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) LogicError("gauss: bad args"); 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(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]); return l; }
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) LogicError("determinant: nonsquare matrix"); if (n == 0) { set(d); return; } 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); 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); return; } } conv(d, det); }
void inv(zz_pE& d, mat_zz_pE& X, const mat_zz_pE& A) { long n = A.NumRows(); if (A.NumCols() != n) LogicError("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(); 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(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); return; } } 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); }
// 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); }