long CRT(mat_ZZ& gg, ZZ& a, const mat_zz_p& G)
{
long n = gg.NumRows();
long m = gg.NumCols();
if (G.NumRows() != n || G.NumCols() != m)
Error("CRT: dimension mismatch");
long p = zz_p::modulus();
ZZ new_a;
mul(new_a, a, p);
long a_inv;
a_inv = rem(a, p);
a_inv = InvMod(a_inv, p);
long p1;
p1 = p >> 1;
ZZ a1;
RightShift(a1, a, 1);
long p_odd = (p & 1);
long modified = 0;
long h;
ZZ g;
long i, j;
for (i = 0; i < n; i++) {
for (j = 0; j < m; j++) {
if (!CRTInRange(gg[i][j], a)) {
modified = 1;
rem(g, gg[i][j], a);
if (g > a1) sub(g, g, a);
}
else
g = gg[i][j];
h = rem(g, p);
h = SubMod(rep(G[i][j]), h, p);
h = MulMod(h, a_inv, p);
if (h > p1)
h = h - p;
if (h != 0) {
modified = 1;
if (!p_odd && g > 0 && (h == p1))
MulSubFrom(g, a, h);
else
MulAddTo(g, a, h);
}
gg[i][j] = g;
}
}
a = new_a;
return modified;
}
void solve(ZZ& d_out, vec_ZZ& x_out,
const mat_ZZ& A, const vec_ZZ& b,
long deterministic)
{
long n = A.NumRows();
if (A.NumCols() != n)
Error("solve: nonsquare matrix");
if (b.length() != n)
Error("solve: dimension mismatch");
if (n == 0) {
set(d_out);
x_out.SetLength(0);
return;
}
zz_pBak zbak;
zbak.save();
ZZ_pBak Zbak;
Zbak.save();
vec_ZZ x(INIT_SIZE, n);
ZZ d, d1;
ZZ d_prod, x_prod;
set(d_prod);
set(x_prod);
long d_instable = 1;
long x_instable = 1;
long check = 0;
long gp_cnt = 0;
vec_ZZ y, b1;
long i;
long bound = 2+DetBound(A);
for (i = 0; ; i++) {
if ((check || IsZero(d)) && !d_instable) {
if (NumBits(d_prod) > bound) {
break;
}
else if (!deterministic &&
bound > 1000 && NumBits(d_prod) < 0.25*bound) {
ZZ P;
long plen = 90 + NumBits(max(bound, NumBits(d)));
GenPrime(P, plen, 90 + 2*NumBits(gp_cnt++));
ZZ_p::init(P);
mat_ZZ_p AA;
conv(AA, A);
ZZ_p dd;
determinant(dd, AA);
if (CRT(d, d_prod, rep(dd), P))
d_instable = 1;
else
break;
}
}
zz_p::FFTInit(i);
long p = zz_p::modulus();
mat_zz_p AA;
conv(AA, A);
if (!check) {
vec_zz_p bb, xx;
conv(bb, b);
zz_p dd;
solve(dd, xx, AA, bb);
d_instable = CRT(d, d_prod, rep(dd), p);
if (!IsZero(dd)) {
mul(xx, xx, dd);
x_instable = CRT(x, x_prod, xx);
}
else
x_instable = 1;
if (!d_instable && !x_instable) {
mul(y, x, A);
mul(b1, b, d);
if (y == b1) {
d1 = d;
check = 1;
}
}
}
else {
zz_p dd;
determinant(dd, AA);
d_instable = CRT(d, d_prod, rep(dd), p);
}
}
if (check && d1 != d) {
mul(x, x, d);
ExactDiv(x, d1);
}
d_out = d;
if (check) x_out = x;
zbak.restore();
Zbak.restore();
}
void solve1(ZZ& d_out, vec_ZZ& x_out, const mat_ZZ& A, const vec_ZZ& b)
{
long n = A.NumRows();
if (A.NumCols() != n)
Error("solve1: nonsquare matrix");
if (b.length() != n)
Error("solve1: dimension mismatch");
if (n == 0) {
set(d_out);
x_out.SetLength(0);
return;
}
ZZ num_bound, den_bound;
hadamard(num_bound, den_bound, A, b);
if (den_bound == 0) {
clear(d_out);
return;
}
zz_pBak zbak;
zbak.save();
long i;
long j;
ZZ prod;
prod = 1;
mat_zz_p B;
for (i = 0; ; i++) {
zz_p::FFTInit(i);
mat_zz_p AA, BB;
zz_p dd;
conv(AA, A);
inv(dd, BB, AA);
if (dd != 0) {
transpose(B, BB);
break;
}
mul(prod, prod, zz_p::modulus());
if (prod > den_bound) {
d_out = 0;
return;
}
}
long max_A_len = MaxBits(A);
long use_double_mul1 = 0;
long use_double_mul2 = 0;
long double_limit = 0;
if (max_A_len + NTL_SP_NBITS + NumBits(n) <= NTL_DOUBLE_PRECISION-1)
use_double_mul1 = 1;
if (!use_double_mul1 && max_A_len+NTL_SP_NBITS+2 <= NTL_DOUBLE_PRECISION-1) {
use_double_mul2 = 1;
double_limit = (1L << (NTL_DOUBLE_PRECISION-1-max_A_len-NTL_SP_NBITS));
}
long use_long_mul1 = 0;
long use_long_mul2 = 0;
long long_limit = 0;
if (max_A_len + NTL_SP_NBITS + NumBits(n) <= NTL_BITS_PER_LONG-1)
use_long_mul1 = 1;
if (!use_long_mul1 && max_A_len+NTL_SP_NBITS+2 <= NTL_BITS_PER_LONG-1) {
use_long_mul2 = 1;
long_limit = (1L << (NTL_BITS_PER_LONG-1-max_A_len-NTL_SP_NBITS));
}
if (use_double_mul1 && use_long_mul1)
use_long_mul1 = 0;
else if (use_double_mul1 && use_long_mul2)
use_long_mul2 = 0;
else if (use_double_mul2 && use_long_mul1)
use_double_mul2 = 0;
else if (use_double_mul2 && use_long_mul2) {
if (long_limit > double_limit)
use_double_mul2 = 0;
else
use_long_mul2 = 0;
}
double **double_A;
double *double_h;
typedef double *double_ptr;
if (use_double_mul1 || use_double_mul2) {
double_h = NTL_NEW_OP double[n];
double_A = NTL_NEW_OP double_ptr[n];
if (!double_h || !double_A) Error("solve1: out of mem");
for (i = 0; i < n; i++) {
double_A[i] = NTL_NEW_OP double[n];
if (!double_A[i]) Error("solve1: out of mem");
}
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
double_A[j][i] = to_double(A[i][j]);
}
void determinant(ZZ& rres, const mat_ZZ& a, long deterministic)
{
long n = a.NumRows();
if (a.NumCols() != n)
Error("determinant: nonsquare matrix");
if (n == 0) {
set(rres);
return;
}
zz_pBak zbak;
zbak.save();
ZZ_pBak Zbak;
Zbak.save();
long instable = 1;
long gp_cnt = 0;
long bound = 2+DetBound(a);
ZZ res, prod;
clear(res);
set(prod);
long i;
for (i = 0; ; i++) {
if (NumBits(prod) > bound)
break;
if (!deterministic &&
!instable && bound > 1000 && NumBits(prod) < 0.25*bound) {
ZZ P;
long plen = 90 + NumBits(max(bound, NumBits(res)));
GenPrime(P, plen, 90 + 2*NumBits(gp_cnt++));
ZZ_p::init(P);
mat_ZZ_p A;
conv(A, a);
ZZ_p t;
determinant(t, A);
if (CRT(res, prod, rep(t), P))
instable = 1;
else
break;
}
zz_p::FFTInit(i);
long p = zz_p::modulus();
mat_zz_p A;
conv(A, a);
zz_p t;
determinant(t, A);
instable = CRT(res, prod, rep(t), p);
}
rres = res;
zbak.restore();
Zbak.restore();
}
void solve1(ZZ& d_out, vec_ZZ& x_out, const mat_ZZ& A, const vec_ZZ& b)
{
long n = A.NumRows();
if (A.NumCols() != n)
LogicError("solve1: nonsquare matrix");
if (b.length() != n)
LogicError("solve1: dimension mismatch");
if (n == 0) {
set(d_out);
x_out.SetLength(0);
return;
}
ZZ num_bound, den_bound;
hadamard(num_bound, den_bound, A, b);
if (den_bound == 0) {
clear(d_out);
return;
}
zz_pBak zbak;
zbak.save();
long i;
long j;
ZZ prod;
prod = 1;
mat_zz_p B;
for (i = 0; ; i++) {
zz_p::FFTInit(i);
mat_zz_p AA, BB;
zz_p dd;
conv(AA, A);
inv(dd, BB, AA);
if (dd != 0) {
transpose(B, BB);
break;
}
mul(prod, prod, zz_p::modulus());
if (prod > den_bound) {
d_out = 0;
return;
}
}
long max_A_len = MaxBits(A);
long use_double_mul1 = 0;
long use_double_mul2 = 0;
long double_limit = 0;
if (max_A_len + NTL_SP_NBITS + NumBits(n) <= NTL_DOUBLE_PRECISION-1)
use_double_mul1 = 1;
if (!use_double_mul1 && max_A_len+NTL_SP_NBITS+2 <= NTL_DOUBLE_PRECISION-1) {
use_double_mul2 = 1;
double_limit = (1L << (NTL_DOUBLE_PRECISION-1-max_A_len-NTL_SP_NBITS));
}
long use_long_mul1 = 0;
long use_long_mul2 = 0;
long long_limit = 0;
if (max_A_len + NTL_SP_NBITS + NumBits(n) <= NTL_BITS_PER_LONG-1)
use_long_mul1 = 1;
if (!use_long_mul1 && max_A_len+NTL_SP_NBITS+2 <= NTL_BITS_PER_LONG-1) {
use_long_mul2 = 1;
long_limit = (1L << (NTL_BITS_PER_LONG-1-max_A_len-NTL_SP_NBITS));
}
if (use_double_mul1 && use_long_mul1)
use_long_mul1 = 0;
else if (use_double_mul1 && use_long_mul2)
use_long_mul2 = 0;
else if (use_double_mul2 && use_long_mul1)
use_double_mul2 = 0;
else if (use_double_mul2 && use_long_mul2) {
if (long_limit > double_limit)
use_double_mul2 = 0;
else
use_long_mul2 = 0;
}
double **double_A=0;
double *double_h=0;
Unique2DArray<double> double_A_store;
UniqueArray<double> double_h_store;
if (use_double_mul1 || use_double_mul2) {
double_h_store.SetLength(n);
double_h = double_h_store.get();
double_A_store.SetDims(n, n);
double_A = double_A_store.get();
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
double_A[j][i] = to_double(A[i][j]);
}
long **long_A=0;
long *long_h=0;
Unique2DArray<long> long_A_store;
UniqueArray<long> long_h_store;
if (use_long_mul1 || use_long_mul2) {
long_h_store.SetLength(n);
long_h = long_h_store.get();
long_A_store.SetDims(n, n);
long_A = long_A_store.get();
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
long_A[j][i] = to_long(A[i][j]);
}
vec_ZZ x;
x.SetLength(n);
vec_zz_p h;
h.SetLength(n);
vec_ZZ e;
e = b;
vec_zz_p ee;
vec_ZZ t;
t.SetLength(n);
prod = 1;
ZZ bound1;
mul(bound1, num_bound, den_bound);
mul(bound1, bound1, 2);
while (prod <= bound1) {
conv(ee, e);
mul(h, B, ee);
if (use_double_mul1) {
for (i = 0; i < n; i++)
double_h[i] = to_double(rep(h[i]));
double_MixedMul1(t, double_h, double_A, n);
}
else if (use_double_mul2) {
for (i = 0; i < n; i++)
double_h[i] = to_double(rep(h[i]));
double_MixedMul2(t, double_h, double_A, n, double_limit);
}
else if (use_long_mul1) {
for (i = 0; i < n; i++)
long_h[i] = to_long(rep(h[i]));
long_MixedMul1(t, long_h, long_A, n);
}
else if (use_long_mul2) {
for (i = 0; i < n; i++)
long_h[i] = to_long(rep(h[i]));
long_MixedMul2(t, long_h, long_A, n, long_limit);
}
else
MixedMul(t, h, A); // t = h*A
SubDiv(e, t, zz_p::modulus()); // e = (e-t)/p
MulAdd(x, prod, h); // x = x + prod*h
mul(prod, prod, zz_p::modulus());
}
vec_ZZ num, denom;
ZZ d, d_mod_prod, tmp1;
num.SetLength(n);
denom.SetLength(n);
d = 1;
d_mod_prod = 1;
for (i = 0; i < n; i++) {
rem(x[i], x[i], prod);
MulMod(x[i], x[i], d_mod_prod, prod);
if (!ReconstructRational(num[i], denom[i], x[i], prod,
num_bound, den_bound))
LogicError("solve1 internal error: rat recon failed!");
mul(d, d, denom[i]);
if (i != n-1) {
if (denom[i] != 1) {
div(den_bound, den_bound, denom[i]);
mul(bound1, num_bound, den_bound);
mul(bound1, bound1, 2);
div(tmp1, prod, zz_p::modulus());
while (tmp1 > bound1) {
prod = tmp1;
div(tmp1, prod, zz_p::modulus());
}
rem(tmp1, denom[i], prod);
rem(d_mod_prod, d_mod_prod, prod);
MulMod(d_mod_prod, d_mod_prod, tmp1, prod);
}
}
}
tmp1 = 1;
for (i = n-1; i >= 0; i--) {
mul(num[i], num[i], tmp1);
mul(tmp1, tmp1, denom[i]);
}
x_out.SetLength(n);
for (i = 0; i < n; i++) {
x_out[i] = num[i];
}
d_out = d;
}
void CharPoly(ZZX& gg, const mat_ZZ& a, long deterministic)
{
long n = a.NumRows();
if (a.NumCols() != n)
LogicError("CharPoly: nonsquare matrix");
if (n == 0) {
set(gg);
return;
}
if (n == 1) {
ZZ t;
SetX(gg);
negate(t, a(1, 1));
SetCoeff(gg, 0, t);
return;
}
long bound = 2 + CharPolyBound(a);
zz_pBak bak;
bak.save();
ZZ_pBak bak1;
bak1.save();
ZZX g;
ZZ prod;
clear(g);
set(prod);
long i;
long instable = 1;
long gp_cnt = 0;
for (i = 0; ; i++) {
if (NumBits(prod) > bound)
break;
if (!deterministic &&
!instable && bound > 1000 && NumBits(prod) < 0.25*bound) {
long plen = 90 + NumBits(max(bound, MaxBits(g)));
ZZ P;
GenPrime(P, plen, 90 + 2*NumBits(gp_cnt++));
ZZ_p::init(P);
mat_ZZ_p A;
ZZ_pX G;
conv(A, a);
CharPoly(G, A);
if (CRT(g, prod, G))
instable = 1;
else
break;
}
zz_p::FFTInit(i);
mat_zz_p A;
zz_pX G;
conv(A, a);
CharPoly(G, A);
instable = CRT(g, prod, G);
}
gg = g;
bak.restore();
bak1.restore();
}
static
long LLL_XD(mat_ZZ& B, mat_ZZ* U, xdouble delta, long deep,
LLLCheckFct check)
{
long m = B.NumRows();
long n = B.NumCols();
long i, j;
long new_m, dep, quit;
xdouble s;
ZZ MU;
xdouble mu1;
xdouble t1;
ZZ T1;
init_red_fudge();
if (U) ident(*U, m);
xdouble **B1; // approximates B
typedef xdouble *xdoubleptr;
B1 = NTL_NEW_OP xdoubleptr[m+1];
if (!B1) Error("LLL_XD: out of memory");
for (i = 1; i <= m; i++) {
B1[i] = NTL_NEW_OP xdouble[n+1];
if (!B1[i]) Error("LLL_XD: out of memory");
}
xdouble **mu;
mu = NTL_NEW_OP xdoubleptr[m+1];
if (!mu) Error("LLL_XD: out of memory");
for (i = 1; i <= m; i++) {
mu[i] = NTL_NEW_OP xdouble[m+1];
if (!mu[i]) Error("LLL_XD: out of memory");
}
xdouble *c; // squared lengths of Gramm-Schmidt basis vectors
c = NTL_NEW_OP xdouble[m+1];
if (!c) Error("LLL_XD: out of memory");
xdouble *b; // squared lengths of basis vectors
b = NTL_NEW_OP xdouble[m+1];
if (!b) Error("LLL_XD: out of memory");
for (i = 1; i <=m; i++)
for (j = 1; j <= n; j++)
conv(B1[i][j], B(i, j));
for (i = 1; i <= m; i++) {
b[i] = InnerProduct(B1[i], B1[i], n);
}
new_m = ll_LLL_XD(B, U, delta, deep, check, B1, mu, b, c, m, 1, quit);
dep = m - new_m;
m = new_m;
if (dep > 0) {
// for consistency, we move all of the zero rows to the front
for (i = 0; i < m; i++) {
swap(B(m+dep-i), B(m-i));
if (U) swap((*U)(m+dep-i), (*U)(m-i));
}
}
// clean-up
for (i = 1; i <= m+dep; i++) {
delete [] B1[i];
}
delete [] B1;
for (i = 1; i <= m+dep; i++) {
delete [] mu[i];
}
delete [] mu;
delete [] c;
delete [] b;
return m;
}
static
long BKZ_XD(mat_ZZ& BB, mat_ZZ* UU, xdouble delta,
long beta, long prune, LLLCheckFct check)
{
long m = BB.NumRows();
long n = BB.NumCols();
long m_orig = m;
long i, j;
ZZ MU;
xdouble t1;
ZZ T1;
xdouble *tp;
init_red_fudge();
mat_ZZ B;
B = BB;
B.SetDims(m+1, n);
xdouble **B1; // approximates B
typedef xdouble *xdoubleptr;
B1 = NTL_NEW_OP xdoubleptr[m+2];
if (!B1) Error("BKZ_XD: out of memory");
for (i = 1; i <= m+1; i++) {
B1[i] = NTL_NEW_OP xdouble[n+1];
if (!B1[i]) Error("BKZ_XD: out of memory");
}
xdouble **mu;
mu = NTL_NEW_OP xdoubleptr[m+2];
if (!mu) Error("BKZ_XD: out of memory");
for (i = 1; i <= m+1; i++) {
mu[i] = NTL_NEW_OP xdouble[m+1];
if (!mu[i]) Error("BKZ_XD: out of memory");
}
xdouble *c; // squared lengths of Gramm-Schmidt basis vectors
c = NTL_NEW_OP xdouble[m+2];
if (!c) Error("BKZ_XD: out of memory");
xdouble *b; // squared lengths of basis vectors
b = NTL_NEW_OP xdouble[m+2];
if (!b) Error("BKZ_XD: out of memory");
xdouble cbar;
xdouble *ctilda;
ctilda = NTL_NEW_OP xdouble[m+2];
if (!ctilda) Error("BKZ_XD: out of memory");
xdouble *vvec;
vvec = NTL_NEW_OP xdouble[m+2];
if (!vvec) Error("BKZ_XD: out of memory");
xdouble *yvec;
yvec = NTL_NEW_OP xdouble[m+2];
if (!yvec) Error("BKZ_XD: out of memory");
xdouble *uvec;
uvec = NTL_NEW_OP xdouble[m+2];
if (!uvec) Error("BKZ_XD: out of memory");
xdouble *utildavec;
utildavec = NTL_NEW_OP xdouble[m+2];
if (!utildavec) Error("BKZ_XD: out of memory");
long *Deltavec;
Deltavec = NTL_NEW_OP long[m+2];
if (!Deltavec) Error("BKZ_XD: out of memory");
long *deltavec;
deltavec = NTL_NEW_OP long[m+2];
if (!deltavec) Error("BKZ_XD: out of memory");
mat_ZZ Ulocal;
mat_ZZ *U;
if (UU) {
Ulocal.SetDims(m+1, m);
for (i = 1; i <= m; i++)
conv(Ulocal(i, i), 1);
U = &Ulocal;
}
else
U = 0;
long quit;
long new_m;
long z, jj, kk;
long s, t;
long h;
xdouble eta;
for (i = 1; i <=m; i++)
for (j = 1; j <= n; j++)
conv(B1[i][j], B(i, j));
for (i = 1; i <= m; i++) {
b[i] = InnerProduct(B1[i], B1[i], n);
}
// cerr << "\n";
// cerr << "first LLL\n";
m = ll_LLL_XD(B, U, delta, 0, check, B1, mu, b, c, m, 1, quit);
double tt;
double enum_time = 0;
unsigned long NumIterations = 0;
unsigned long NumTrivial = 0;
unsigned long NumNonTrivial = 0;
unsigned long NumNoOps = 0;
long verb = verbose;
verbose = 0;
if (m < m_orig) {
for (i = m_orig+1; i >= m+2; i--) {
// swap i, i-1
swap(B(i), B(i-1));
if (U) swap((*U)(i), (*U)(i-1));
}
}
long clean = 1;
if (!quit && m > 1) {
// cerr << "continuing\n";
if (beta > m) beta = m;
if (prune > 0)
ComputeBKZConstant(beta, prune);
z = 0;
jj = 0;
while (z < m-1) {
jj++;
kk = min(jj+beta-1, m);
if (jj == m) {
jj = 1;
kk = beta;
clean = 1;
}
if (verb) {
tt = GetTime();
if (tt > LastTime + LLLStatusInterval)
BKZStatus(tt, enum_time, NumIterations, NumTrivial,
NumNonTrivial, NumNoOps, m, B);
}
// ENUM
double tt1;
if (verb) {
tt1 = GetTime();
}
if (prune > 0)
ComputeBKZThresh(&c[jj], kk-jj+1);
cbar = c[jj];
utildavec[jj] = uvec[jj] = 1;
yvec[jj] = vvec[jj] = 0;
Deltavec[jj] = 0;
s = t = jj;
deltavec[jj] = 1;
for (i = jj+1; i <= kk+1; i++) {
ctilda[i] = uvec[i] = utildavec[i] = yvec[i] = 0;
Deltavec[i] = 0;
vvec[i] = 0;
deltavec[i] = 1;
}
long enum_cnt = 0;
while (t <= kk) {
if (verb) {
enum_cnt++;
if (enum_cnt > 100000) {
enum_cnt = 0;
tt = GetTime();
if (tt > LastTime + LLLStatusInterval) {
enum_time += tt - tt1;
tt1 = tt;
BKZStatus(tt, enum_time, NumIterations, NumTrivial,
NumNonTrivial, NumNoOps, m, B);
}
}
}
ctilda[t] = ctilda[t+1] +
(yvec[t]+utildavec[t])*(yvec[t]+utildavec[t])*c[t];
if (prune > 0 && t > jj) {
eta = BKZThresh(t-jj);
}
else
eta = 0;
if (ctilda[t] < cbar - eta) {
if (t > jj) {
t--;
t1 = 0;
for (i = t+1; i <= s; i++) {
t1 += utildavec[i]*mu[i][t];
}
yvec[t] = t1;
t1 = -t1;
if (t1 >= 0)
t1 = ceil(t1-0.5);
else
t1 = floor(t1+0.5);
utildavec[t] = vvec[t] = t1;
Deltavec[t] = 0;
if (utildavec[t] > -yvec[t])
deltavec[t] = -1;
else
deltavec[t] = 1;
}
else {
cbar = ctilda[jj];
for (i = jj; i <= kk; i++) {
uvec[i] = utildavec[i];
}
}
}
else {
t++;
s = max(s, t);
if (t < s) Deltavec[t] = -Deltavec[t];
if (Deltavec[t]*deltavec[t] >= 0) Deltavec[t] += deltavec[t];
utildavec[t] = vvec[t] + Deltavec[t];
}
}
if (verb) {
tt1 = GetTime() - tt1;
enum_time += tt1;
}
NumIterations++;
h = min(kk+1, m);
if ((delta-8*red_fudge)*c[jj] > cbar) {
clean = 0;
// we treat the case that the new vector is b_s (jj < s <= kk)
// as a special case that appears to occur most of the time.
s = 0;
for (i = jj+1; i <= kk; i++) {
if (uvec[i] != 0) {
if (s == 0)
s = i;
else
s = -1;
}
}
if (s == 0) Error("BKZ_XD: internal error");
if (s > 0) {
// special case
NumTrivial++;
for (i = s; i > jj; i--) {
// swap i, i-1
swap(B(i-1), B(i));
if (U) swap((*U)(i-1), (*U)(i));
tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp;
t1 = b[i-1]; b[i-1] = b[i]; b[i] = t1;
}
// cerr << "special case\n";
new_m = ll_LLL_XD(B, U, delta, 0, check,
B1, mu, b, c, h, jj, quit);
if (new_m != h) Error("BKZ_XD: internal error");
if (quit) break;
}
else {
// the general case
NumNonTrivial++;
for (i = 1; i <= n; i++) conv(B(m+1, i), 0);
if (U) {
for (i = 1; i <= m_orig; i++)
conv((*U)(m+1, i), 0);
}
for (i = jj; i <= kk; i++) {
if (uvec[i] == 0) continue;
conv(MU, uvec[i]);
RowTransform2(B(m+1), B(i), MU);
if (U) RowTransform2((*U)(m+1), (*U)(i), MU);
}
for (i = m+1; i >= jj+1; i--) {
// swap i, i-1
swap(B(i-1), B(i));
if (U) swap((*U)(i-1), (*U)(i));
tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp;
t1 = b[i-1]; b[i-1] = b[i]; b[i] = t1;
}
for (i = 1; i <= n; i++)
conv(B1[jj][i], B(jj, i));
b[jj] = InnerProduct(B1[jj], B1[jj], n);
if (b[jj] == 0) Error("BKZ_XD: internal error");
// remove linear dependencies
// cerr << "general case\n";
new_m = ll_LLL_XD(B, U, delta, 0, 0, B1, mu, b, c, kk+1, jj, quit);
if (new_m != kk) Error("BKZ_XD: internal error");
// remove zero vector
for (i = kk+2; i <= m+1; i++) {
// swap i, i-1
swap(B(i-1), B(i));
if (U) swap((*U)(i-1), (*U)(i));
tp = B1[i-1]; B1[i-1] = B1[i]; B1[i] = tp;
t1 = b[i-1]; b[i-1] = b[i]; b[i] = t1;
}
quit = 0;
if (check) {
for (i = 1; i <= kk; i++)
if ((*check)(B(i))) {
quit = 1;
break;
}
}
if (quit) break;
if (h > kk) {
// extend reduced basis
new_m = ll_LLL_XD(B, U, delta, 0, check,
B1, mu, b, c, h, h, quit);
if (new_m != h) Error("BKZ_XD: internal error");
if (quit) break;
}
}
z = 0;
}
else {
// LLL_XD
// cerr << "progress\n";
NumNoOps++;
if (!clean) {
new_m =
ll_LLL_XD(B, U, delta, 0, check, B1, mu, b, c, h, h, quit);
if (new_m != h) Error("BKZ_XD: internal error");
if (quit) break;
}
z++;
}
}
}
if (verb) {
BKZStatus(GetTime(), enum_time, NumIterations, NumTrivial, NumNonTrivial,
NumNoOps, m, B);
}
// clean up
if (m_orig > m) {
// for consistency, we move zero vectors to the front
for (i = m+1; i <= m_orig; i++) {
swap(B(i), B(i+1));
if (U) swap((*U)(i), (*U)(i+1));
}
for (i = 0; i < m; i++) {
swap(B(m_orig-i), B(m-i));
if (U) swap((*U)(m_orig-i), (*U)(m-i));
}
}
B.SetDims(m_orig, n);
BB = B;
if (U) {
U->SetDims(m_orig, m_orig);
*UU = *U;
}
for (i = 1; i <= m_orig+1; i++) {
delete [] B1[i];
}
delete [] B1;
for (i = 1; i <= m_orig+1; i++) {
delete [] mu[i];
}
delete [] mu;
delete [] c;
delete [] b;
delete [] ctilda;
delete [] vvec;
delete [] yvec;
delete [] uvec;
delete [] utildavec;
delete [] Deltavec;
delete [] deltavec;
return m;
}
static
long G_LLL_XD(mat_ZZ& B, mat_ZZ* U, xdouble delta, long deep,
LLLCheckFct check)
{
long m = B.NumRows();
long n = B.NumCols();
long i, j;
long new_m, dep, quit;
xdouble s;
ZZ MU;
xdouble mu1;
xdouble t1;
ZZ T1;
init_red_fudge();
if (U) ident(*U, m);
xdouble **B1; // approximates B
typedef xdouble *xdoubleptr;
B1 = NTL_NEW_OP xdoubleptr[m+1];
if (!B1) Error("G_LLL_XD: out of memory");
for (i = 1; i <= m; i++) {
B1[i] = NTL_NEW_OP xdouble[n+1];
if (!B1[i]) Error("G_LLL_XD: out of memory");
}
xdouble **mu;
mu = NTL_NEW_OP xdoubleptr[m+1];
if (!mu) Error("G_LLL_XD: out of memory");
for (i = 1; i <= m; i++) {
mu[i] = NTL_NEW_OP xdouble[n+2];
if (!mu[i]) Error("G_LLL_XD: out of memory");
}
xdouble **aux;
aux = NTL_NEW_OP xdoubleptr[m+1];
if (!aux) Error("G_LLL_XD: out of memory");
for (i = 1; i <= m; i++) {
aux[i] = NTL_NEW_OP xdouble[n+1];
if (!aux[i]) Error("G_LLL_XD: out of memory");
}
for (i = 1; i <=m; i++)
for (j = 1; j <= n; j++)
conv(B1[i][j], B(i, j));
GivensCache_XD cache(m, n);
new_m =
ll_G_LLL_XD(B, U, delta, deep, check, B1, mu, aux, m, 1, quit, cache);
dep = m - new_m;
m = new_m;
if (dep > 0) {
// for consistency, we move all of the zero rows to the front
for (i = 0; i < m; i++) {
swap(B(m+dep-i), B(m-i));
if (U) swap((*U)(m+dep-i), (*U)(m-i));
}
}
// clean-up
for (i = 1; i <= m+dep; i++) {
delete [] B1[i];
}
delete [] B1;
for (i = 1; i <= m+dep; i++) {
delete [] mu[i];
}
delete [] mu;
for (i = 1; i <= m+dep; i++) {
delete [] aux[i];
}
delete [] aux;
return m;
}
long LatticeSolve(vec_ZZ& x, const mat_ZZ& A, const vec_ZZ& y, long reduce)
{
long n = A.NumRows();
long m = A.NumCols();
if (y.length() != m)
Error("LatticeSolve: dimension mismatch");
if (reduce < 0 || reduce > 2)
Error("LatticeSolve: bad reduce parameter");
if (IsZero(y)) {
x.SetLength(n);
clear(x);
return 1;
}
mat_ZZ A1, U1;
ZZ det2;
long im_rank, ker_rank;
A1 = A;
im_rank = image(det2, A1, U1);
ker_rank = n - im_rank;
mat_ZZ A2, U2;
long new_rank;
long i;
A2.SetDims(im_rank + 1, m);
for (i = 1; i <= im_rank; i++)
A2(i) = A1(ker_rank + i);
A2(im_rank + 1) = y;
new_rank = image(det2, A2, U2);
if (new_rank != im_rank ||
(U2(1)(im_rank+1) != 1 && U2(1)(im_rank+1) != -1))
return 0;
vec_ZZ x1;
x1.SetLength(im_rank);
for (i = 1; i <= im_rank; i++)
x1(i) = U2(1)(i);
if (U2(1)(im_rank+1) == 1)
negate(x1, x1);
vec_ZZ x2, tmp;
x2.SetLength(n);
clear(x2);
tmp.SetLength(n);
for (i = 1; i <= im_rank; i++) {
mul(tmp, U1(ker_rank+i), x1(i));
add(x2, x2, tmp);
}
if (reduce == 0) {
x = x2;
return 1;
}
else if (reduce == 1) {
U1.SetDims(ker_rank+1, n);
U1(ker_rank+1) = x2;
image(det2, U1);
x = U1(ker_rank + 1);
return 1;
}
else if (reduce == 2) {
U1.SetDims(ker_rank, n);
LLL(det2, U1);
U1.SetDims(ker_rank+1, n);
U1(ker_rank+1) = x2;
image(det2, U1);
x = U1(ker_rank + 1);
return 1;
}
return 0;
}
static
long LLL(vec_ZZ& D, mat_ZZ& B, mat_ZZ* U, long a, long b, long verbose)
{
long m = B.NumRows();
long n = B.NumCols();
long force_reduce = 1;
vec_long P;
P.SetLength(m);
D.SetLength(m+1);
D[0] = 1;
vec_vec_ZZ lam;
lam.SetLength(m);
long j;
for (j = 1; j <= m; j++)
lam(j).SetLength(m);
if (U) ident(*U, m);
long s = 0;
long k = 1;
long max_k = 0;
while (k <= m) {
if (k > max_k) {
IncrementalGS(B, P, D, lam, s, k);
max_k = k;
}
if (k == 1) {
force_reduce = 1;
k++;
continue;
}
if (force_reduce)
for (j = k-1; j >= 1; j--)
reduce(k, j, B, P, D, lam, U);
if (P(k-1) != 0 &&
(P(k) == 0 ||
SwapTest(D[P(k)], D[P(k)-1], D[P(k)-2], lam(k)(P(k)-1), a, b))) {
force_reduce = swap(k, B, P, D, lam, U, max_k, verbose);
k--;
}
else {
force_reduce = 1;
k++;
}
}
D.SetLength(s+1);
return s;
}
static
long LLL_XD(mat_ZZ& B, mat_ZZ* U, xdouble delta, long deep,
LLLCheckFct check)
{
long m = B.NumRows();
long n = B.NumCols();
long i, j;
long new_m, dep, quit;
xdouble s;
ZZ MU;
xdouble mu1;
xdouble t1;
ZZ T1;
init_red_fudge();
if (U) ident(*U, m);
Unique2DArray<xdouble> B1_store;
B1_store.SetDimsFrom1(m+1, n+1);
xdouble **B1 = B1_store.get(); // approximates B
Unique2DArray<xdouble> mu_store;
mu_store.SetDimsFrom1(m+1, m+1);
xdouble **mu = mu_store.get();
UniqueArray<xdouble> c_store;
c_store.SetLength(m+1);
xdouble *c = c_store.get(); // squared lengths of Gramm-Schmidt basis vectors
UniqueArray<xdouble> b_store;
b_store.SetLength(m+1);
xdouble *b = b_store.get(); // squared lengths of basis vectors
for (i = 1; i <=m; i++)
for (j = 1; j <= n; j++)
conv(B1[i][j], B(i, j));
for (i = 1; i <= m; i++) {
b[i] = InnerProduct(B1[i], B1[i], n);
}
new_m = ll_LLL_XD(B, U, delta, deep, check, B1, mu, b, c, m, 1, quit);
dep = m - new_m;
m = new_m;
if (dep > 0) {
// for consistency, we move all of the zero rows to the front
for (i = 0; i < m; i++) {
swap(B(m+dep-i), B(m-i));
if (U) swap((*U)(m+dep-i), (*U)(m-i));
}
}
return m;
}
RR ReductionQualityChecker::computeHermiteFactor(mat_ZZ &mat) {
RR numerator = findShortestNormVector(mat);
RR denominator = pow(abs(determinant(mat)), (1./mat.NumRows()));
RR result = numerator/denominator;
return result;
}
