modpol.cpp

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//
// Program to generate Modular Polynomials mod p, as required for fast
// implementations of the Schoof-Elkies-Atkins algorithm
// for counting points on an elliptic curve Y^2=X^3 + A.X + B mod p
//
// Implemented entirely from the description provided in:
// 1. "Distributed Computation of the number of points on an elliptic curve
//    over a finite prime field", Buchmann, Mueller, & Shoup, SFB 124-TP D5
//    Report 03/95, April 1995, Universitat des Saarlandes, and
// 2. "Counting the number of points on elliptic curves over finite fields
//    of characteristic greater than three", Lehmann, Maurer, Mueller & Shoup,
//    Proc. 1st Algorithmic Number Theory Symposium (ANTS), pp 60-70, 1994
//
// Both papers are available on the Web from Volker Mueller's home page
// www.informatik.th-darmstadt.de/TI/Mitarbeiter/vmueller.html
//
// Also strongly recommended is the book 
//
// 3. "Elliptic Curves in Cryptography"
//     by Blake, Seroussi and Smart, London Mathematical Society Lecture Note 
//     Series 265, Cambridge University Press. ISBN 0 521 65374 6 
//
// The programs's output for each prime in the range is a bivariate polynomial 
// in X and Y, which can optionally be stored to disk. Some informative output
// is generated just to convince you that it is still working, and to give an 
// idea of progress.
//
// This program is a composite of the "mueller" and "process" applications.
// It generates the modular polynomials, pre-reduced wrt to a specified prime
// modulus. This may be the only feasible way to do it on a small computer 
// system, for which the "mueller" application is too resource intensive. 
//
// Although less memory intensive than "mueller", problems may still arise.
// See mueller.cpp for a description of the -s2, -s3 and -s6 flags
//
// .pol file format 
// <modulus>,<prime>,<1st coef>,<1st power of X>,<1st power of Y>,<2nd coef>...
// Each polynomial ends wih zero powers of X and Y.
//
// For example
// modpol -d -f 2#512 0 500 -o test512.pol
//
// If appending to a file with the -a flag, make sure and use the same 
// prime modulus as used to create the file originally - no check is made
//
// generates the test512.pol file directly, given the range 0 to 500 and 
// using the first prime modulus it can find less than 2^512. This file can 
// then be used directly with the "sea" application
//
// Copyright Shamus Software Ltd., 1999 
//

#include <iostream>
#include <fstream>
#include <cstring>
#include <iomanip>
#include "ps_zzn.h"      // power series class

using namespace std;

extern int psN;          // power series are modulo x^psN

BOOL fout;
BOOL append;
Miracl precision=20;

ofstream mueller;

// Code to parse formula in command line
// This code isn't mine, but its public domain
// Shamefully I forget the source
//
// NOTE: It may be necessary on some platforms to change the operators * and #
//

#define TIMES '*'
#define RAISE '#'

Big tt;
static char *ss;

void eval_power (Big& oldn,Big& n,char op)
{
        if (op) n=pow(oldn,toint(n));    // power(oldn,size(n),n,n);
}

void eval_product (Big& oldn,Big& n,char op)
{
        switch (op)
        {
        case TIMES:
                n*=oldn; 
                break;
        case '/':
                n=oldn/n;
                break;
        case '%':
                n=oldn%n;
        }
}

void eval_sum (Big& oldn,Big& n,char op)
{
        switch (op)
        {
        case '+':
                n+=oldn;
                break;
        case '-':
                n=oldn-n;
        }
}

void eval (void)
{
        Big oldn[3];
        Big n;
        int i;
        char oldop[3];
        char op;
        char minus;
        for (i=0;i<3;i++)
        {
            oldop[i]=0;
        }
LOOP:
        while (*ss==' ')
        ss++;
        if (*ss=='-')    /* Unary minus */
        {
        ss++;
        minus=1;
        }
        else
        minus=0;
        while (*ss==' ')
        ss++;
        if (*ss=='(' || *ss=='[' || *ss=='{')    /* Number is subexpression */
        {
        ss++;
        eval ();
        n=tt;
        }
        else            /* Number is decimal value */
        {
        for (i=0;ss[i]>='0' && ss[i]<='9';i++)
                ;
        if (!i)         /* No digits found */
        {
                cout <<  "Error - invalid number" << endl;
                exit (20);
        }
        op=ss[i];
        ss[i]=0;
        n=atoi(ss);
        ss+=i;
        *ss=op;
        }
        if (minus) n=-n;
        do
        op=*ss++;
        while (op==' ');
        if (op==0 || op==')' || op==']' || op=='}')
        {
        eval_power (oldn[2],n,oldop[2]);
        eval_product (oldn[1],n,oldop[1]);
        eval_sum (oldn[0],n,oldop[0]);
        tt=n;
        return;
        }
        else
        {
        if (op==RAISE)
        {
                eval_power (oldn[2],n,oldop[2]);
                oldn[2]=n;
                oldop[2]=RAISE;
        }
        else
        {
                if (op==TIMES || op=='/' || op=='%')
                {
                eval_power (oldn[2],n,oldop[2]);
                oldop[2]=0;
                eval_product (oldn[1],n,oldop[1]);
                oldn[1]=n;
                oldop[1]=op;
                }
                else
                {
                if (op=='+' || op=='-')
                {
                        eval_power (oldn[2],n,oldop[2]);
                        oldop[2]=0;
                        eval_product (oldn[1],n,oldop[1]);
                        oldop[1]=0;
                        eval_sum (oldn[0],n,oldop[0]);
                        oldn[0]=n;
                        oldop[0]=op;
                }
                else    /* Error - invalid operator */
                {
                        cout <<  "Error - invalid operator" << endl;
                        exit (20);
                }
                }
        }
        }
        goto LOOP;
}

//
// When summing the Zk^n 0<=k<L (1. page 3, top), most terms cancel out,  
// leaving only every L-th term
//

Ps_ZZn phase(Ps_ZZn &z, int L,int off)
{ // Keep L times every L-th element in the Power Series
    Ps_ZZn w;
    term_ps_zzn *pos=NULL;
    int i,k;
    k=off+z.first();
    for (i=off;i<psN;i+=L,k+=L)
    {
        pos=w.addterm(L*z.coeff(k),k,pos);
    }
    return w;
}

void mueller_pol(int L,int s)
{ // Calculate Modular Polynomial for prime L 
  // s is smallest int such that s*(L-1)/12 is integer
    int i,j,n,v;
  
    Ps_ZZn klein,flt,zlt,x,y,z,f,jlt[500],c[1000],ps[1000];


// First calculate v, and hence psN - number of terms in Power Series
// 2. page 5 1st para

    cout << "preliminaries" << flush;

    v=s*(L-1)/12;
    psN=v+2;

//
// calculate Klein=j(tau) from its definition
// Numerator x... 
//
// 1. page 2
//
    for (n=1;n<psN;n++)
    {
        Ps_ZZn a,b,t;
        a.addterm((ZZn)n*n*n,n);     // a=n^3*x^n
        b.addterm((ZZn)1,0);
        b.addterm((ZZn)-1,n);
        t=a/b;
        x+=t;
    }
    x=(ZZn)240*x;
    x.addterm((ZZn)1,0);

    x=pow(x,3);
// Denominator y...
    y=eta();
    y=pow(y,24);

    klein=x/y;
    cout << "." << flush;
    klein.divxn(1);         // divides power series by x^parameter

    psN*=L;
//    cout << "psN= " << psN << endl;
    klein=power(klein,L);   // this substitutes x^L for x in the power series
    cout << "." << flush;

// Find Fl(t), Numerator z= Dedekind eta function 
// This has a simple repeating pattern of coefficients, and so costs nothing
// to calculate 1. page 2 bottom
    

    z=eta();
// Denominator y=n(Lt)...
    y=power(z,L);
    y=(ZZn)1/y;       // y has only psN/L terms.
    cout << "." << flush;
    z*=y;            // z has psN terms 
    flt=pow(z,2*s);   // ^2*s - expensive
    cout << "." << flush;
    flt.divxn(v);     // times x^-v

    ZZn w=pow((ZZn)L,s);
    y=power(flt,L);
    cout << "." << flush;
    zlt=w/y;          // l^s/Fl(lt) - cheap - psN/L terms in power series
    cout << "." << endl;
    y.clear();
    x.clear();

    ps[0]=L+1;
//
// Calculate Power Sums. Note that f and flt are very large objects
// with psN terms. Most other power series are in "compressed" form
// with "only" psN/L terms
//
// 1. page 3
//

    cout << "Power Sum   = " << flush;
    z=1;
    f=1;
    for (i=1;i<=L+1;i++)
    {
        cout << setw(3) << i << flush;
        f*=flt;    // expensive. In place multiplication discourages C++
                   // from moving large objects about
        z=z*zlt;   // cheap
        ps[i]=phase(f,L,(i*v)%L)+z;
        cout << "\b\b\b" << flush;
    }

    cout << setw(3) << L+1 << endl;

    f.clear();
    z.clear();
    flt.clear();
    zlt.clear();

    cout << "Coefficient = " << flush;
    c[0]=1;
//