
a3 = (odd(b6) ? 1 : 0);
a2 = (b2-a1*a1)/4;
a4 = (b4-a1*a3)/2;
a6 = (b6-a3*a3)/4;
}
else
{
cout << " ## invalid invariants, reading as null curve\n";
a1=0; a2=0; a3=0;a4=0; a6=0;
}
}
break;
case '[':
// cout<<"Reading [a1,a2,a3,a4,a6]...\n";
is >> a1 >> c;
if(c!=',')
{
cout << "syntax error on curve input" << endl;
return;
}
is >> a2 >> c;
if(c!=',')
{
cout << "syntax error on curve input" << endl;
return;
}
is >> a3 >> c;
if(c!=',')
{
cout << "syntax error on curve input" << endl;
return;
}
is >> a4 >> c;
if(c!=',')
{
cout << "syntax error on curve input" << endl;
return;
}
is >> a6 >> c;
if(c!=']')
{
cout << "syntax error on curve input" << endl;
return;
}
// cout<<"["<<a1<<","<<a2<<","<<a3<<","<<a4<<","<<a6<<"]"<<endl;
break;
default:
// cout<<"Reading a1 a2 a3 a4 a6 ...\n";
is.unget();
is >> a1 >> a2 >> a3 >> a4 >> a6;
// cout<<"["<<a1<<","<<a2<<","<<a3<<","<<a4<<","<<a6<<"]"<<endl;
}
}
// puts out TeX-ed equation of curve
void Curve::tex_print(ostream &os) const
{
os << "$y^2" ;
if(a1==0){
;
} else {
if(a1==1) os << " + xy" ;
else if(a1==-1) os << " - xy" ;
else if(a1 > 0) os << " +" << a1 << "xy" ;
else os << " " << a1 << " xy" ;
}
if(a3==0){
;
} else {
if(a3==1) os << " + y" ;
else if(a3==-1) os << " - y" ;
else if(a3 > 0) os << " +" << a3 << "y" ;
else os << " " << a3 << " y" ;
}
os << " = x^3" ;
if(a2==0){
;
} else {
if(a2==1) os << " + x^2" ;
else if(a2==-1) os << " - x^2" ;
else if(a2 > 0) os << " +" << a2 << "x^2" ;
else os << " " << a2 << " x^2" ;
}
if(a4==0){
;
} else {
if(a4==1) os << " + x" ;
else if(a4==-1) os << " - x" ;
else if(a4 > 0) os << " +" << a4 << "x" ;
else os << " " << a4 << " x" ;
}
if(a6==0){
;
} else {
if(a6==1) os << " + 1" ;
else if(a6==-1) os << " - 1" ;
else if(a6 > 0) os << " +" << a6 ;
else os << " " << a6 ;
}
os << "$" ;
return ;
}
// end of file: curve.cc
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