longinteger_domain.cpp

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// longinteger_domain.cpp
//
// Anton Betten
//
// started as longinteger.cpp:  October 26, 2002
// moved here: January 23, 2015
 
 
 
#include "foundations.h"
 
using namespace std;
 
 
 
namespace orbiter {
namespace layer1_foundations {
namespace ring_theory {
 
 
 
int longinteger_domain::compare(
        longinteger_object &a, longinteger_object &b)
{
    int r;
    
    if (a.sign() != b.sign()) {
        if (a.sign())
            return -1;
        else
            return 1;
        }
    r = compare_unsigned(a, b);
    if (a.sign())
        return -1 * r;
    else
        return r;
}
 
int longinteger_domain::compare_unsigned(
        longinteger_object &a, longinteger_object &b)
// returns -1 if a < b, 0 if a = b,
// and 1 if a > b, treating a and b as unsigned.
{
    int i, l;
    char ai, bi;
    
    l = MAXIMUM(a.len(), b.len());
    for (i = l - 1; i >= 0; i--) {
        if (i < a.len())
            ai = a.rep()[i];
        else
            ai = 0;
        if (i < b.len())
            bi = b.rep()[i];
        else
            bi = 0;
        if (ai < bi)
            return -1;
        else if (ai > bi)
            return 1;
        }
    return 0;
}
 
int longinteger_domain::is_less_than(longinteger_object &a, longinteger_object &b)
{
    if (compare_unsigned(a, b) == -1)
        return TRUE;
    else
        return FALSE;
}
 
void longinteger_domain::subtract_signless(
        longinteger_object &a, longinteger_object &b,
        longinteger_object &c)
// c = a - b, assuming a > b
{
    int i;
    char ai, bi, carry;
    
    c.freeself();
    c.sign() = FALSE;
    c.len() = a.len();
    c.rep() = NEW_char(c.len());
    for (i = 0; i < c.len(); i++)
        c.rep()[i] = 0;
 
    carry = 0;
    for (i = 0; i < a.len(); i++) {
        if (i < b.len())
            bi = b.rep()[i];
        else
            bi = 0;
        bi += carry;
        ai = a.rep()[i];
        if (bi > ai) {
            ai += 10;
            carry = 1;
            }
        else
            carry = 0;
        c.rep()[i] = ai - bi;
        }
    c.normalize();
}
 
void longinteger_domain::subtract_signless_in_place(
        longinteger_object &a, longinteger_object &b)
// a := a - b, assuming a > b
{
    int i;
    char ai, bi, carry;
    
    carry = 0;
    for (i = 0; i < a.len(); i++) {
        if (i < b.len()) {
            bi = b.rep()[i];
        }
        else {
            bi = 0;
        }
        bi += carry;
        ai = a.rep()[i];
        if (bi > ai) {
            ai += 10;
            carry = 1;
        }
        else {
            carry = 0;
        }
        a.rep()[i] = ai - bi;
    }
    //c.normalize();
}
 
void longinteger_domain::add(
        longinteger_object &a, longinteger_object &b,
        longinteger_object &c)
{
    int cmp, carry, i, ai, bi, ci;
    
    c.freeself();
    c.len() = MAXIMUM(a.len(), b.len()) + 1;
    if ((a.sign() && b.sign()) || (!a.sign() && !b.sign())) {
        c.sign() = a.sign();
    }
    else {
        // mixed signs: subtraction 
        cmp = compare_unsigned(a, b);
        if (cmp < 0) {
            // |a| < |b|
            
            subtract_signless(b, a, c);
            c.sign() = b.sign();
            c.normalize();
            return;
        }
        else if (cmp > 0) {  
            // |a| > |b|
        
            subtract_signless(a, b, c);
            c.sign() = a.sign();
            c.normalize();
            return;
        }
        else {
            // |a| = |b|
            c.zero();
            return;
        }
    }
    c.rep() = NEW_char(c.len());
    for (i = 0; i < c.len(); i++) {
        c.rep()[i] = 0;
    }
 
    carry = 0;
    for (i = 0; i < c.len(); i++) {
        if (i < a.len()) {
            ai = a.rep()[i];
        }
        else {
            ai = 0;
        }
        if (i < b.len()) {
            bi = b.rep()[i];
        }
        else {
            bi = 0;
        }
        ci = ai + bi + carry;
        if (ci >= 10) {
            carry = 1;
        }
        else {
            carry = 0;
        }
        c.rep()[i] = ci % 10;
    }
    c.normalize();
}
 
void longinteger_domain::add_mod(longinteger_object &a,
    longinteger_object &b, longinteger_object &c,
    longinteger_object &m, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    longinteger_object d, q;
 
    if (f_v ) {
        cout << "longinteger_domain::add_mod "
                "a=" << a << " b=" << b << " m=" << m << endl;
    }
    add(a, b, d);
    integral_division(
        d, m,
        q, c,
        0 /*verbose_level*/);
    if (f_v ) {
        cout << "longinteger_domain::add_mod "
                "a=" << a << " +  b=" << b << " mod m=" << m << " is " << c << endl;
    }
 
}
 
void longinteger_domain::add_in_place(
        longinteger_object &a, longinteger_object &b)
// a := a + b
{
    longinteger_object C;
    
    add(a, b, C);
    C.assign_to(a);
}
 
void longinteger_domain::subtract_in_place(
        longinteger_object &a, longinteger_object &b)
// a := a - b
{
    longinteger_object c;
 
    if (is_less_than(a, b)) {
        subtract_signless(b, a, c);
        mult_integer_in_place(c, -1);
    }
    else {
        subtract_signless(a, b, c);
    }
    c.assign_to(a);
}
 
void longinteger_domain::add_int_in_place(
        longinteger_object &a, long int b)
// a := a + b
{
    longinteger_object C;
    longinteger_object B;
 
    B.create(b, __FILE__, __LINE__);
    add(a, B, C);
    C.assign_to(a);
}
 
 
void longinteger_domain::mult(
        longinteger_object &a, longinteger_object &b,
        longinteger_object &c)
{
    int i, j;
    char ai, bj, d, carry;
    int f_v = FALSE;
    
    if (a.is_zero() || b.is_zero()) {
        c.create(0, __FILE__, __LINE__);
        return;
    }
    if ((a.sign() && !b.sign()) || (!a.sign() && b.sign())) {
        c.sign() = TRUE;
    }
    else {
        c.sign() = FALSE;
    }
    
    c.freeself();
    c.len() = a.len() + b.len() + 2;
    c.rep() = NEW_char(c.len());
    for (i = 0; i < c.len(); i++) {
        c.rep()[i] = 0;
    }
    
    if (f_v) {
        cout << "longinteger_domain::mult a=";
        print_digits(a.rep(), a.len());
        cout << "b=";
        print_digits(b.rep(), b.len());
        cout << endl;
    }
    for (j = 0; j < b.len(); j++) {
        bj = b.rep()[j];
        carry = 0;
        for (i = 0; i < a.len(); i++) {
            ai = a.rep()[i];
            d = ai * bj + carry + c.rep()[i + j];
            if (d >= 100) {
                cout << "longinteger:mult error: d >= 100" << endl;
                exit(1);
            }
            carry = d / 10;
            c.rep()[i + j] = d % 10;
            if (f_v) {
                cout << "c[" << i + j << "]=" << d % 10 << "="
                        << (char)('0' + c.rep()[i + j]) << endl;
            }
        }
        if (carry) {
            c.rep()[j + a.len()] = carry;
            if (f_v) {
                cout << "c[" << j + a.len() << "]=" << carry << "="
                        << (char)('0' + carry) << endl;
            }
        }
    }
    if (f_v) {
        cout << "longinteger_domain::mult c=";
        print_digits(c.rep(), c.len());
        cout << endl;
    }
    c.normalize();
    if (f_v) {
        cout << "longinteger_domain::mult after normalize, c=";
        print_digits(c.rep(), c.len());
        cout << endl;
    }
}
 
void longinteger_domain::mult_in_place(
        longinteger_object &a, longinteger_object &b)
{
    longinteger_object C;
    
    mult(a, b, C);
    C.assign_to(a);
}
 
 
void longinteger_domain::mult_integer_in_place(
        longinteger_object &a, int b)
{
    longinteger_object B, C;
    
    B.create(b, __FILE__, __LINE__);
    mult(a, B, C);
    C.assign_to(a);
}
 
static int do_division(longinteger_domain &D,
        longinteger_object &r, longinteger_object table[10])
{
    int i, cmp;
    
    for (i = 9; i >= 0; i--) {
        cmp = D.compare(r, table[i]);
        if (cmp >= 0) {
            return i;
        }
    }
    cout << "do_division we should never reach this point" << endl;
    exit(1);
}
 
void longinteger_domain::mult_mod(longinteger_object &a, 
    longinteger_object &b, longinteger_object &c, 
    longinteger_object &m, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    int f_vv = (verbose_level >= 2);
    int i, j, l;
    char ai, bj, d, carry;
    //longinteger_object table[10], cc;
    longinteger_object q, r, c0;
    
    if (f_v ) {
        cout << "longinteger_domain::mult_mod "
                "a=" << a << " b=" << b << " m=" << m << endl;
    }
    if (a.is_zero() || b.is_zero()) {
        c.create(0, __FILE__, __LINE__);
        return;
    }
    if (compare_unsigned(a, m) >= 0) {
        cout << a << " * " << b << endl;
        cout << "longinteger_domain::mult_mod a >= m" << endl;
        exit(1);
    }
    if (compare_unsigned(b, m) >= 0) {
        cout << a << " * " << b << endl;
        cout << "longinteger_domain::mult_mod b >= m" << endl;
        exit(1);
    }
    if ((a.sign() && !b.sign()) || (!a.sign() && b.sign())) {
        c.sign() = TRUE;
    }
    else {
        c.sign() = FALSE;
    }
 
#if 0
    for (i = 0; i < 10; i++) {
        cc.create(i);
        mult(m, cc, table[i]);
    }
#endif
    
    if (f_v) {
        cout << "longinteger_domain::mult_mod "
                "calling c.freeself" << endl;
    }
    c.freeself();
    if (f_v) {
        cout << "longinteger_domain::mult_mod"
                "after c.freeself" << endl;
    }
    l = m.len();
    c.len() = (int) (l + 2);
    c.rep() = NEW_char(c.len());
    for (i = 0; i < c.len(); i++) {
        c.rep()[i] = 0;
    }
    c.assign_to(c0);
    
    for (j = b.len() - 1; j >= 0; j--) {
        if (f_vv) {
            cout << "j=" << j << endl;
        }
        bj = b.rep()[j];
        carry = 0;
        for (i = 0; i < l; i++) {
            if (i < a.len()) {
                ai = a.rep()[i];
            }
            else {
                ai = 0;
            }
            d = ai * bj + carry + c.rep()[i];
            //cout << (int) ai << " * " << (int) bj << " + "
            // << (int)carry << " + " << (int) c.rep()[i]
            // << " = " << (int)d << endl;
            if (d >= 100) {
                cout << "longinteger:mult_mod error: d >= 100" << endl;
                exit(1);
            }
            carry = d / 10;
            c.rep()[i] = d % 10;
        }
        if (carry) {
            d = c.rep()[i] + carry;
            carry = d / 10;
            c.rep()[i] = d % 10;
            i++;
        }
        if (carry) {
            d = c.rep()[i] + carry;
            carry = d / 10;
            c.rep()[i] = d % 10;
            if (carry) {
                cout << "longinteger_domain::mult_mod "
                        "error: carry" << endl;
                exit(1);
            }
        }
        if (f_vv) {
            cout << "longinteger_domain::mult_mod "
                    "c=" << c << " len " << c.len() << endl;
        }
        integral_division(c, m, q, r, 0/*verbose_level - 1*/);
        //h = do_division(*this, c, table);
        //subtract_signless_in_place(c, table[h]);
        if (f_vv) {
            cout << "longinteger_domain::mult_mod "
                    "r=" << r << " len " << r.len() << endl;
        }
        c0.assign_to(c);
        if (f_vv) {
            cout << "longinteger_domain::mult_mod"
                    "c=c0=" << c << " len " << c.len() << endl;
        }
        if (j) {
            c.rep()[0] = 0;
            for (i = 0; i < r.len(); i++) {
                if (i + 1 < c.len()) {
                    c.rep()[i + 1] = r.rep()[i];
                }
            }
            i++;
            for (; i < c.len(); i++) {
                c.rep()[i] = 0;
            }
        }
        else {
            for (i = 0; i < r.len(); i++) {
                if (i < c.len()) {
                    c.rep()[i] = r.rep()[i];
                }
            }
            for (; i < c.len(); i++) {
                c.rep()[i] = 0;
            }
        }
        if (f_vv) {
            cout << "c=" << c << " len " << c.len() << endl;
        }
    }
    c.normalize();
    if (f_v) {
        cout << "longinteger_domain::mult_mod " << a << " * " << b
                << " = " << c << " mod " << m << endl;
    }
}
 
void longinteger_domain::multiply_up(
        longinteger_object &a, int *x, int len, int verbose_level)
{
    longinteger_object b, c;
    int i;
    int f_v = (verbose_level >= 1);
 
    if (f_v) {
        cout << "longinteger_domain::multiply_up" << endl;
    }
    a.one();
    if (f_v) {
        cout << "longinteger_domain::multiply_up "
                "a=" << a << endl;
    }
    for (i = 0; i < len; i++) {
        if (x[i] == 1) {
            continue;
        }
        b.create(x[i], __FILE__, __LINE__);
        if (f_v) {
            cout << "longinteger_domain::multiply_up "
                    "i=" << i << " x[i]=" << x[i]
                    << " b=" << b << endl;
        }
        mult(a, b, c);
        if (f_v) {
            cout << "longinteger_domain::multiply_up "
                    "i=" << i << " x[i]=" << x[i]
                    << " c=" << c << endl;
        }
        c.assign_to(a);
        if (f_v) {
            cout << "longinteger_domain::multiply_up "
                    "i=" << i << " x[i]=" << x[i]
                    << " a=" << a << endl;
        }
        //cout << "*" << x[i] << "=" << a << endl;
    }
    if (f_v) {
        cout << "longinteger_domain::multiply_up done" << endl;
    }
}
 
void longinteger_domain::multiply_up_lint(
        longinteger_object &a, long int *x, int len, int verbose_level)
{
    longinteger_object b, c;
    int i;
    int f_v = (verbose_level >= 1);
 
    if (f_v) {
        cout << "longinteger_domain::multiply_up_lint" << endl;
    }
    a.one();
    if (f_v) {
        cout << "longinteger_domain::multiply_up_lint "
                "a=" << a << endl;
    }
    for (i = 0; i < len; i++) {
        if (x[i] == 1) {
            continue;
        }
        b.create(x[i], __FILE__, __LINE__);
        if (f_v) {
            cout << "longinteger_domain::multiply_up_lint "
                    "i=" << i << " x[i]=" << x[i]
                    << " b=" << b << endl;
        }
        mult(a, b, c);
        if (f_v) {
            cout << "longinteger_domain::multiply_up_lint "
                    "i=" << i << " x[i]=" << x[i]
                    << " c=" << c << endl;
        }
        c.assign_to(a);
        if (f_v) {
            cout << "longinteger_domain::multiply_up_lint "
                    "i=" << i << " x[i]=" << x[i]
                    << " a=" << a << endl;
        }
        //cout << "*" << x[i] << "=" << a << endl;
    }
    if (f_v) {
        cout << "longinteger_domain::multiply_up_lint done" << endl;
    }
}
 
int longinteger_domain::quotient_as_int(
        longinteger_object &a, longinteger_object &b)
{
    longinteger_object q, r;
 
    integral_division(a, b, q, r, 0);
    return q.as_int();
}
 
long int longinteger_domain::quotient_as_lint(
        longinteger_object &a, longinteger_object &b)
{
    longinteger_object q, r;
 
    integral_division(a, b, q, r, 0);
    return q.as_lint();
}
 
void longinteger_domain::integral_division_exact(
        longinteger_object &a, longinteger_object &b,
        longinteger_object &a_over_b)
{
    longinteger_object r;
 
    integral_division(a, b, a_over_b, r, 0);
    if (!r.is_zero()) {
        cout << "longinteger_domain::integral_division_exact "
                "b does not divide a" << endl;
        exit(1);
    }
}
 
void longinteger_domain::integral_division(
    longinteger_object &a, longinteger_object &b, 
    longinteger_object &q, longinteger_object &r,
    int verbose_level)
{
    int f_v = (verbose_level >= 1);
    longinteger_object table[10], c;
    int i, l, ql;
    
    if (f_v) {
        cout << "longinteger_domain::integral_division "
                "dividing a=" << a << " by b=" << b << endl;
    }
    if (a.sign()) {
        cout << "longinteger_domain::integral_division "
                "a is negative" << endl;
        exit(1);
    }
    if (b.sign()) {
        cout << "longinteger_domain::integral_division "
                "b is negative" << endl;
        exit(1);
    }
    if (compare(a, b) == -1) {
        q.zero();
        a.assign_to(r);
        return;
    }
    
    for (i = 0; i < 10; i++) {
        c.create(i, __FILE__, __LINE__);
        mult(b, c, table[i]);
    }
    q.freeself();
    r.freeself();
    
    // load r with leading b.len() digits of a: 
    r.sign() = FALSE;
    r.len() = b.len() + 1;
    r.rep() = NEW_char(r.len());
    l = a.len() - b.len();
    for (i = 0; i < b.len(); i++) {
        r.rep()[i] = a.rep()[l + i];
    }
    r.rep()[b.len()] = 0;
    
    // allocate q of length a.len() - b.len() + 1
    q.sign() = FALSE;
    q.len() = a.len() - b.len() + 1;
    q.rep() = NEW_char(q.len());
    for (i = 0; i < q.len(); i++) {
        q.rep()[i] = 0;
    }
    
    // main loop containing the divisions:
    for ( ; l >= 0; l--) {
        ql = do_division(*this, r, table);
        q.rep()[l] = ql;
        subtract_signless(r, table[ql], c);
        if (f_v) {
            cout << "l=" << l << " r=" << r
                    << " ql=" << ql << " c=" << c << endl;
        }
        
        if (l == 0) {
            break;
        }
        
        // put c into r, shift up by one digit 
        // and append the next digit a[l-1].
        for (i = 0; i < c.len(); i++) {
            r.rep()[i + 1] = c.rep()[i];
        }
        for (i++ ; i < r.len(); i++) {
            r.rep()[i] = 0;
        }
        r.rep()[0] = a.rep()[l - 1];
    }
    c.assign_to(r);
    r.normalize();
    q.normalize();
}
 
void longinteger_domain::integral_division_by_int(
    longinteger_object &a,
    int b, longinteger_object &q, int &r)
{
    longinteger_object B, R;
    int verbose_level = 0;
 
    B.create(b, __FILE__, __LINE__);
    integral_division(a, B, q, R, verbose_level);
    r = R.as_int();
}
 
void longinteger_domain::integral_division_by_lint(
    longinteger_object &a,
    long int b, longinteger_object &q, long int &r)
{
    longinteger_object B, R;
    int verbose_level = 0;
 
    B.create(b, __FILE__, __LINE__);
    integral_division(a, B, q, R, verbose_level);
    r = R.as_lint();
}
 
void longinteger_domain::extended_gcd(
    longinteger_object &a,
    longinteger_object &b, longinteger_object &g, 
    longinteger_object &u, longinteger_object &v, int verbose_level)
// the gcd computed here is always nonnegative
{
    int f_v = (verbose_level >= 1);
    int f_vv = (verbose_level >= 2);
    longinteger_object q, rm1, r, rp1, sm1, s, sp1, tm1, t, tp1, x, y;
    int c;
    
    if (f_v) {
        cout << "longinteger_domain::extended_gcd "
                "a=" << a << " b=" << b << endl;
    }
    if (a.is_zero()) {
        b.assign_to(g);
        u.create(0, __FILE__, __LINE__);
        v.create(1, __FILE__, __LINE__);
        if (g.sign()) {
            g.negate();
            u.negate();
            v.negate();
        }
        return;
    }
    if (b.is_zero()) {
        a.assign_to(g);
        u.create(1, __FILE__, __LINE__);
        v.create(0, __FILE__, __LINE__);
        if (g.sign()) {
            g.negate();
            u.negate();
            v.negate();
        }
        return;
    }
    c = compare_unsigned(a, b);
    if (c < 0) { // |a| < |b|
        extended_gcd(b, a, g, v, u, verbose_level);
        return;
    }
    if (a.sign()) {
        a.negate();
        extended_gcd(a, b, g, u, v, verbose_level);
        a.negate();
        u.negate();
        if (g.sign()) {
            g.negate();
            u.negate();
            v.negate();
        }
        return;
    }
    if (b.sign()) {
        b.negate();
        extended_gcd(a, b, g, u, v, verbose_level);
        b.negate();
        v.negate();
        if (g.sign()) {
            g.negate();
            u.negate();
            v.negate();
        }
        return;
    }
 
    // now a > 0, b > 0 and a >= b
    a.assign_to(rm1);
    b.assign_to(r);
    sm1.create(1, __FILE__, __LINE__);
    tm1.create(0, __FILE__, __LINE__);
    s.create(0, __FILE__, __LINE__);
    t.create(1, __FILE__, __LINE__);
    while (TRUE) {
        integral_division(rm1, r, q, rp1, verbose_level - 1);
        if (rp1.is_zero()) {
            r.assign_to(g);
            s.assign_to(u);
            t.assign_to(v);
            return;
        }
        if (f_vv) {
            rm1.print(cout);
            cout << " = ";
            q.print(cout);
            cout << " * ";
            r.print(cout);
            cout << " + ";
            rp1.print(cout);
            cout << endl;
        }
        q.negate();
        mult(q, s, x);
        mult(q, t, y);
        add(sm1, x, sp1);
        add(tm1, y, tp1);
        r.swap_with(rm1);
        s.swap_with(sm1);
        t.swap_with(tm1);
        rp1.swap_with(r);
        sp1.swap_with(s);
        tp1.swap_with(t);
        if (f_vv) {
            r.print(cout);
            cout << " = ";
            s.print(cout);
            cout << " * ";
            a.print(cout);
            cout << " + ";
            t.print(cout);
            cout << " * ";
            b.print(cout);
            cout << endl;
        }
    }
}
 
int longinteger_domain::logarithm_base_b(
        longinteger_object &a, int b)
{
    int r, l = 0;
    longinteger_object a1, a2;
    
    a.assign_to(a1);
    a1.normalize();
    while (!a1.is_zero()) {
        integral_division_by_int(a1, b, a2, r);
        l++;
        a2.assign_to(a1);
    }
    return l;
}
 
void longinteger_domain::base_b_representation(
        longinteger_object &a, int b, int *&rep, int &len)
{
    int i, r;
    longinteger_object a1, a2;
    
    a.assign_to(a1);
    a1.normalize();
    len = 0;
    while (!a1.is_zero()) {
        integral_division_by_int(a1, b, a2, r);
        len++;
        a2.assign_to(a1);
    }
    a.assign_to(a1);
    a1.normalize();
    rep = NEW_int(len);
    for (i = 0; i < len; i++) {
        integral_division_by_int(a1, b, a2, r);
        rep[i] = r;
        a2.assign_to(a1);
    }
    for (i = 0; i < len; i++) {
        cout << rep[i] << " ";
    }
    cout << endl;
}
 
void longinteger_domain::power_int(
        longinteger_object &a, int n)
{
    longinteger_object b, c, d;
    
    a.assign_to(b);
    c.one();
    while (n) {
        if (n % 2) {
            mult(b, c, d);
            d.assign_to(c);
        }
        mult(b, b, d);
        d.assign_to(b);
        n >>= 1;
    }
    c.assign_to(a);
}
 
void longinteger_domain::power_int_mod(
        longinteger_object &a, int n, longinteger_object &m)
{
    longinteger_object b, c, d;
    
    a.assign_to(b);
    c.one();
    while (n) {
        if (n % 2) {
            mult_mod(b, c, d, m, 0);
            d.assign_to(c);
        }
        mult_mod(b, b, d, m, 0);
        d.assign_to(b);
        n >>= 1;
    }
    c.assign_to(a);
}
 
void longinteger_domain::power_longint_mod(
    longinteger_object &a,
    longinteger_object &n, longinteger_object &m,
    int verbose_level)
{
    int f_v = (verbose_level >= 1);
    int f_vv = (verbose_level >= 2);
    longinteger_object b, c, d, n1, N;
    int r;
    
    a.assign_to(b);
    n.assign_to(N);
    c.one();
    if (f_v) {
        cout << "longinteger_domain::power_longint_mod" << endl;
        cout << "computing " << b << " to the power "
                << n << " mod " << m << ":" << endl;
    }
    while (!N.is_zero()) {
        if (f_vv) {
            cout << "n=" << N << " : " << b << "^"
                    << N << " * " << c << endl;
        }
        integral_division_by_int(N, 2, n1, r);
        if (f_vv) {
            cout << "after division by 2, n1=" << n1
                    << " r=" << r << endl;
        }
        if (r) {
            mult_mod(b, c, d, m, verbose_level);
            if (f_vv) {
                cout << b << " * " << c << " = " << d << endl;
            }
            d.assign_to(c);
        }
        mult_mod(b, b, d, m, verbose_level);
        if (f_vv) {
            cout << b << "^2 = " << d << endl;
        }
        d.assign_to(b);
        n1.assign_to(N);
    }
    c.assign_to(a);
}
 
 
 
 
 
 
void longinteger_domain::square_root(
        longinteger_object &a, longinteger_object &sqrt_a,
        int verbose_level)
{
    int i, la, len;
    char c1;
    int f_v = (verbose_level >= 1);
    longinteger_object a2;
 
    if (a.is_zero()) {
        sqrt_a.create(0, __FILE__, __LINE__);
        return;
    }
    if (a.sign()) {
        cout << "longinteger_domain::square_root a is negative" << endl;
        exit(1);
    }
 
    la = a.len();
    if (ODD(la)) {
        la++;
    }
    len = (la >> 1) + 1;
 
    sqrt_a.freeself();
    sqrt_a.len() = len;
    sqrt_a.rep() = NEW_char(sqrt_a.len());
    for (i = 0; i < sqrt_a.len(); i++) {
        sqrt_a.rep()[i] = 0;
    }
 
    if (f_v) {
        cout << "longinteger_domain::square_root a=";
        print_digits(a.rep(), a.len());
    }
 
    for (i = len - 1; i >= 0; i--) {
        for (c1 = 9; c1 >= 0; c1--) {
            if (f_v) {
                cout << "trying a[" << i << "]=" << (int) c1 << endl;
            }
            sqrt_a.rep()[i] = c1;
            if (f_v) {
                cout << "sqrt_a = " << sqrt_a << endl;
            }
            if (c1) {
                mult(sqrt_a, sqrt_a, a2);
                if (f_v) {
                    cout << "a2 = " << a2 << endl;
                }
                if (compare_unsigned(a2, a) <= 0) {
                    if (f_v) {
                        cout << "success with digit " << i
                                << " : sqrt_a=" << sqrt_a << endl;
                    }
                    break;
                }
            }
        }
        if (f_v) {
            cout << "sqrt_a[" << i << "]=" << sqrt_a.rep()[i] << endl;
        }
    }
 
    if (f_v) {
        cout << "longinteger_domain::square_root c=";
        print_digits(sqrt_a.rep(), sqrt_a.len());
        cout << endl;
    }
    sqrt_a.normalize();
    if (f_v) {
        cout << "longinteger_domain::square_root after normalize, sqrt_a=";
        print_digits(sqrt_a.rep(), sqrt_a.len());
        cout << endl;
    }
}
 
 
 
int longinteger_domain::square_root_mod(int a, int p, int verbose_level)
// solves x^2 = a mod p. Returns x
{
    int f_v = (verbose_level >= 1);
    longinteger_object P;
    longinteger_object A, X, a2, a4, b, X2, Four, Two, mOne;
    int round;
    number_theory::number_theory_domain NT;
 
    if (f_v) {
        cout << "longinteger_domain::square_root_mod" << endl;
    }
    a = a % p;
    if (NT.Jacobi(a, p, 0 /* verbose_level*/) == -1) {
        cout << "longinteger_domain::square_root_mod a is not a square modulo p" << endl;
        cout << "a=" << a << endl;
        cout << "p=" << p << endl;
        exit(1);
    }
    Two.create(2, __FILE__, __LINE__);
    Four.create(4, __FILE__, __LINE__);
    A.create(a, __FILE__, __LINE__);
    P.create(p, __FILE__, __LINE__);
    mOne.create(p - 1, __FILE__, __LINE__);
    if (f_v) {
        cout << "longinteger_domain::square_root_mod A=" << A << endl;
        cout << "longinteger_domain::square_root_mod P=" << P << endl;
        cout << "longinteger_domain::square_root_mod mOne=" << mOne << endl;
    }
    if (p % 4 == 3) {
        A.assign_to(X);
        power_int_mod(X, (p + 1) >> 2, P);
        return X.as_int();
    }
    if (p % 8 == 5) {
        if (f_v) {
            cout << "longinteger_domain::square_root_mod p % 8 == 5" << endl;
        }
        A.assign_to(b);
        power_int_mod(b, (p - 1) >> 2, P);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod a^((p-1)/4)=" << b << endl;
        }
        // cout << "A = " << A << endl;
        // cout << "b = A^(p-1)/4=" << b << endl;
        if (b.is_one()) {
            if (f_v) {
                cout << "longinteger_domain::square_root_mod a^((p-1)/4)=1" << endl;
            }
            A.assign_to(X);
            power_int_mod(X, (p + 3) >> 3, P);
            if (f_v) {
                cout << "longinteger_domain::square_root_mod done" << endl;
            }
            return X.as_int();
            }
        if (compare_unsigned(b, mOne) == 0) {
            if (f_v) {
                cout << "longinteger_domain::square_root_mod a^((p-1)/4)=1" << endl;
            }
 
            //a2.add_mod(A, A, P);
            //a4.add_mod(a2, a2, P);
            //a4.power_int_mod((p - 5) >> 3, P);
            //X.mult_mod(a2, a4, P);
 
            add_mod(A, A, a2, P, 0 /* verbose_level */);
            add_mod(a2, a2, a4, P, 0 /* verbose_level */);
            power_int_mod(a4, (p - 5) >> 3, P);
            mult_mod(a2, a4, X, P, 0 /* verbose_level */);
            if (f_v) {
                cout << "longinteger_domain::square_root_mod done" << endl;
            }
            return X.as_int();
            }
        else {
            cout << "longinteger_domain::square_root_mod p % 8 = 5 and power neq +-1" << endl;
            cout << "power = " << b << endl;
            exit(1);
        }
    }
    if (f_v) {
        cout << "longinteger_domain::square_root_mod p % 8 == 1" << endl;
    }
    // now p % 8 == 1
    // Tonelli / Shanks algorithm:
    int n, r = 0, q, e, m;
    longinteger_object Z, N, Y, B, T, d, AB, Ypower, Bpower, Tmp1;
 
    if (f_v) {
        cout << "longinteger_domain::square_root_mod, Tonelli / Shanks:" << endl;
    }
    q = p - 1;
    if (f_v) {
        cout << "longinteger_domain::square_root_mod q=" << q << endl;
    }
    e = 0;
    while (EVEN(q)) {
        q >>= 1;
        e++;
    }
    if (f_v) {
        cout << "p - 1 = 2^" << e << " * " << q << endl;
    }
 
    // pick n to be a nonsquare mod p:
    for (n = 1; n < p - 1; n++) {
        r = NT.Legendre(n, p, verbose_level - 1);
        if (r == -1) {
            break;
        }
    }
 
    if (f_v) {
        cout << "n=" << n << " p=" << p << " Legendre(n,p)=" << r<< endl;
    }
    N.create(n, __FILE__, __LINE__);
    if (f_v) {
        cout << "longinteger_domain::square_root_mod N=" << N << endl;
    }
    N.assign_to(Z);
    power_int_mod(Z, q, P);
    Z.assign_to(Y);
    if (f_v) {
        cout << "longinteger_domain::square_root_mod Y=N^q=" << Y << endl;
    }
    r = e;
    A.assign_to(X);
    power_int_mod(X, (q - 1) >> 1, P);
    if (f_v) {
        cout << "longinteger_domain::square_root_mod X=" << X << endl;
    }
    mult_mod(X, X, d, P, 0 /* verbose_level */);
    mult_mod(A, d, B, P, 0 /* verbose_level */);
    mult_mod(A, X, Tmp1, P, 0 /* verbose_level */);
    Tmp1.assign_to(X);
    if (f_v) {
        cout << "initialization:" << endl;
    }
    round = 0;
    while (TRUE) {
        if (f_v) {
            cout << "Y=" << Y << endl;
            cout << "r=" << r << endl;
            cout << "X=" << X << endl;
            cout << "B=" << B << endl;
        }
 
 
        mult_mod(X, X, X2, P, 0 /* verbose_level */);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod X2=" << X2 << endl;
        }
        mult_mod(A, B, AB, P, 0 /* verbose_level */);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod AB=" << AB << endl;
            cout << "B=" << B << endl;
        }
 
        if (compare_unsigned(AB, X2) != 0) {
            cout << "loop invariant violated: ab != x^2" << endl;
            cout << "ab=" << d << endl;
            cout << "x^2=" << X2 << endl;
            exit(1);
        }
 
        Y.assign_to(Ypower);
        power_int_mod(Ypower, 1 << (r - 1), P);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod Ypower=" << Ypower << endl;
        }
 
        if (compare_unsigned(Ypower, mOne)) {
            cout << "loop invariant violated: Y^{2^{r-1}} != -1" << endl;
            exit(1);
        }
 
 
 
        B.assign_to(Bpower);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod Bpower (before)=" << Bpower << endl;
        }
        power_int_mod(Bpower, 1 << (r - 1), P);
        if (f_v) {
            cout << "longinteger_domain::square_root_mod Bpower=" << Bpower << endl;
        }
        if (!Bpower.is_one()) {
            cout << "loop invariant violated: B^{2^{r-1}} != 1" << endl;
            exit(1);
        }
 
 
 
 
 
        if (remainder_mod_int(B, p) == 1) {
            m = -1;
        }
        else {
            for (m = 1; ; m++) {
                B.assign_to(d);
                power_int_mod(d, 1 << m, P);
                if (d.is_one()) {
                    break;
                }
                if (m >= r) {
                    cout << "sqrt_mod(), Tonelli / Shanks:" << endl;
                    cout << "error: a is not a quadratic residue mod p" << endl;
                    exit(1);
                }
            }
        }
 
 
        if (f_v) {
            cout << round << " & " << A << " & " << B << " & " << X << " & "
                << X2 << " & " << Y << " & " << r << " & " << AB
                << " & " << Ypower << " & " << Bpower << " & ";
        }
 
        if (m == -1) {
            if (f_v) {
                cout << " & & & & \\\\" << endl;
            }
        }
        else {
            if (f_v) {
                cout << m;
            }
        }
 
        //cout << "m=" << m << endl;
 
        if (m == -1) {
            if (f_v) {
                cout << "longinteger_domain::square_root_mod done" << endl;
            }
            return X.as_int();
        }
 
        if (f_v) {
            cout << "m=" << m << endl;
        }
        Y.assign_to(T);
        power_int_mod(T, 1 << (r - m - 1), P);
        mult_mod(T, T, Y, P, 0 /* verbose_level */);
        r = m; // integers
 
        mult_mod(X, T, Tmp1, P, 0 /* verbose_level */);
        Tmp1.assign_to(X);
 
        mult_mod(B, Y, Tmp1, P, 0 /* verbose_level */);
        Tmp1.assign_to(B);
 
        if (f_v) {
            cout << " & " << Y << " & " << X << " & " << B << " & " << r;
            cout << "\\\\" << endl;
        }
        round++;
        }
    if (f_v) {
        cout << "longinteger_domain::square_root_mod done" << endl;
    }
}
 
void longinteger_domain::create_q_to_the_n(
        longinteger_object &a, int q, int n)
// create (q^n - 1)
{
    a.create(q, __FILE__, __LINE__);
    power_int(a, n);
}
 
 
void longinteger_domain::create_qnm1(
        longinteger_object &a, int q, int n)
// create (q^n - 1)
{
    longinteger_object b, c;
    
    b.create(q, __FILE__, __LINE__);
    power_int(b, n);
    c.create(-1, __FILE__, __LINE__);
    add(b, c, a);
}
 
void longinteger_domain::create_Mersenne(longinteger_object &M, int n)
// $M_n = 2^n - 1$
{
    longinteger_object a, b;
 
    a.create(2, __FILE__, __LINE__);
    b.create(-1, __FILE__, __LINE__);
    power_int(a, n);
    add(a, b, M);
    // cout << "Mersenne number M_" << n << "=" << a << endl;
}
 
void longinteger_domain::create_Fermat(longinteger_object &F, int n)
// $F_n = 2^{2^n} + 1$
{
    longinteger_object a, b;
    int l;
 
    a.create(2, __FILE__, __LINE__);
    b.create(1, __FILE__, __LINE__);
    l = 1 << n;
    // cout << "l=" << l << endl;
    power_int(a, l);
    add(a, b, F);
    // cout << "Fermat number F_" << n << "=" << a << endl;
}
 
 
 
void longinteger_domain::Dedekind_number(longinteger_object &Dnq, int n, int q, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    number_theory::number_theory_domain NT;
    geometry::geometry_global Gg;
    longinteger_object A, S, B;
 
    if (f_v) {
        cout << "longinteger_domain::Dedekind_number n=" << n << " q=" << q << endl;
    }
    int *primes;
    int *exponents;
    int len, cnt;
    long int i, N, h, d, d_prime;
    int *v;
 
    len = NT.factor_int(n, primes, exponents);
 
    v = NEW_int(len);
 
    S.create(0, __FILE__, __LINE__);
 
    N = NT.i_power_j(2, len);
    if (f_v) {
        cout << "\\frac{1}{" << n << "} \\Big( ";
    }
 
    int f_first = TRUE;
    for (i = 0; i < N; i++) {
        if (FALSE) {
            cout << "i=" << i << endl;
        }
        Gg.AG_element_unrank(2, v, 1, len, i);
        d = 1;
        cnt = 0;
        for (h = 0; h < len; h++) {
            if (v[h]) {
                d *= primes[h];
                cnt++;
            }
        }
        d_prime = n / d;
        //a = NT.i_power_j(q, d_prime);
        create_q_to_the_n(A, q, d_prime);
 
        if (FALSE) {
            cout << "d=" << d << " d_prime=" << d_prime << " A=" << A << " S=" << S << endl;
        }
 
        if (ODD(cnt)) {
            if (f_v) {
                cout << " - ";
            }
            subtract_in_place(S, A);
        }
        else {
            if (f_v) {
                if (!f_first) {
                    cout << " + ";
                }
            }
            add_in_place(S, A);
        }
        if (f_first) {
            f_first = FALSE;
        }
        if (f_v) {
            cout << "q^{" << d_prime << "}";
        }
    }
    if (f_v) {
        cout << "\\Big) ";
    }
    if (f_v) {
        cout << endl;
    }
    B.create(n, __FILE__, __LINE__);
 
    if (f_v) {
        cout << "S=" << S << endl;
    }
    integral_division_exact(S, B, Dnq);
 
 
 
    FREE_int(v);
    FREE_int(primes);
    FREE_int(exponents);
 
 
    if (f_v) {
        cout << "longinteger_domain::Dedekind_number done" << endl;
    }
}
 
 
 
int longinteger_domain::is_even(longinteger_object &a)
{
    if (((a.rep()[0] % 2)) == 0)
        return TRUE;
    else
        return FALSE;
}
 
int longinteger_domain::is_odd(longinteger_object &a)
{
    if (is_even(a))
        return FALSE;
    else
        return TRUE;
}
 
 
 
int longinteger_domain::remainder_mod_int(longinteger_object &a, int p)
{
    int r;
    longinteger_object q;
    
    integral_division_by_int(a, p, q, r);
    return r;
}
 
int longinteger_domain::multiplicity_of_p(longinteger_object &a, 
    longinteger_object &residue, int p)
{
    int r, n = 0;
    longinteger_object q;
    
    if (a.is_zero()) {
        cout << "longinteger_domain::multiplicity_of_p a = 0" << endl;
        exit(1);
    }
    a.assign_to(residue);
    while (!residue.is_one()) {
        integral_division_by_int(residue, p, q, r);
        if (r)
            break;
        n++;
        q.assign_to(residue);
    }
    return n;
}
 
long int longinteger_domain::smallest_primedivisor(
    longinteger_object &a,
    int p_min, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    int f_vv = (verbose_level >= 2);
    longinteger_object n, n1, q, pp;
    long int p, r, cnt = 0;
    
    if (f_v) {
        cout << "longinteger_domain::smallest_primedivisor " << a
                << " p_min=" << p_min << endl;
    }
    a.assign_to(n);
    
    if (p_min == 0) {
        p_min = 2;
    }
    if (p_min < 0) {
        p_min = - p_min;
    }
    if (p_min <= 2) {
        if (is_even(n)) {
            return 2;
        }
        p_min = 3;
    }
    if (p_min <= 3) {
        if (remainder_mod_int(n, 3) == 0) {
            return 3;
        }
        p_min = 5;
    }
    if (EVEN(p_min)) {
        p_min--;
    }
    p = p_min;
    while (TRUE) {
        cnt++;
        if (f_vv) {
            cout << "longinteger_domain::smallest_primedivisor n=" << n
                    << " trying p=" << p << endl;
        }
        n.assign_to(n1);
        integral_division_by_lint(n1, p, q, r);
        if (f_vv && (cnt % 1) == 0) {
            cout << "longinteger_domain::smallest_primedivisor n=" << n1
                << " trying p=" << p << " q=" << q
                << " r=" << r << endl;
            cnt = 0;
        }
        if (r == 0) {
            return p;
        }
        pp.create(p, __FILE__, __LINE__);
        if (compare(q, pp) < 0) {
            break;
        }
        p += 2;
    }
    if (f_v) {
        cout << "longinteger_domain::smallest_primedivisor "
                "the number is prime" << endl;
    }
    return 0;
}
 
void longinteger_domain::factor_into_longintegers(
    longinteger_object &a,
    int &nb_primes, longinteger_object *&primes, 
    int *&exponents, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    //int f_vv = (verbose_level >= 2);
    longinteger_object n, q, pp, r;
    longinteger_domain D;
    long int p, last_prime = 0;
    int i;
    number_theory::number_theory_domain NT;
    
    if (f_v) {
        cout << "longinteger_domain::factor_into_longintegers factoring " << a << endl;
    }
    if (a.is_zero()) {
        cout << "longinteger_domain::factor_into_longintegers a is zero" << endl;
        exit(1);
    }
    if (a.is_one()) {
        nb_primes = 0;
        primes = NEW_OBJECTS(longinteger_object, 1);
        exponents = NEW_int(1);
        return;
    }
    a.assign_to(n);
    p = smallest_primedivisor(n, last_prime, verbose_level);
    if (p == 0) {
        p = n.as_lint();
    }
    pp.create(p, __FILE__, __LINE__);
    primes = NEW_OBJECTS(longinteger_object, 1);
    exponents = NEW_int(1);
    nb_primes = 1;
    pp.assign_to(primes[0]);
    exponents[0] = 1;
    last_prime = p;
    D.integral_division(n, pp, q, r, verbose_level - 1);
    if (!r.is_zero()) {
        cout << "longinteger_domain::factor_into_longintegers "
                "factor does not divide" << endl;
    }
    q.assign_to(n);
    while (!n.is_one()) {
        if (f_v) {
            cout << "longinteger_domain::factor_into_longintegers "
                    "remaining factor: " << n << endl;
        }
        p = smallest_primedivisor(n, last_prime, verbose_level);
        // if p == 0: n is prime
        
        if (p == 0) {
            p = n.as_lint();
        }
        if (p == last_prime) {
            exponents[nb_primes - 1]++;
            pp.create(p, __FILE__, __LINE__);
        }
        else {
            longinteger_object *pr = NEW_OBJECTS(
                longinteger_object, nb_primes + 1);
            int *ex = NEW_int(nb_primes + 1);
            for (i = 0; i < nb_primes; i++) {
                primes[i].assign_to(pr[i]);
                ex[i] = exponents[i];
            }
            FREE_OBJECTS(primes);
            FREE_int(exponents);
            primes = pr;
            exponents = ex;
            if (p) {
                pp.create(p, __FILE__, __LINE__);
            }
            else {
                n.assign_to(pp);
            }
            pp.assign_to(primes[nb_primes]);
            exponents[nb_primes] = 1;
            nb_primes++;
            last_prime = p;
        }
        
        D.integral_division(n, pp, q, r, verbose_level - 1);
        if (!r.is_zero()) {
            cout << "longinteger_domain::factor_into_longintegers "
                    "factor does not divide" << endl;
        }
        q.assign_to(n);
        if (f_v) {
            cout << "partial factorization: " << a << " = ";
            NT.print_longfactorization(nb_primes, primes, exponents);
            cout << "   * " << n;
            cout << endl;
        }
 
    }
    if (f_v) {
        cout << "longinteger_domain::factor_into_longintegers "
                "prime factorization of " << a << " = ";
        NT.print_longfactorization(nb_primes, primes, exponents);
        cout << endl;
    }
}
 
void longinteger_domain::factor(longinteger_object &a, 
    int &nb_primes, int *&primes, int *&exponents, 
    int verbose_level)
{
    int f_v = (verbose_level >= 1);
    longinteger_object n, q;
    int p, last_prime = 2, i, r;
    number_theory::number_theory_domain NT;
    
    if (f_v) {
        cout << "factoring " << a << endl;
    }
    if (a.is_zero()) {
        cout << "longinteger_domain::factor a is zero" << endl;
        exit(1);
    }
    if (a.is_one()) {
        nb_primes = 0;
        primes = NEW_int(1);
        exponents = NEW_int(1);
        return;
    }
    a.assign_to(n);
    p = smallest_primedivisor(n, last_prime, verbose_level);
    if (p == 0) {
        p = n.as_int();
    }
    primes = NEW_int(1);
    exponents = NEW_int(1);
    nb_primes = 1;
    primes[0] = p;
    exponents[0] = 1;
    last_prime = p;
    integral_division_by_int(n, p, q, r);
    q.assign_to(n);
    while (!n.is_one()) {
        if (f_v) {
            cout << "remaining factor: " << n << endl;
        }
        p = smallest_primedivisor(n, last_prime, verbose_level);
        // if p == 0: n is prime
        
        if (p == 0) {
            p = n.as_int();
        }
        
        if (p == last_prime) {
            exponents[nb_primes - 1]++;
        }
        else {
            int *pr = NEW_int(nb_primes + 1);
            int *ex = NEW_int(nb_primes + 1);
            for (i = 0; i < nb_primes; i++) {
                pr[i] = primes[i];
                ex[i] = exponents[i];
            }
            FREE_int(primes);
            FREE_int(exponents);
            primes = pr;
            exponents = ex;
            primes[nb_primes] = p;
            exponents[nb_primes] = 1;
            nb_primes++;
            last_prime = p;
        }
        
        if (f_v) {
            cout << "dividing " << n << " by " << p << endl;
        }
        integral_division_by_int(n, p, q, r);
        q.assign_to(n);
        if (f_v) {
            cout << "partial factorization: " << a << " = ";
            NT.print_factorization(nb_primes, primes, exponents);
            cout << "   * " << n;
            cout << endl;
        }
    }
    if (f_v) {
        cout << "factor(): " << a << " = ";
        NT.print_factorization(nb_primes, primes, exponents);
        cout << endl;
    }
}
 
int longinteger_domain::jacobi(longinteger_object &a, 
    longinteger_object &m, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    int f_vv = (verbose_level >= 2);
    longinteger_object a1, m1, a2, m2, a3, m3;
    longinteger_object u, v, g, q, r, res, minus_one;
    int n, rr, r1, t1, t2;
    
    if (f_v) {
        cout << "longinteger_domain::jacobi(" << a << " over " << m << ")" << endl;
    }
    a.assign_to(a1);
    m.assign_to(m1);
    r1 = 1;
    
    minus_one.create(-1, __FILE__, __LINE__);
    extended_gcd(a1, m1, g, u, v, verbose_level - 2);
    if (!g.is_one_or_minus_one()) {
        return 0;
    }
    
    while (TRUE) {
        // we now have
        // jacobi(a, m) = r1 * jacobi(a1, m1);
        integral_division(a1, m1, q, r, verbose_level - 2);
        if (f_vv) {
            cout << r1 << " * Jacobi(" << r << ", "
                    << m1 << ")" << endl;
        }
        n = multiplicity_of_p(r, res, 2);
        res.assign_to(a1);
        if (f_vv) {
            cout << r1 << " * Jacobi( 2^" << n << " * "
                    << a1 << ", " << m1 << ")" << endl;
        }
        if (ODD(n)) {
            // t = (m1 * m1 - 1) >> 3 = (m1 * m1 - 1) / 8
            // multiply r1 by  (-1) to the power t:
            rr = remainder_mod_int(m1, 8);
            if (rr == 3 || rr == 5) {
                r1 = -r1; // note ABS(r1) == 1
            }
        }
        if (f_vv) {
            cout << r1 << " * Jacobi(" << a1 << ", " << m1 << ")" << endl;
        }
        if (a1.is_one_or_minus_one()) {
            break;
        }
        // reciprocity:
        add(a1, minus_one, a2);
        add(m1, minus_one, m2);
        integral_division_by_int(a2, 2, a3, rr);
        integral_division_by_int(m2, 2, m3, rr);
        integral_division_by_int(a3, 2, a2, t1);
        integral_division_by_int(m3, 2, m2, t2);
        a1.assign_to(a2);
        m1.assign_to(a1);
        a2.assign_to(m1);
        if (ODD(t1) && ODD(t2)) {
            r1 = -r1;
        }
        if (f_vv) {
            cout << r1 << " * Jacobi(" << a1 << ", " << m1 << ")" << endl;
        }
    }
    if (f_v) {
        cout << "jacobi(" << a << ", " << m << ") = " << r1 << endl;
    }
    return r1;
}
 
void longinteger_domain::random_number_less_than_n(
    longinteger_object &n, longinteger_object &r)
{
    int i, l, rr;
    //char *n_rep;
    char *r_rep;
    orbiter_kernel_system::os_interface Os;
    
    l = n.len();
    n.assign_to(r);
    //n_rep = n.rep();
    r_rep = r.rep();
    while (TRUE) {
        for (i = l - 1; i >= 0; i--) {
            rr = Os.random_integer(10);
            r_rep[i] = (char) rr;
        }
        r.normalize();
        if (compare_unsigned(r, n) < 0) {
            break;
        }
    }
}
 
void longinteger_domain::random_number_with_n_decimals(
    longinteger_object &R, int n, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    char *str;
    int i;
    orbiter_kernel_system::os_interface Os;
 
    if (f_v) {
        cout << "longinteger_domain::random_number_with_n_decimals" << endl;
    }
    str = NEW_char(n + 1);
    for (i = 0; i <= n; i++) {
        str[n - i] = '0' + Os.random_integer(10);
        if (i == n) {
            str[n - i] = '1' + Os.random_integer(9);
        }
    }
    str[n] = 0;
 
 
    if (f_v) {
        cout << "longinteger_domain::random_number_with_n_decimals "
                "random number = " << str << endl;
    }
 
    R.create_from_base_10_string(str);
 
    FREE_char(str);
 
    if (f_v) {
        cout << "longinteger_domain::random_number_with_n_decimals done" << endl;
    }
}
 
 
void longinteger_domain::matrix_product(
        longinteger_object *A, longinteger_object *B,
        longinteger_object *&C, int Am, int An, int Bn)
{
    int i, j, k;
    longinteger_object a, b, c;
    
    for (i = 0; i < Am; i++) {
        for (j = 0; j < Bn; j++) {
            c.create(0, __FILE__, __LINE__);
            for (k = 0; k < An; k++) {
                mult(A[i * An + k], B[k * Bn + j], a);
                add(a, c, b);
                b.assign_to(c);
            }
            c.assign_to(C[i * Bn + j]);
        }
    }
}
 
void longinteger_domain::matrix_entries_integral_division_exact(
    longinteger_object *A, longinteger_object &b,
    int Am, int An)
{
    int i, j;
    longinteger_object q, r;
    int verbose_level = 0;
    
    for (i = 0; i < Am; i++) {
        for (j = 0; j < An; j++) {
            integral_division(A[i * An + j], b, q, r, verbose_level);
            if (!r.is_zero()) {
                cout << "integral division: " << b
                    << " does not divide " << A[i * An + j] << endl;
                cout << "i=" << i << " j=" << j << endl;
                exit(1);
            }
            q.assign_to(A[i * An + j]);
        }
    }
}
 
void longinteger_domain::matrix_print_GAP(
        ostream &ost, longinteger_object *A,
        int Am, int An)
{
    int i, j;
    
    ost << "[";
    for (i = 0; i < Am; i++) {
        ost << "[";
        for (j = 0; j < An; j++) {
            ost << A[i * An + j];
            if (j < An - 1) {
                ost << ",";
            }
            ost << " ";
        }
        ost << "]";
        if (i < An - 1) {
            ost << ",";
        }
        ost << endl;
    }
    ost << "];" << endl;
}
 
void longinteger_domain::matrix_print_tex(
        ostream &ost, longinteger_object *A,
        int Am, int An)
{
    int i, j;
    
    ost << "\\begin{array}{*{" << An << "}{r}}" << endl;
    for (i = 0; i < Am; i++) {
        for (j = 0; j < An; j++) {
            ost << A[i * An + j];
            if (j < An - 1) {
                ost << " & ";
            }
        }
        ost << "\\\\" << endl;
    }
    ost << "\\end{array}" << endl;
}
 
void longinteger_domain::power_mod(
    char *aa, char *bb, char *nn,
    longinteger_object &result, int verbose_level)
{
    int f_v = (verbose_level >= 1);
    longinteger_object a, b, c, d, n;
    
    a.create_from_base_10_string(aa, 0);
    c.create_from_base_10_string(aa, 0);
    b.create_from_base_10_string(bb, 0);
    d.create_from_base_10_string(bb, 0);
    n.create_from_base_10_string(nn, 0);
    power_longint_mod(c, d, n, verbose_level - 1);
    c.assign_to(result);
    if (f_v) {
        cout << "longinteger_domain::power_mod:" << aa
                << " ^ " << bb << " == " << result
                << " mod " << nn << endl;
    }
}
 
 
void longinteger_domain::factorial(
        longinteger_object &result, int n)
{
    int *x;
    int i;
    
    x = NEW_int(n);
    for (i = 0; i < n; i++) {
        x[i] = i + 1;
    }
    multiply_up(result, x, n, 0 /* verbose_level */);
    FREE_int(x);
}
 
void longinteger_domain::group_order_PGL(
        longinteger_object &result,
        int n, int q, int f_semilinear)
{
    long int *x;
    int i, l;
    int p, e;
    number_theory::number_theory_domain NT;
    
    NT.factor_prime_power(q, p, e);
    l = n;
    if (f_semilinear) {
        l++;
    }
    
    x = NEW_lint(l);
    for (i = 0; i < n; i++) {
        x[i] = NT.i_power_j_lint(q, n) - NT.i_power_j_lint(q, i);
        if (i == 0) {
            x[i] = x[i] / (q - 1);
        }
    }
    if (f_semilinear) {
        x[n] = e;
    }
 
#if 0
    cout << "longinteger_domain::group_order_PGL "
            "factors of |PGL(n,q)| = ";
    int_vec_print(cout, x, l);
    cout << endl;
#endif
 
    multiply_up_lint(result, x, l, 0 /* verbose_level */);
    FREE_lint(x);
}
 
 
 
 
 
void longinteger_domain::square_root_floor(longinteger_object &a,
        longinteger_object &x, int verbose_level)
{
    int f_v = (verbose_level >= 1);
 
    if (f_v) {
        cout << "longinteger_domain::square_root_floor" << endl;
    }
    x.freeself();
    longinteger_object Y, YY;
    int la, l, len;
    char c1;
 
    a.normalize();
    if (a.sign()) {
        cout << "longinteger_domain::square_root_floor "
                "no square root, the number is negative" << endl;
        exit(1);
    }
    if (a.is_zero()) {
        x.create(0, __FILE__, __LINE__);
        return;
    }
 
    la = a.len();
    if (ODD(la)) {
        la++;
    }
    len = (la >> 1) + 1;
    Y.sign() = FALSE;
    Y.len() = len;
    Y.rep() = NEW_char(len);
 
    //Y.allocate_empty(len);
    //Y.s_sign() = FALSE;
    for (l = 0; l < len; l++) {
        Y.ith(l) = (char) 0;
    }
 
    for (l = len - 1; l >= 0; l--) {
        for (c1 = 9; c1 >= 0; c1--) {
            Y.ith(l) = c1;
            mult(Y, Y, YY);
 
            if (compare_unsigned(YY, a) <= 0) {
                break;
            }
        }
    }
    Y.normalize();
    Y.swap_with(x);
    if (f_v) {
        cout << "longinteger_domain::square_root_floor done" << endl;
    }
}
 
 
 
 
void longinteger_domain::print_digits(char *rep, int len)
{
    for (int h = 0; h < len; h++) {
        cout << (char)('0' + rep[h]) << " ";
    }
}
 
 
 
 
}}}