void power(mat_zz_pE& X, const mat_zz_pE& A, const ZZ& e)
{
if (A.NumRows() != A.NumCols()) Error("power: non-square matrix");
if (e == 0) {
ident(X, A.NumRows());
return;
}
mat_zz_pE T1, T2;
long i, k;
k = NumBits(e);
T1 = A;
for (i = k-2; i >= 0; i--) {
sqr(T2, T1);
if (bit(e, i))
mul(T1, T2, A);
else
T1 = T2;
}
if (e < 0)
inv(X, T1);
else
X = T1;
}
void transpose(mat_zz_pE& X, const mat_zz_pE& A)
{
long n = A.NumRows();
long m = A.NumCols();
long i, j;
if (&X == & A) {
if (n == m)
for (i = 1; i <= n; i++)
for (j = i+1; j <= n; j++)
swap(X(i, j), X(j, i));
else {
mat_zz_pE tmp;
tmp.SetDims(m, n);
for (i = 1; i <= n; i++)
for (j = 1; j <= m; j++)
tmp(j, i) = A(i, j);
X.kill();
X = tmp;
}
}
else {
X.SetDims(m, n);
for (i = 1; i <= n; i++)
for (j = 1; j <= m; j++)
X(j, i) = A(i, j);
}
}
void mul_aux(mat_zz_pE& X, const mat_zz_pE& A, const mat_zz_pE& B)
{
long n = A.NumRows();
long l = A.NumCols();
long m = B.NumCols();
if (l != B.NumRows())
Error("matrix mul: dimension mismatch");
X.SetDims(n, m);
long i, j, k;
zz_pX acc, tmp;
for (i = 1; i <= n; i++) {
for (j = 1; j <= m; j++) {
clear(acc);
for(k = 1; k <= l; k++) {
mul(tmp, rep(A(i,k)), rep(B(k,j)));
add(acc, acc, tmp);
}
conv(X(i,j), acc);
}
}
}
void negate(mat_zz_pE& X, const mat_zz_pE& A)
{
long n = A.NumRows();
long m = A.NumCols();
X.SetDims(n, m);
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= m; j++)
negate(X(i,j), A(i,j));
}
void mul(mat_zz_pE& X, const mat_zz_pE& A, const zz_pE& b_in)
{
zz_pE b = b_in;
long n = A.NumRows();
long m = A.NumCols();
X.SetDims(n, m);
long i, j;
for (i = 0; i < n; i++)
for (j = 0; j < m; j++)
mul(X[i][j], A[i][j], b);
}
void mul(mat_zz_pE& X, const mat_zz_pE& A, long b_in)
{
newNTL_zz_pRegister(b);
b = b_in;
long n = A.NumRows();
long m = A.NumCols();
X.SetDims(n, m);
long i, j;
for (i = 0; i < n; i++)
for (j = 0; j < m; j++)
mul(X[i][j], A[i][j], b);
}
void sub(mat_zz_pE& X, const mat_zz_pE& A, const mat_zz_pE& B)
{
long n = A.NumRows();
long m = A.NumCols();
if (B.NumRows() != n || B.NumCols() != m)
Error("matrix sub: dimension mismatch");
X.SetDims(n, m);
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= m; j++)
sub(X(i,j), A(i,j), B(i,j));
}
void convert(vector< vector<ZZX> >& X, const mat_zz_pE& A)
{
long n = A.NumRows();
X.resize(n);
for (long i = 0; i < n; i++)
convert(X[i], A[i]);
}
void convert(mat_zz_pE& X, const vector< vector<ZZX> >& A)
{
long n = A.size();
if (n == 0) {
long m = X.NumCols();
X.SetDims(0, m);
return;
}
long m = A[0].size();
X.SetDims(n, m);
for (long i = 0; i < n; i++)
convert(X[i], A[i]);
}
void clear(mat_zz_pE& x)
{
long n = x.NumRows();
long i;
for (i = 0; i < n; i++)
clear(x[i]);
}
long IsDiag(const mat_zz_pE& A, long n, const zz_pE& d)
{
if (A.NumRows() != n || A.NumCols() != n)
return 0;
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= n; j++)
if (i != j) {
if (!IsZero(A(i, j))) return 0;
}
else {
if (A(i, j) != d) return 0;
}
return 1;
}
NTL_START_IMPL
void add(mat_zz_pE& X, const mat_zz_pE& A, const mat_zz_pE& B)
{
long n = A.NumRows();
long m = A.NumCols();
if (B.NumRows() != n || B.NumCols() != m)
LogicError("matrix add: dimension mismatch");
X.SetDims(n, m);
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= m; j++)
add(X(i,j), A(i,j), B(i,j));
}
void mul(vec_zz_pE& x, const mat_zz_pE& A, const vec_zz_pE& b)
{
if (&b == &x || A.position1(x) != -1) {
vec_zz_pE tmp;
mul_aux(tmp, A, b);
x = tmp;
}
else
mul_aux(x, A, b);
}
long IsZero(const mat_zz_pE& a)
{
long n = a.NumRows();
long i;
for (i = 0; i < n; i++)
if (!IsZero(a[i]))
return 0;
return 1;
}
// prime power solver
// A is an n x n matrix, we compute its inverse mod p^r. An error is raised
// if A is not inverible mod p. zz_p::modulus() is assumed to be p^r, for
// p prime, r >= 1. Also zz_pE::modulus() is assumed to be initialized.
void ppInvert(mat_zz_pE& X, const mat_zz_pE& A, long p, long r)
{
if (r == 1) { // use native inversion from NTL
inv(X, A); // X = A^{-1}
return;
}
// begin by inverting A modulo p
// convert to ZZX for a safe transaltion to mod-p objects
vector< vector<ZZX> > tmp;
convert(tmp, A);
{ // open a new block for mod-p computation
ZZX G;
convert(G, zz_pE::modulus());
zz_pBak bak_pr; bak_pr.save(); // backup the mod-p^r moduli
zz_pEBak bak_prE; bak_prE.save();
zz_p::init(p); // Set the mod-p moduli
zz_pE::init(conv<zz_pX>(G));
mat_zz_pE A1, Inv1;
convert(A1, tmp); // Recover A as a mat_zz_pE object modulo p
inv(Inv1, A1); // Inv1 = A^{-1} (mod p)
convert(tmp, Inv1); // convert to ZZX for transaltion to a mod-p^r object
} // mod-p^r moduli restored on desctuction of bak_pr and bak_prE
mat_zz_pE XX;
convert(XX, tmp); // XX = A^{-1} (mod p)
// Now lift the solution modulo p^r
// Compute the "correction factor" Z, s.t. XX*A = I - p*Z (mod p^r)
long n = A.NumRows();
const mat_zz_pE I = ident_mat_zz_pE(n); // identity matrix
mat_zz_pE Z = I - XX*A;
convert(tmp, Z); // Conver to ZZX to divide by p
for (long i=0; i<n; i++) for (long j=0; j<n; j++) tmp[i][j] /= p;
convert(Z, tmp); // convert back to a mod-p^r object
// The inverse of A is ( I+(pZ)+(pZ)^2+...+(pZ)^{r-1} )*XX (mod p^r). We use
// O(log r) products to copmute it as (I+pZ)* (I+(pZ)^2)* (I+(pZ)^4)*...* XX
long e = NextPowerOfTwo(r); // 2^e is smallest power of two >= r
Z *= p; // = pZ
mat_zz_pE prod = I + Z; // = I + pZ
for (long i=1; i<e; i++) {
sqr(Z, Z); // = (pZ)^{2^i}
prod *= (I+Z); // = sum_{j=0}^{2^{i+1}-1} (pZ)^j
}
mul(X, prod, XX); // X = A^{-1} mod p^r
assert(X*A == I);
}
void ident(mat_zz_pE& X, long n)
{
X.SetDims(n, n);
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= n; j++)
if (i == j)
set(X(i, j));
else
clear(X(i, j));
}
void diag(mat_zz_pE& X, long n, const zz_pE& d_in)
{
zz_pE d = d_in;
X.SetDims(n, n);
long i, j;
for (i = 1; i <= n; i++)
for (j = 1; j <= n; j++)
if (i == j)
X(i, j) = d;
else
clear(X(i, j));
}
void buildLinPolyMatrix(mat_zz_pE& M, long p)
{
long d = zz_pE::degree();
M.SetDims(d, d);
for (long j = 0; j < d; j++)
conv(M[0][j], zz_pX(j, 1));
for (long i = 1; i < d; i++)
for (long j = 0; j < d; j++)
M[i][j] = power(M[i-1][j], p);
}
namespaceanon
void mul_aux(vec_zz_pE& x, const mat_zz_pE& A, const vec_zz_pE& b)
{
long n = A.NumRows();
long l = A.NumCols();
if (l != b.length())
Error("matrix mul: dimension mismatch");
x.SetLength(n);
long i, k;
zz_pX acc, tmp;
for (i = 1; i <= n; i++) {
clear(acc);
for (k = 1; k <= l; k++) {
mul(tmp, rep(A(i,k)), rep(b(k)));
add(acc, acc, tmp);
}
conv(x(i), acc);
}
}
namespaceanon
void mul_aux(vec_zz_pE& x, const vec_zz_pE& a, const mat_zz_pE& B)
{
long n = B.NumRows();
long l = B.NumCols();
if (n != a.length())
Error("matrix mul: dimension mismatch");
x.SetLength(l);
long i, k;
zz_pX acc, tmp;
for (i = 1; i <= l; i++) {
clear(acc);
for (k = 1; k <= n; k++) {
mul(tmp, rep(a(k)), rep(B(k,i)));
add(acc, acc, tmp);
}
conv(x(i), acc);
}
}
void determinant(zz_pE& d, const mat_zz_pE& M_in)
{
long k, n;
long i, j;
long pos;
zz_pX t1, t2;
zz_pX *x, *y;
const zz_pXModulus& p = zz_pE::modulus();
n = M_in.NumRows();
if (M_in.NumCols() != n)
Error("determinant: nonsquare matrix");
if (n == 0) {
set(d);
return;
}
vec_zz_pX *M = newNTL_NEW_OP vec_zz_pX[n];
for (i = 0; i < n; i++) {
M[i].SetLength(n);
for (j = 0; j < n; j++) {
M[i][j].rep.SetMaxLength(2*deg(p)-1);
M[i][j] = rep(M_in[i][j]);
}
}
zz_pX det;
set(det);
for (k = 0; k < n; k++) {
pos = -1;
for (i = k; i < n; i++) {
rem(t1, M[i][k], p);
M[i][k] = t1;
if (pos == -1 && !IsZero(t1))
pos = i;
}
if (pos != -1) {
if (k != pos) {
swap(M[pos], M[k]);
negate(det, det);
}
MulMod(det, det, M[k][k], p);
// make M[k, k] == -1 mod p, and make row k reduced
InvMod(t1, M[k][k], p);
negate(t1, t1);
for (j = k+1; j < n; j++) {
rem(t2, M[k][j], p);
MulMod(M[k][j], t2, t1, p);
}
for (i = k+1; i < n; i++) {
// M[i] = M[i] + M[k]*M[i,k]
t1 = M[i][k]; // this is already reduced
x = M[i].elts() + (k+1);
y = M[k].elts() + (k+1);
for (j = k+1; j < n; j++, x++, y++) {
// *x = *x + (*y)*t1
mul(t2, *y, t1);
add(*x, *x, t2);
}
}
}
else {
clear(d);
goto done;
}
}
conv(d, det);
done:
delete[] M;
}
long gauss(mat_zz_pE& M_in, long w)
{
long k, l;
long i, j;
long pos;
zz_pX t1, t2, t3;
zz_pX *x, *y;
long n = M_in.NumRows();
long m = M_in.NumCols();
if (w < 0 || w > m)
Error("gauss: bad args");
const zz_pXModulus& p = zz_pE::modulus();
vec_zz_pX *M = newNTL_NEW_OP vec_zz_pX[n];
for (i = 0; i < n; i++) {
M[i].SetLength(m);
for (j = 0; j < m; j++) {
M[i][j].rep.SetMaxLength(2*deg(p)-1);
M[i][j] = rep(M_in[i][j]);
}
}
l = 0;
for (k = 0; k < w && l < n; k++) {
pos = -1;
for (i = l; i < n; i++) {
rem(t1, M[i][k], p);
M[i][k] = t1;
if (pos == -1 && !IsZero(t1)) {
pos = i;
}
}
if (pos != -1) {
swap(M[pos], M[l]);
InvMod(t3, M[l][k], p);
negate(t3, t3);
for (j = k+1; j < m; j++) {
rem(M[l][j], M[l][j], p);
}
for (i = l+1; i < n; i++) {
// M[i] = M[i] + M[l]*M[i,k]*t3
MulMod(t1, M[i][k], t3, p);
clear(M[i][k]);
x = M[i].elts() + (k+1);
y = M[l].elts() + (k+1);
for (j = k+1; j < m; j++, x++, y++) {
// *x = *x + (*y)*t1
mul(t2, *y, t1);
add(t2, t2, *x);
*x = t2;
}
}
l++;
}
}
for (i = 0; i < n; i++)
for (j = 0; j < m; j++)
conv(M_in[i][j], M[i][j]);
delete [] M;
return l;
}
// prime power solver
// zz_p::modulus() is assumed to be p^r, for p prime, r >= 1
// A is an n x n matrix, b is a length n (row) vector,
// and a solution for the matrix-vector equation x A = b is found.
// If A is not inverible mod p, then error is raised.
void ppsolve(vec_zz_pE& x, const mat_zz_pE& A, const vec_zz_pE& b,
long p, long r)
{
if (r == 1) {
zz_pE det;
solve(det, x, A, b);
if (det == 0) Error("ppsolve: matrix not invertible");
return;
}
long n = A.NumRows();
if (n != A.NumCols())
Error("ppsolve: matrix not square");
if (n == 0)
Error("ppsolve: matrix of dimension 0");
zz_pContext pr_context;
pr_context.save();
zz_pEContext prE_context;
prE_context.save();
zz_pX G = zz_pE::modulus();
ZZX GG = to_ZZX(G);
vector< vector<ZZX> > AA;
convert(AA, A);
vector<ZZX> bb;
convert(bb, b);
zz_pContext p_context(p);
p_context.restore();
zz_pX G1 = to_zz_pX(GG);
zz_pEContext pE_context(G1);
pE_context.restore();
// we are now working mod p...
// invert A mod p
mat_zz_pE A1;
convert(A1, AA);
mat_zz_pE I1;
zz_pE det;
inv(det, I1, A1);
if (det == 0) {
Error("ppsolve: matrix not invertible");
}
vec_zz_pE b1;
convert(b1, bb);
vec_zz_pE y1;
y1 = b1 * I1;
vector<ZZX> yy;
convert(yy, y1);
// yy is a solution mod p
for (long k = 1; k < r; k++) {
// lift solution yy mod p^k to a solution mod p^{k+1}
pr_context.restore();
prE_context.restore();
// we are now working mod p^r
vec_zz_pE d, y;
convert(y, yy);
d = b - y * A;
vector<ZZX> dd;
convert(dd, d);
long pk = power_long(p, k);
vector<ZZX> ee;
div(ee, dd, pk);
p_context.restore();
pE_context.restore();
// we are now working mod p
vec_zz_pE e1;
convert(e1, ee);
vec_zz_pE z1;
z1 = e1 * I1;
vector<ZZX> zz, ww;
convert(zz, z1);
mul(ww, zz, pk);
add(yy, yy, ww);
}
pr_context.restore();
prE_context.restore();
convert(x, yy);
assert(x*A == b);
}
void kernel(mat_zz_pE& X, const mat_zz_pE& A)
{
long m = A.NumRows();
long n = A.NumCols();
mat_zz_pE M;
long r;
transpose(M, A);
r = gauss(M);
X.SetDims(m-r, m);
long i, j, k, s;
zz_pX t1, t2;
zz_pE T3;
vec_long D;
D.SetLength(m);
for (j = 0; j < m; j++) D[j] = -1;
vec_zz_pE inverses;
inverses.SetLength(m);
j = -1;
for (i = 0; i < r; i++) {
do {
j++;
} while (IsZero(M[i][j]));
D[j] = i;
inv(inverses[j], M[i][j]);
}
for (k = 0; k < m-r; k++) {
vec_zz_pE& v = X[k];
long pos = 0;
for (j = m-1; j >= 0; j--) {
if (D[j] == -1) {
if (pos == k)
set(v[j]);
else
clear(v[j]);
pos++;
}
else {
i = D[j];
clear(t1);
for (s = j+1; s < m; s++) {
mul(t2, rep(v[s]), rep(M[i][s]));
add(t1, t1, t2);
}
conv(T3, t1);
mul(T3, T3, inverses[j]);
negate(v[j], T3);
}
}
}
}
void inv(zz_pE& d, mat_zz_pE& X, const mat_zz_pE& A)
{
long n = A.NumRows();
if (A.NumCols() != n)
Error("inv: nonsquare matrix");
if (n == 0) {
set(d);
X.SetDims(0, 0);
return;
}
long i, j, k, pos;
zz_pX t1, t2;
zz_pX *x, *y;
const zz_pXModulus& p = zz_pE::modulus();
vec_zz_pX *M = newNTL_NEW_OP vec_zz_pX[n];
for (i = 0; i < n; i++) {
M[i].SetLength(2*n);
for (j = 0; j < n; j++) {
M[i][j].rep.SetMaxLength(2*deg(p)-1);
M[i][j] = rep(A[i][j]);
M[i][n+j].rep.SetMaxLength(2*deg(p)-1);
clear(M[i][n+j]);
}
set(M[i][n+i]);
}
zz_pX det;
set(det);
for (k = 0; k < n; k++) {
pos = -1;
for (i = k; i < n; i++) {
rem(t1, M[i][k], p);
M[i][k] = t1;
if (pos == -1 && !IsZero(t1)) {
pos = i;
}
}
if (pos != -1) {
if (k != pos) {
swap(M[pos], M[k]);
negate(det, det);
}
MulMod(det, det, M[k][k], p);
// make M[k, k] == -1 mod p, and make row k reduced
InvMod(t1, M[k][k], p);
negate(t1, t1);
for (j = k+1; j < 2*n; j++) {
rem(t2, M[k][j], p);
MulMod(M[k][j], t2, t1, p);
}
for (i = k+1; i < n; i++) {
// M[i] = M[i] + M[k]*M[i,k]
t1 = M[i][k]; // this is already reduced
x = M[i].elts() + (k+1);
y = M[k].elts() + (k+1);
for (j = k+1; j < 2*n; j++, x++, y++) {
// *x = *x + (*y)*t1
mul(t2, *y, t1);
add(*x, *x, t2);
}
}
}
else {
clear(d);
goto done;
}
}
X.SetDims(n, n);
for (k = 0; k < n; k++) {
for (i = n-1; i >= 0; i--) {
clear(t1);
for (j = i+1; j < n; j++) {
mul(t2, rep(X[j][k]), M[i][j]);
add(t1, t1, t2);
}
sub(t1, t1, M[i][n+k]);
conv(X[i][k], t1);
}
}
conv(d, det);
done:
delete[] M;
}
long gauss(mat_zz_pE& M)
{
return gauss(M, M.NumCols());
}
static
void solve_impl(zz_pE& d, vec_zz_pE& X, const mat_zz_pE& A, const vec_zz_pE& b, bool trans)
{
long n = A.NumRows();
if (A.NumCols() != n)
LogicError("solve: nonsquare matrix");
if (b.length() != n)
LogicError("solve: dimension mismatch");
if (n == 0) {
set(d);
X.SetLength(0);
return;
}
long i, j, k, pos;
zz_pX t1, t2;
zz_pX *x, *y;
const zz_pXModulus& p = zz_pE::modulus();
UniqueArray<vec_zz_pX> M_store;
M_store.SetLength(n);
vec_zz_pX *M = M_store.get();
for (i = 0; i < n; i++) {
M[i].SetLength(n+1);
if (trans) {
for (j = 0; j < n; j++) {
M[i][j].rep.SetMaxLength(2*deg(p)-1);
M[i][j] = rep(A[j][i]);
}
}
else {
for (j = 0; j < n; j++) {
M[i][j].rep.SetMaxLength(2*deg(p)-1);
M[i][j] = rep(A[i][j]);
}
}
M[i][n].rep.SetMaxLength(2*deg(p)-1);
M[i][n] = rep(b[i]);
}
zz_pX det;
set(det);
for (k = 0; k < n; k++) {
pos = -1;
for (i = k; i < n; i++) {
rem(t1, M[i][k], p);
M[i][k] = t1;
if (pos == -1 && !IsZero(t1)) {
pos = i;
}
}
if (pos != -1) {
if (k != pos) {
swap(M[pos], M[k]);
negate(det, det);
}
MulMod(det, det, M[k][k], p);
// make M[k, k] == -1 mod p, and make row k reduced
InvMod(t1, M[k][k], p);
negate(t1, t1);
for (j = k+1; j <= n; j++) {
rem(t2, M[k][j], p);
MulMod(M[k][j], t2, t1, p);
}
for (i = k+1; i < n; i++) {
// M[i] = M[i] + M[k]*M[i,k]
t1 = M[i][k]; // this is already reduced
x = M[i].elts() + (k+1);
y = M[k].elts() + (k+1);
for (j = k+1; j <= n; j++, x++, y++) {
// *x = *x + (*y)*t1
mul(t2, *y, t1);
add(*x, *x, t2);
}
}
}
else {
clear(d);
return;
}
}
X.SetLength(n);
for (i = n-1; i >= 0; i--) {
clear(t1);
for (j = i+1; j < n; j++) {
mul(t2, rep(X[j]), M[i][j]);
add(t1, t1, t2);
}
sub(t1, t1, M[i][n]);
conv(X[i], t1);
}
conv(d, det);
}