void ProjectPowers(vec_ZZ_p& x, const vec_ZZ_p& a, long k,
const ZZ_pXArgument& H, const ZZ_pXModulus& F)
{
long n = F.n;
if (a.length() > n || k < 0)
LogicError("ProjectPowers: bad args");
if (NTL_OVERFLOW(k, 1, 0))
ResourceError("ProjectPowers: excessive args");
long m = H.H.length()-1;
long l = (k+m-1)/m - 1;
ZZ_pXMultiplier M;
build(M, H.H[m], F);
vec_ZZ_p s(INIT_SIZE, n);
s = a;
StripZeroes(s);
x.SetLength(k);
for (long i = 0; i <= l; i++) {
long m1 = min(m, k-i*m);
ZZ_p* w = &x[i*m];
for (long j = 0; j < m1; j++)
InnerProduct(w[j], H.H[j].rep, s);
if (i < l)
UpdateMap(s, s, M, F);
}
}
static void StripZeroes(vec_ZZ_p& x)
{
long n = x.length();
while (n > 0 && IsZero(x[n-1]))
n--;
x.SetLength(n);
}
void negate(vec_ZZ_p& x, const vec_ZZ_p& a)
{
long n = a.length();
x.SetLength(n);
long i;
for (i = 0; i < n; i++)
negate(x[i], a[i]);
}
void sub(vec_ZZ_p& x, const vec_ZZ_p& a, const vec_ZZ_p& b)
{
long n = a.length();
if (b.length() != n) LogicError("vector sub: dimension mismatch");
x.SetLength(n);
long i;
for (i = 0; i < n; i++)
sub(x[i], a[i], b[i]);
}
void mul(vec_ZZ_p& x, const vec_ZZ_p& a, long b_in)
{
NTL_ZZ_pRegister(b);
b = b_in;
long n = a.length();
x.SetLength(n);
long i;
for (i = 0; i < n; i++)
mul(x[i], a[i], b);
}
void FindRoots(vec_ZZ_p& x, const ZZ_pX& ff)
{
ZZ_pX f = ff;
if (!IsOne(LeadCoeff(f)))
Error("FindRoots: bad args");
x.SetMaxLength(deg(f));
x.SetLength(0);
RecFindRoots(x, f);
}
void conv(vec_ZZ_p& x, const vec_ZZ& a)
{
long i, n;
n = a.length();
x.SetLength(n);
ZZ_p* xp = x.elts();
const ZZ* ap = a.elts();
for (i = 0; i < n; i++)
conv(xp[i], ap[i]);
}
void InnerProduct(ZZ_p& x, const vec_ZZ_p& a, const vec_ZZ_p& b)
{
long n = min(a.length(), b.length());
long i;
NTL_ZZRegister(accum);
NTL_ZZRegister(t);
clear(accum);
for (i = 0; i < n; i++) {
mul(t, rep(a[i]), rep(b[i]));
add(accum, accum, t);
}
conv(x, accum);
}
void eval(vec_ZZ_p& b, const ZZ_pX& f, const vec_ZZ_p& a)
// naive algorithm: repeats Horner
{
if (&b == &f.rep) {
vec_ZZ_p bb;
eval(bb, f, a);
b = bb;
return;
}
long m = a.length();
b.SetLength(m);
long i;
for (i = 0; i < m; i++)
eval(b[i], f, a[i]);
}
void clear(vec_ZZ_p& x)
{
long n = x.length();
long i;
for (i = 0; i < n; i++)
clear(x[i]);
}
void InnerProduct(ZZ_pX& x, const vec_ZZ_p& v, long low, long high,
const vec_ZZ_pX& H, long n, ZZVec& t)
{
NTL_ZZRegister(s);
long i, j;
for (j = 0; j < n; j++)
clear(t[j]);
high = min(high, v.length()-1);
for (i = low; i <= high; i++) {
const vec_ZZ_p& h = H[i-low].rep;
long m = h.length();
const ZZ& w = rep(v[i]);
for (j = 0; j < m; j++) {
mul(s, w, rep(h[j]));
add(t[j], t[j], s);
}
}
x.rep.SetLength(n);
for (j = 0; j < n; j++)
conv(x.rep[j], t[j]);
x.normalize();
}
static
void ComputeTraceVec(vec_ZZ_p& S, const ZZ_pXModulus& F)
{
if (!F.UseFFT) {
PlainTraceVec(S, F.f);
return;
}
long i;
long n = F.n;
FFTRep R;
ZZ_pX P, g;
g.rep.SetLength(n-1);
for (i = 1; i < n; i++)
mul(g.rep[n-i-1], F.f.rep[n-i], i);
g.normalize();
ToFFTRep(R, g, F.l);
mul(R, R, F.HRep);
FromFFTRep(P, R, n-2, 2*n-4);
S.SetLength(n);
S[0] = n;
for (i = 1; i < n; i++)
negate(S[i], coeff(P, n-1-i));
}
void PlainUpdateMap(vec_ZZ_p& xx, const vec_ZZ_p& a,
const ZZ_pX& b, const ZZ_pX& f)
{
long n = deg(f);
long i, m;
if (IsZero(b)) {
xx.SetLength(0);
return;
}
m = n-1 - deg(b);
vec_ZZ_p x(INIT_SIZE, n);
for (i = 0; i <= m; i++)
InnerProduct(x[i], a, b.rep, i);
if (deg(b) != 0) {
ZZ_pX c(INIT_SIZE, n);
LeftShift(c, b, m);
for (i = m+1; i < n; i++) {
MulByXMod(c, c, f);
InnerProduct(x[i], a, c.rep);
}
}
xx = x;
}
void PlainTraceVec(vec_ZZ_p& S, const ZZ_pX& ff)
{
if (deg(ff) <= 0)
LogicError("TraceVec: bad args");
ZZ_pX f;
f = ff;
MakeMonic(f);
long n = deg(f);
S.SetLength(n);
if (n == 0)
return;
long k, i;
ZZ acc, t;
ZZ_p t1;
S[0] = n;
for (k = 1; k < n; k++) {
mul(acc, rep(f.rep[n-k]), k);
for (i = 1; i < k; i++) {
mul(t, rep(f.rep[n-i]), rep(S[k-i]));
add(acc, acc, t);
}
conv(t1, acc);
negate(S[k], t1);
}
}
void conv(vec_ZZ& x, const vec_ZZ_p& a)
{
long n = a.length();
x.SetLength(n);
long i;
for (i = 0; i < n; i++)
x[i] = rep(a[i]);
}
void VectorCopy(vec_ZZ_p& x, const vec_ZZ_p& a, long n)
{
if (n < 0) LogicError("VectorCopy: negative length");
if (NTL_OVERFLOW(n, 1, 0)) ResourceError("overflow in VectorCopy");
long m = min(n, a.length());
x.SetLength(n);
long i;
for (i = 0; i < m; i++)
x[i] = a[i];
for (i = m; i < n; i++)
clear(x[i]);
}
void ProjectPowers(vec_ZZ_p& x, const vec_ZZ_p& a, long k,
const ZZ_pX& h, const ZZ_pXModulus& F)
{
if (a.length() > F.n || k < 0) LogicError("ProjectPowers: bad args");
if (k == 0) {
x.SetLength(0);
return;
}
long m = SqrRoot(k);
ZZ_pXArgument H;
build(H, h, F, m);
ProjectPowers(x, a, k, H, F);
}
void FastTraceVec(vec_ZZ_p& S, const ZZ_pX& f)
{
long n = deg(f);
if (n <= 0)
LogicError("FastTraceVec: bad args");
if (n == 0) {
S.SetLength(0);
return;
}
if (n == 1) {
S.SetLength(1);
set(S[0]);
return;
}
long i;
ZZ_pX f1;
f1.rep.SetLength(n-1);
for (i = 0; i <= n-2; i++)
f1.rep[i] = f.rep[n-i];
f1.normalize();
ZZ_pX f2;
f2.rep.SetLength(n-1);
for (i = 0; i <= n-2; i++)
mul(f2.rep[i], f.rep[n-1-i], i+1);
f2.normalize();
ZZ_pX f3;
InvTrunc(f3, f1, n-1);
MulTrunc(f3, f3, f2, n-1);
S.SetLength(n);
S[0] = n;
for (i = 1; i < n; i++)
negate(S[i], coeff(f3, i-1));
}
long IsZero(const vec_ZZ_p& a)
{
long n = a.length();
long i;
for (i = 0; i < n; i++)
if (!IsZero(a[i]))
return 0;
return 1;
}
void MinPolySeq(ZZ_pX& h, const vec_ZZ_p& a, long m)
{
if (m < 0) LogicError("MinPoly: bad args");
if (NTL_OVERFLOW(m, 1, 0)) LogicError("MinPoly: bad args");
if (a.length() < 2*m) LogicError("MinPoly: sequence too short");
if (m > NTL_ZZ_pX_BERMASS_CROSSOVER)
GCDMinPolySeq(h, a, m);
else
BerlekampMassey(h, a, m);
}
static
void FindFactors(vec_ZZ_pX& factors, const ZZ_pX& f, const ZZ_pX& g,
const vec_ZZ_p& roots)
{
long r = roots.length();
factors.SetMaxLength(r);
factors.SetLength(0);
RecFindFactors(factors, f, g, roots, 0, r-1);
}
void InnerProduct(ZZ_p& x, const vec_ZZ_p& a, const vec_ZZ_p& b,
long offset)
{
if (offset < 0) LogicError("InnerProduct: negative offset");
if (NTL_OVERFLOW(offset, 1, 0))
ResourceError("InnerProduct: offset too big");
long n = min(a.length(), b.length()+offset);
long i;
NTL_ZZRegister(accum);
NTL_ZZRegister(t);
clear(accum);
for (i = offset; i < n; i++) {
mul(t, rep(a[i]), rep(b[i-offset]));
add(accum, accum, t);
}
conv(x, accum);
}
// inner product (svec_ZZ and svec_ZZ_p)
inline void InnerProduct(ZZ_p& result, const svec_ZZ& a, const vec_ZZ_p& b) {
if (a.length()!=b.length()) {
cerr<<"InnerProduct() length mismatch\n";
exit(1);
}
clear(result);
long an = a.nvalues();
const svec_ZZ::index_t* ai = a.indices();
const svec_ZZ::value_t* av = a.values();
for (long i=0; i<an; ++i)
result += to_ZZ_p(av[i])*b[ai[i]];
}
static
void mul_aux(vec_ZZ_p& x, const vec_ZZ_p& a, const mat_ZZ_p& 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 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);
}
}
static
void mul_aux(vec_ZZ_p& x, const mat_ZZ_p& A, const vec_ZZ_p& 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 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);
}
}
static
void IterFindFactors(vec_ZZ_pX& factors, const ZZ_pX& f,
const ZZ_pX& g, const vec_ZZ_p& roots)
{
long r = roots.length();
long i;
ZZ_pX h;
factors.SetLength(r);
for (i = 0; i < r; i++) {
sub(h, g, roots[i]);
GCD(factors[i], f, h);
}
}
static
void RecFindRoots(vec_ZZ_p& x, const ZZ_pX& f)
{
if (deg(f) == 0) return;
if (deg(f) == 1) {
long k = x.length();
x.SetLength(k+1);
negate(x[k], ConstTerm(f));
return;
}
ZZ_pX h;
ZZ_p r;
ZZ p1;
RightShift(p1, ZZ_p::modulus(), 1);
{
ZZ_pXModulus F;
build(F, f);
do {
random(r);
PowerXPlusAMod(h, r, p1, F);
add(h, h, -1);
GCD(h, h, f);
} while (deg(h) <= 0 || deg(h) == deg(f));
}
RecFindRoots(x, h);
div(h, f, h);
RecFindRoots(x, h);
}
void solve(ZZ_p& d, vec_ZZ_p& X,
const mat_ZZ_p& A, const vec_ZZ_p& b)
{
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);
X.SetLength(0);
return;
}
long i, j, k, pos;
ZZ t1, t2;
ZZ *x, *y;
const ZZ& p = ZZ_p::modulus();
vec_ZZVec M;
sqr(t1, p);
mul(t1, t1, n);
M.SetLength(n);
for (i = 0; i < n; i++) {
M[i].SetSize(n+1, t1.size());
for (j = 0; j < n; j++)
M[i][j] = rep(A[j][i]);
M[i][n] = rep(b[i]);
}
ZZ 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]);
NegateMod(det, det, p);
}
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);
NegateMod(t1, t1, p);
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);
}
void interpolate(ZZ_pX& f, const vec_ZZ_p& a, const vec_ZZ_p& b)
{
long m = a.length();
if (b.length() != m) LogicError("interpolate: vector length mismatch");
if (m == 0) {
clear(f);
return;
}
vec_ZZ_p prod;
prod = a;
ZZ_p t1, t2;
long k, i;
vec_ZZ_p res;
res.SetLength(m);
for (k = 0; k < m; k++) {
const ZZ_p& aa = a[k];
set(t1);
for (i = k-1; i >= 0; i--) {
mul(t1, t1, aa);
add(t1, t1, prod[i]);
}
clear(t2);
for (i = k-1; i >= 0; i--) {
mul(t2, t2, aa);
add(t2, t2, res[i]);
}
inv(t1, t1);
sub(t2, b[k], t2);
mul(t1, t1, t2);
for (i = 0; i < k; i++) {
mul(t2, prod[i], t1);
add(res[i], res[i], t2);
}
res[k] = t1;
if (k < m-1) {
if (k == 0)
negate(prod[0], prod[0]);
else {
negate(t1, a[k]);
add(prod[k], t1, prod[k-1]);
for (i = k-1; i >= 1; i--) {
mul(t2, prod[i], t1);
add(prod[i], t2, prod[i-1]);
}
mul(prod[0], prod[0], t1);
}
}
}
while (m > 0 && IsZero(res[m-1])) m--;
res.SetLength(m);
f.rep = res;
}
void BuildFromRoots(ZZ_pX& x, const vec_ZZ_p& a)
{
long n = a.length();
if (n == 0) {
set(x);
return;
}
long k0 = NextPowerOfTwo(NTL_ZZ_pX_FFT_CROSSOVER);
long crossover = 1L << k0;
if (n <= crossover) {
x.rep.SetMaxLength(n+1);
x.rep = a;
IterBuild(&x.rep[0], n);
x.rep.SetLength(n+1);
SetCoeff(x, n);
return;
}
long k = NextPowerOfTwo(n);
long m = 1L << k;
long i, j;
long l, width;
ZZ_pX b(INIT_SIZE, m+1);
b.rep = a;
b.rep.SetLength(m+1);
for (i = n; i < m; i++)
clear(b.rep[i]);
set(b.rep[m]);
FFTRep R1(INIT_SIZE, k), R2(INIT_SIZE, k);
ZZ_p t1, one;
set(one);
vec_ZZ_p G(INIT_SIZE, crossover), H(INIT_SIZE, crossover);
ZZ_p *g = G.elts();
ZZ_p *h = H.elts();
ZZ_p *tmp;
for (i = 0; i < m; i+= crossover) {
for (j = 0; j < crossover; j++)
negate(g[j], b.rep[i+j]);
if (k0 > 0) {
for (j = 0; j < crossover; j+=2) {
mul(t1, g[j], g[j+1]);
add(g[j+1], g[j], g[j+1]);
g[j] = t1;
}
}
for (l = 1; l < k0; l++) {
width = 1L << l;
for (j = 0; j < crossover; j += 2*width)
mul(&h[j], &g[j], &g[j+width], width);
tmp = g; g = h; h = tmp;
}
for (j = 0; j < crossover; j++)
b.rep[i+j] = g[j];
}
for (l = k0; l < k; l++) {
width = 1L << l;
for (i = 0; i < m; i += 2*width) {
t1 = b.rep[i+width];
set(b.rep[i+width]);
ToFFTRep(R1, b, l+1, i, i+width);
b.rep[i+width] = t1;
t1 = b.rep[i+2*width];
set(b.rep[i+2*width]);
ToFFTRep(R2, b, l+1, i+width, i+2*width);
b.rep[i+2*width] = t1;
mul(R1, R1, R2);
FromFFTRep(&b.rep[i], R1, 0, 2*width-1);
sub(b.rep[i], b.rep[i], one);
}
}
x.rep.SetLength(n+1);
long delta = m-n;
for (i = 0; i <= n; i++)
x.rep[i] = b.rep[i+delta];
// no need to normalize
}