mueller.cpp

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//
// Program to generate Modular Polynomials, 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.
//
// .raw file format 
// <prime>,<1st coef>,<1st power of X>,<1st power of Y>,<2nd coef>...
// Each polynomial ends wih zero powers of X and Y.
//
// NOTE: This program is very hard on memory. In places "obvious" speed
// optimizations have not been applied, if they are memory intensive. But in
// any case 64Mb is really a minimal requirement to generate enough 
// polynomials for serious work. 
//
// The time/memory requirements for a particular prime L depend on the value
// s, defined as the smallest integer such that s.(L-1) is divisible by 12.
// The value of s can be 1, 2, 3 or 6. The bigger s, the worse the time/memory
// requirements, and the bigger the coefficients in the polynomial. 
// The flags -s2, -s3, and -s6 cause the primes in the specified 
// range to be skipped if, for example, s>=3. 
//
// Four things can go wrong. An "Out of Space message means that you have run
// out of virtual memory. The "control panel" on your operating system should 
// enable you to fix this. A "Number too big" error means that the value
// specified in the first parameter of the call to mirsys() has proven to be 
// too small. This also means that you have pushed the program beyond the 
// limits for which we have tested it (well done!). If your hard disk 
// "trashes" continually, and processing slows to a crawl, you have run out of 
// physical memory. Buy more memory for your computer, or a new computer.
// Another problem may be I/O buffer overflow. Use set_io_buffer_size() to
// specify a larger I/O buffer
//
// With 96 Mb of RAM, what works for me (so far, running over a long weekend) 
// is
//
// mueller 0 180 -o mueller.raw
// mueller 180 300 -s6 -a mueller.raw
//
// and it should be possible to continue, for example, like
// 
// mueller 300 400 -s3 -a mueller.raw
// mueller 400 500 -s2 -a mueller.raw
//
// This creates a file mueller.raw containing modular polynomials 
// for 69 primes
//
// Of course different ranges can be covered simultaneously on multiple 
// computers, if you have them.
//
// Alternatively, if these problems should prove insurmountable - see the
// "modpol" application
//
// This program would be a lot faster if the coefficients were calculated mod
// many 32-bit primes, and then combined via the CRT.
//
// Copyright Shamus Software Ltd., 1999 
//

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

using namespace std;

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

BOOL fout;
ofstream mueller;

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

Ps_Big phase(Ps_Big &z, int L)
{ // Keep L times every L-th element in the Power Series
    Ps_Big w;
    term_ps_big *pos=NULL;
    int i,k,zf;

// k should be first coefficient a multiple of L 

    zf=z.first();
    if (zf%L==0) k=zf;
    else
    {
            k=(zf/L)*L;
            if (zf>=0) k+=L; 
    }

    for (;k<psN;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_Big 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 << "v= " << s*(L-1)/12 << endl;
    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_Big a,b,t;
        a.addterm(n*n*n,n);     // a=n^3*x^n
        b.addterm(1,0);
        b.addterm(-1,n);
        t=a/b;
        x+=t;
    }
    x=(Big)240*x;
    x.addterm(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=(Big)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
                      // This compensates for the missing q^(1/24) part of eta

    Big w=pow((Big)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 << "flt= " << flt << endl;

    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)+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;
//
// Newton's Identities - Calculate coefficients from Power Sums
//
// from a Web page somewhere and 3. page 54
//

    for (i=1;i<=L+1;i++)
    {
        cout << setw(3) << i << flush;
        c[i]=0;
        for (j=1;j<=i;j++)
              c[i]+=(ps[j]*c[i-j]);   // cheap, but lots of them

        c[i]=(-c[i])/i;
        cout << "\b\b\b" << flush;
    }

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

    for (i=0;i<=L+1;i++) ps[i].clear(); // reclaim space
//
// Get powers of j(Lt)^i, i=1 to v
// These will be needed to determine the exponent of Y in each
// coefficient of the Modular Polynomial
//
    jlt[0]=1;
    jlt[1]=klein;

    for (i=2;i<=v;i++)
        jlt[i]=jlt[i-1]*klein;    // cheap
//
// Find Modular Polynomial, format it, and output
//
// 2. page 5, middle "Hl(X) = ..."
//
    cout << "\nG" << L << "(X,Y) = X^" << L+1 ;

    if (fout) 
    {
        mueller << L << endl;
        mueller << 1 << "\n" << L+1 << "\n" << 0 << endl;
    }

    for (i=1;i<=L+1;i++)
    {
        Big cf;
        BOOL brackets,first;
        first=TRUE;
        brackets=FALSE;
        z=c[i];            

// idea is to reduce this to an integer
// by subtracting j(Lt)^k as necessary
// The power of k required is then
// the coefficient of Y^k in G(X,Y)
// The first coefficient of c[i] tells us which j(Lt)^k to try

        if (z.first()!=0) 
        {
            brackets=TRUE;
            cout << "+(" ;
        }
   // coefficient may be a polynomial in Y
        while (z.first()!=0)    
        {
            int j=(-z.first()/L);    // index into jlt
            cf=z.coeff(z.first());   // get coefficient to be cancelled
            if (fout) mueller << cf << "\n" << L+1-i << "\n" << j << endl;
            z-=(jlt[j]*cf);
            if (cf>0 && (!first || !brackets)) cout << "+";
            first=FALSE;
            if (cf==1) cout << "Y";
            if (cf==-1) cout << "-Y";
            if (abs(cf)!=1) cout << cf << "*Y";
            if (j!=1) cout << "^" << j;
        }
        cf=z.coeff(0);
        if (fout) mueller << cf << "\n" << L+1-i << "\n" << 0 << endl;

        if (cf>0) cout << "+";
        if (brackets) cout << cf << ")*X";
        else 
        {
            if (i==L+1) 
            {
                cout << cf;
                continue;
            }
            if (cf==1) cout << "X";
            if (cf==-1) cout << "-X";
            if (abs(cf)!=1) cout << cf << "*X";
        }

        if (i!=L) cout << "^" << L+1-i ;
  // all other coefficients should now be zero
        if (z.coeff(L)!=0) 
        { // check next coefficient is zero
            cout << "\n\n Sanity Check Failed " << endl;
            exit(0);
        }
    }
    for (i=0;i<=L+1;i++) c[i].clear(); // reclaim space
    for (i=0;i<=v;i++) jlt[i].clear();
    cout << endl;

    fft_reset();
}

int main(int argc,char **argv)
{
    miracl *mip;
    int i,j,s,lo,hi,p,ip,skip;
    int primes[200];
    BOOL gothi,gotlo;
    argv++; argc--;

    if (argc<1)
    {
        cout << "Incorrect usage" << endl;
        cout << "Program generates Modular Polynomials, for use by fast Schoof-Elkies-Atkins" << endl;
        cout << "program for counting points on an elliptic curve" << endl;
        cout << "mueller <low number> <high number>" << endl;
        cout << "where the numbers define a range. The program will attempt to" << endl;
        cout << "find the Modular Polynomials for all primes in this range" << endl;
        cout << "To output polynomials to a file use flag -o <filename>" << endl;
        cout << "e.g. mueller 0 10 -o mueller.raw" << endl;
        cout << "Alternatively to append to a file use flag -a <filename>" << endl;
        cout << "NOTE: Program is both memory and time intensive" << endl;
        cout << "To skip \"difficult\" primes, use -s2, -s3 or -s6" << endl;
        cout << "where -s2 skips most and -s6 skips least" << endl;
        cout << "See source code file for details" << endl;
        cout << "Files generated from different ranges may be combined in an" << endl;
        cout << "obvious way using a standard text editor" << endl;
        cout << "\nFreeware from Shamus Software, Dublin, Ireland" << endl;
        cout << "Full C++ source code and MIRACL multiprecision library available" << endl;
        cout << "http://indigo.ie/~mscott for details" << endl;
        cout << "or email [e-mail]" << endl;

        return 0;
    }
    if (argc<2)
    {
        cout << "Error in command line" << endl;
        return 0;
    }
    ip=0;
    skip=12;
    fout=FALSE;
    gothi=gotlo=FALSE;
    while (ip<argc)
    {
        if (!fout && strcmp(argv[ip],"-o")==0)
        {
            ip++;
            if (ip<argc)
            {
                fout=TRUE;
                mueller.open(argv[ip++]);
                continue;
            }
            else
            {
                cout << "Error in command line" << endl;
                return 0;
            }
        }
        if (!fout && strcmp(argv[ip],"-a")==0)
        {
            ip++;
            if (ip<argc)
            {
                fout=TRUE;
                mueller.open(argv[ip++],ios::app);
                continue;
            }
            else
            {
                cout << "Error in command line" << endl;
                return 0;
            }
        }
        if (skip==12 && strcmp(argv[ip],"-s2")==0)
        {
            ip++;
            skip=2;
            continue;
        }
        if (skip==12 && strcmp(argv[ip],"-s3")==0)
        {
            ip++;
            skip=3;
            continue;
        }
        if (skip==12 && strcmp(argv[ip],"-s6")==0)
        {
            ip++;
            skip=6;
            continue;
        }
        if (!gotlo)
        {
            lo=atoi(argv[ip++]);
            gotlo=TRUE;
            continue;
        }
        if (!gothi)
        {
            hi=atoi(argv[ip++]);
            gothi=TRUE;
            continue;
        }
        cout << "Error in command line" << endl;
        return 0;
    }
    if (!gothi || !gotlo)
    {
        cout << "Error in command line" << endl;
        return 0;
    }
    if (lo>hi || hi>1000)
    {
        cout << "Invalid range specified" << endl;
        return 0;
    }

    mip=mirsys(20,0);

    gprime(1000);         // get all primes < 1000
    for (i=0;;i++)
    {
        p=mip->PRIMES[i];
        primes[i]=p;
        if (p==0) break;
    }
    mirexit();

    for (j=0,i=1;;i++)       // lets go
    { 
        p=primes[i];
        if (p==0) break;
        if (p<lo) continue;
        if (p>hi) break;
        for (s=1;;s++)
            if (s*(p-1)%12==0) break;
        if (s>=skip) continue;

 // p*s/6 seems to be a fortuitous upper bound (?)
 // on the size of integer coefficients generated.
 // This MAY need to be increased if "number 
 // too big" errors accur.

        mirsys(1+(p*s)/6,0);  
        set_io_buffer_size(4096);
        j++;
        cout << "prime " << j << " = " << p << " (s=" << s << ")" << endl; 
        cout << 32*(1+(p*s)/6) << " bits reserved for each coefficient" << endl;
        mueller_pol(p,s); 
        mirexit();
    }
    cout << endl;
    if (j==0) cout << "No primes processed in the specified range" << endl;
    if (j==1) cout << "One prime processed in the specified range" << endl;
    if (j>1) cout << j << " primes processed in the specified range" << endl;
    return 0;   
}