X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Ffactor.cpp;h=204010be3750294640434dc28da698f9f452db69;hp=3a012405ffc2934ae9d91358261a039101343e2e;hb=edc92b7a463993da62357fb4afad053e8c6d0771;hpb=114449ae6f2cd3151d9b8342c570db021a9e892c

diff --git a/ginac/factor.cpp b/ginac/factor.cpp
index 3a012405..204010be 100644
--- a/ginac/factor.cpp
+++ b/ginac/factor.cpp
@@ -1,7 +1,12 @@
 /** @file factor.cpp
  *
- *  Polynomial factorization routines.
- *  Only univariate at the moment and completely non-optimized!
+ *  Polynomial factorization code (implementation).
+ *
+ *  Algorithms used can be found in
+ *    [W1]  An Improved Multivariate Polynomial Factoring Algorithm,
+ *          P.S.Wang, Mathematics of Computation, Vol. 32, No. 144 (1978) 1215--1231.
+ *    [GCL] Algorithms for Computer Algebra,
+ *          K.O.Geddes, S.R.Czapor, G.Labahn, Springer Verlag, 1992.
  */
 
 /*
@@ -22,6 +27,8 @@
  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
+//#define DEBUGFACTOR
+
 #include "factor.h"
 
 #include "ex.h"
@@ -36,589 +43,589 @@
 #include "add.h"
 
 #include <algorithm>
+#include <cmath>
+#include <limits>
 #include <list>
 #include <vector>
+#ifdef DEBUGFACTOR
+#include <ostream>
+#endif
 using namespace std;
 
 #include <cln/cln.h>
 using namespace cln;
 
-//#define DEBUGFACTOR
-
-#ifdef DEBUGFACTOR
-#include <ostream>
-#endif // def DEBUGFACTOR
-
 namespace GiNaC {
 
+#ifdef DEBUGFACTOR
+#define DCOUT(str) cout << #str << endl
+#define DCOUTVAR(var) cout << #var << ": " << var << endl
+#define DCOUT2(str,var) cout << #str << ": " << var << endl
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif
+
+// anonymous namespace to hide all utility functions
 namespace {
 
-typedef vector<cl_MI> Vec;
-typedef vector<Vec> VecVec;
-
+typedef vector<cl_MI> mvec;
 #ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Vec& v)
+ostream& operator<<(ostream& o, const vector<int>& v)
 {
-	Vec::const_iterator i = v.begin(), end = v.end();
+	vector<int>::const_iterator i = v.begin(), end = v.end();
 	while ( i != end ) {
 		o << *i++ << " ";
 	}
 	return o;
 }
-#endif // def DEBUGFACTOR
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const VecVec& v)
+ostream& operator<<(ostream& o, const vector<cl_I>& v)
 {
-	VecVec::const_iterator i = v.begin(), end = v.end();
+	vector<cl_I>::const_iterator i = v.begin(), end = v.end();
 	while ( i != end ) {
-		o << *i++ << endl;
+		o << *i << "[" << i-v.begin() << "]" << " ";
+		++i;
 	}
 	return o;
 }
-#endif // def DEBUGFACTOR
-
-struct Term
-{
-	cl_MI c;          // coefficient
-	unsigned int exp; // exponent >=0
-};
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Term& t)
+ostream& operator<<(ostream& o, const vector<cl_MI>& v)
 {
-	if ( t.exp ) {
-		o << "(" << t.c << ")x^" << t.exp;
+	vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
+	while ( i != end ) {
+		o << *i << "[" << i-v.begin() << "]" << " ";
+		++i;
 	}
-	else {
-		o << "(" << t.c << ")";
+	return o;
+}
+ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
+{
+	vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
+	while ( i != end ) {
+		o << i-v.begin() << ": " << *i << endl;
+		++i;
 	}
 	return o;
 }
-#endif // def DEBUGFACTOR
+#endif
 
-struct UniPoly
+////////////////////////////////////////////////////////////////////////////////
+// modular univariate polynomial code
+
+typedef std::vector<cln::cl_MI> umodpoly;
+typedef std::vector<cln::cl_I> upoly;
+typedef vector<umodpoly> upvec;
+
+// COPY FROM UPOLY.HPP
+
+// CHANGED size_t -> int !!!
+template<typename T> static int degree(const T& p)
 {
-	cl_modint_ring R;
-	list<Term> terms;  // highest exponent first
-
-	UniPoly(const cl_modint_ring& ring) : R(ring) { }
-	UniPoly(const cl_modint_ring& ring, const ex& poly, const ex& x) : R(ring)
-	{ 
-		// assert: poly is in Z[x]
-		Term t;
-		for ( int i=poly.degree(x); i>=poly.ldegree(x); --i ) {
-			int coeff = ex_to<numeric>(poly.coeff(x,i)).to_int();
-			if ( coeff ) {
-				t.c = R->canonhom(coeff);
-				if ( !zerop(t.c) ) {
-					t.exp = i;
-					terms.push_back(t);
-				}
-			}
-		}
-	}
-	UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
-	{
-		Term t;
-		for ( unsigned int i=0; i<v.size(); ++i ) {
-			if ( !zerop(v[i]) ) {
-				t.c = v[i];
-				t.exp = i;
-				terms.push_front(t);
-			}
-		}
+	return p.size() - 1;
+}
+
+template<typename T> static typename T::value_type lcoeff(const T& p)
+{
+	return p[p.size() - 1];
+}
+
+static bool normalize_in_field(umodpoly& a)
+{
+	if (a.size() == 0)
+		return true;
+	if ( lcoeff(a) == a[0].ring()->one() ) {
+		return true;
 	}
-	unsigned int degree() const
-	{
-		if ( terms.size() ) {
-			return terms.front().exp;
+
+	const cln::cl_MI lc_1 = recip(lcoeff(a));
+	for (std::size_t k = a.size(); k-- != 0; )
+		a[k] = a[k]*lc_1;
+	return false;
+}
+
+template<typename T> static void
+canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
+{
+	if (p.empty())
+		return;
+
+	std::size_t i = p.size() - 1;
+	// Be fast if the polynomial is already canonicalized
+	if (!zerop(p[i]))
+		return;
+
+	if (hint < p.size())
+		i = hint;
+
+	bool is_zero = false;
+	do {
+		if (!zerop(p[i])) {
+			++i;
+			break;
 		}
-		else {
-			return 0;
+		if (i == 0) {
+			is_zero = true;
+			break;
 		}
+		--i;
+	} while (true);
+
+	if (is_zero) {
+		p.clear();
+		return;
 	}
-	bool zero() const { return (terms.size() == 0); }
-	const cl_MI operator[](unsigned int deg) const
-	{
-		list<Term>::const_iterator i = terms.begin(), end = terms.end();
-		for ( ; i != end; ++i ) {
-			if ( i->exp == deg ) {
-				return i->c;
-			}
-			if ( i->exp < deg ) {
-				break;
-			}
-		}
-		return R->zero();
+
+	p.erase(p.begin() + i, p.end());
+}
+
+// END COPY FROM UPOLY.HPP
+
+static void expt_pos(umodpoly& a, unsigned int q)
+{
+	if ( a.empty() ) return;
+	cl_MI zero = a[0].ring()->zero(); 
+	int deg = degree(a);
+	a.resize(degree(a)*q+1, zero);
+	for ( int i=deg; i>0; --i ) {
+		a[i*q] = a[i];
+		a[i] = zero;
 	}
-	void set(unsigned int deg, const cl_MI& c)
-	{
-		list<Term>::iterator i = terms.begin(), end = terms.end();
-		while ( i != end ) {
-			if ( i->exp == deg ) {
-				if ( !zerop(c) ) {
-					i->c = c;
-				}
-				else {
-					terms.erase(i);
-				}
-				return;
-			}
-			if ( i->exp < deg ) {
-				break;
-			}
-			++i;
+}
+
+template<typename T>
+static T operator+(const T& a, const T& b)
+{
+	int sa = a.size();
+	int sb = b.size();
+	if ( sa >= sb ) {
+		T r(sa);
+		int i = 0;
+		for ( ; i<sb; ++i ) {
+			r[i] = a[i] + b[i];
 		}
-		if ( !zerop(c) ) {
-			Term t;
-			t.c = c;
-			t.exp = deg;
-			terms.insert(i, t);
+		for ( ; i<sa; ++i ) {
+			r[i] = a[i];
 		}
+		canonicalize(r);
+		return r;
 	}
-	ex to_ex(const ex& x, bool symmetric = true) const
-	{
-		ex r;
-		list<Term>::const_iterator i = terms.begin(), end = terms.end();
-		if ( symmetric ) {
-			numeric mod(R->modulus);
-			numeric halfmod = (mod-1)/2;
-			for ( ; i != end; ++i ) {
-				numeric n(R->retract(i->c));
-				if ( n > halfmod ) {
-					r += pow(x, i->exp) * (n-mod);
-				}
-				else {
-					r += pow(x, i->exp) * n;
-				}
-			}
+	else {
+		T r(sb);
+		int i = 0;
+		for ( ; i<sa; ++i ) {
+			r[i] = a[i] + b[i];
 		}
-		else {
-			for ( ; i != end; ++i ) {
-				r += pow(x, i->exp) * numeric(R->retract(i->c));
-			}
+		for ( ; i<sb; ++i ) {
+			r[i] = b[i];
 		}
+		canonicalize(r);
 		return r;
 	}
-	void unit_normal()
-	{
-		if ( terms.size() ) {
-			if ( terms.front().c != R->one() ) {
-				list<Term>::iterator i = terms.begin(), end = terms.end();
-				cl_MI cont = i->c;
-				i->c = R->one();
-				while ( ++i != end ) {
-					i->c = div(i->c, cont);
-					if ( zerop(i->c) ) {
-						terms.erase(i);
-					}
-				}
-			}
-		}
-	}
-	cl_MI unit() const
-	{
-		return terms.front().c;
-	}
-	void divide(const cl_MI& x)
-	{
-		list<Term>::iterator i = terms.begin(), end = terms.end();
-		for ( ; i != end; ++i ) {
-			i->c = div(i->c, x);
-			if ( zerop(i->c) ) {
-				terms.erase(i);
-			}
+}
+
+template<typename T>
+static T operator-(const T& a, const T& b)
+{
+	int sa = a.size();
+	int sb = b.size();
+	if ( sa >= sb ) {
+		T r(sa);
+		int i = 0;
+		for ( ; i<sb; ++i ) {
+			r[i] = a[i] - b[i];
 		}
-	}
-	void reduce_exponents(unsigned int prime)
-	{
-		list<Term>::iterator i = terms.begin(), end = terms.end();
-		while ( i != end ) {
-			if ( i->exp > 0 ) {
-				// assert: i->exp is multiple of prime
-				i->exp /= prime;
-			}
-			++i;
+		for ( ; i<sa; ++i ) {
+			r[i] = a[i];
 		}
+		canonicalize(r);
+		return r;
 	}
-	void deriv(UniPoly& d) const
-	{
-		list<Term>::const_iterator i = terms.begin(), end = terms.end();
-		while ( i != end ) {
-			if ( i->exp ) {
-				cl_MI newc = i->c * i->exp;
-				if ( !zerop(newc) ) {
-					Term t;
-					t.c = newc;
-					t.exp = i->exp-1;
-					d.terms.push_back(t);
-				}
-			}
-			++i;
+	else {
+		T r(sb);
+		int i = 0;
+		for ( ; i<sa; ++i ) {
+			r[i] = a[i] - b[i];
 		}
-	}
-	bool operator<(const UniPoly& o) const
-	{
-		if ( terms.size() != o.terms.size() ) {
-			return terms.size() < o.terms.size();
-		}
-		list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
-		list<Term>::const_iterator i2 = o.terms.begin();
-		while ( i1 != end ) {
-			if ( i1->exp != i2->exp ) {
-				return i1->exp < i2->exp;
-			}
-			if ( i1->c != i2->c ) {
-				return R->retract(i1->c) < R->retract(i2->c);
-			}
-			++i1; ++i2;
+		for ( ; i<sb; ++i ) {
+			r[i] = -b[i];
 		}
-		return true;
+		canonicalize(r);
+		return r;
 	}
-	bool operator==(const UniPoly& o) const
-	{
-		if ( terms.size() != o.terms.size() ) {
-			return false;
-		}
-		list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
-		list<Term>::const_iterator i2 = o.terms.begin();
-		while ( i1 != end ) {
-			if ( i1->exp != i2->exp ) {
-				return false;
-			}
-			if ( i1->c != i2->c ) {
-				return false;
-			}
-			++i1; ++i2;
+}
+
+static upoly operator*(const upoly& a, const upoly& b)
+{
+	upoly c;
+	if ( a.empty() || b.empty() ) return c;
+
+	int n = degree(a) + degree(b);
+	c.resize(n+1, 0);
+	for ( int i=0 ; i<=n; ++i ) {
+		for ( int j=0 ; j<=i; ++j ) {
+			if ( j > degree(a) || (i-j) > degree(b) ) continue;
+			c[i] = c[i] + a[j] * b[i-j];
 		}
-		return true;
-	}
-	bool operator!=(const UniPoly& o) const
-	{
-		bool res = !(*this == o);
-		return res;
 	}
-};
+	canonicalize(c);
+	return c;
+}
 
-static UniPoly operator*(const UniPoly& a, const UniPoly& b)
+static umodpoly operator*(const umodpoly& a, const umodpoly& b)
 {
-	unsigned int n = a.degree()+b.degree();
-	UniPoly c(a.R);
-	Term t;
-	for ( unsigned int i=0 ; i<=n; ++i ) {
-		t.c = a.R->zero();
-		for ( unsigned int j=0 ; j<=i; ++j ) {
-			t.c = t.c + a[j] * b[i-j];
-		}
-		if ( !zerop(t.c) ) {
-			t.exp = i;
-			c.terms.push_front(t);
+	umodpoly c;
+	if ( a.empty() || b.empty() ) return c;
+
+	int n = degree(a) + degree(b);
+	c.resize(n+1, a[0].ring()->zero());
+	for ( int i=0 ; i<=n; ++i ) {
+		for ( int j=0 ; j<=i; ++j ) {
+			if ( j > degree(a) || (i-j) > degree(b) ) continue;
+			c[i] = c[i] + a[j] * b[i-j];
 		}
 	}
+	canonicalize(c);
 	return c;
 }
 
-static UniPoly operator-(const UniPoly& a, const UniPoly& b)
+static upoly operator*(const upoly& a, const cl_I& x)
 {
-	list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
-	list<Term>::const_iterator ib = b.terms.begin(), bend = b.terms.end();
-	UniPoly c(a.R);
-	while ( ia != aend && ib != bend ) {
-		if ( ia->exp > ib->exp ) {
-			c.terms.push_back(*ia);
-			++ia;
-		}
-		else if ( ia->exp < ib->exp ) {
-			c.terms.push_back(*ib);
-			c.terms.back().c = -c.terms.back().c;
-			++ib;
-		}
-		else {
-			Term t;
-			t.exp = ia->exp;
-			t.c = ia->c - ib->c;
-			if ( !zerop(t.c) ) {
-				c.terms.push_back(t);
-			}
-			++ia; ++ib;
-		}
+	if ( zerop(x) ) {
+		upoly r;
+		return r;
 	}
-	while ( ia != aend ) {
-		c.terms.push_back(*ia);
-		++ia;
+	upoly r(a.size());
+	for ( size_t i=0; i<a.size(); ++i ) {
+		r[i] = a[i] * x;
 	}
-	while ( ib != bend ) {
-		c.terms.push_back(*ib);
-		c.terms.back().c = -c.terms.back().c;
-		++ib;
+	return r;
+}
+
+static upoly operator/(const upoly& a, const cl_I& x)
+{
+	if ( zerop(x) ) {
+		upoly r;
+		return r;
 	}
-	return c;
+	upoly r(a.size());
+	for ( size_t i=0; i<a.size(); ++i ) {
+		r[i] = exquo(a[i],x);
+	}
+	return r;
 }
 
-static UniPoly operator-(const UniPoly& a)
+static umodpoly operator*(const umodpoly& a, const cl_MI& x)
 {
-	list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
-	UniPoly c(a.R);
-	while ( ia != aend ) {
-		c.terms.push_back(*ia);
-		c.terms.back().c = -c.terms.back().c;
-		++ia;
+	umodpoly r(a.size());
+	for ( size_t i=0; i<a.size(); ++i ) {
+		r[i] = a[i] * x;
 	}
-	return c;
+	canonicalize(r);
+	return r;
 }
 
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPoly& t)
+static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
 {
-	list<Term>::const_iterator i = t.terms.begin(), end = t.terms.end();
-	if ( i == end ) {
-		o << "0";
-		return o;
+	// assert: e is in Z[x]
+	int deg = e.degree(x);
+	up.resize(deg+1);
+	int ldeg = e.ldegree(x);
+	for ( ; deg>=ldeg; --deg ) {
+		up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
 	}
-	for ( ; i != end; ) {
-		o << *i++;
-		if ( i != end ) {
-			o << " + ";
-		}
+	for ( ; deg>=0; --deg ) {
+		up[deg] = 0;
 	}
-	return o;
+	canonicalize(up);
 }
-#endif // def DEBUGFACTOR
 
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const list<UniPoly>& t)
+static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
 {
-	list<UniPoly>::const_iterator i = t.begin(), end = t.end();
-	o << "{" << endl;
-	for ( ; i != end; ) {
-		o << *i++ << endl;
+	int deg = degree(e);
+	ump.resize(deg+1);
+	for ( ; deg>=0; --deg ) {
+		ump[deg] = R->canonhom(e[deg]);
 	}
-	o << "}" << endl;
-	return o;
+	canonicalize(ump);
 }
-#endif // def DEBUGFACTOR
 
-typedef vector<UniPoly> UniPolyVec;
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
+{
+	// assert: e is in Z[x]
+	int deg = e.degree(x);
+	ump.resize(deg+1);
+	int ldeg = e.ldegree(x);
+	for ( ; deg>=ldeg; --deg ) {
+		cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
+		ump[deg] = R->canonhom(coeff);
+	}
+	for ( ; deg>=0; --deg ) {
+		ump[deg] = R->zero();
+	}
+	canonicalize(ump);
+}
 
-struct UniFactor
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
 {
-	UniPoly p;
-	unsigned int exp;
+	umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
+}
 
-	UniFactor(const cl_modint_ring& ring) : p(ring) { }
-	UniFactor(const UniPoly& p_, unsigned int exp_) : p(p_), exp(exp_) { }
-	bool operator<(const UniFactor& o) const
-	{
-		return p < o.p;
+static ex upoly_to_ex(const upoly& a, const ex& x)
+{
+	if ( a.empty() ) return 0;
+	ex e;
+	for ( int i=degree(a); i>=0; --i ) {
+		e += numeric(a[i]) * pow(x, i);
 	}
-};
+	return e;
+}
 
-struct UniFactorVec
+static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
 {
-	vector<UniFactor> factors;
-
-	void unique()
-	{
-		sort(factors.begin(), factors.end());
-		if ( factors.size() > 1 ) {
-			vector<UniFactor>::iterator i = factors.begin();
-			vector<UniFactor>::const_iterator cmp = factors.begin()+1;
-			vector<UniFactor>::iterator end = factors.end();
-			while ( cmp != end ) {
-				if ( i->p != cmp->p ) {
-					++i;
-					++cmp;
-				}
-				else {
-					i->exp += cmp->exp;
-					++cmp;
-				}
-			}
-			if ( i != end-1 ) {
-				factors.erase(i+1, end);
-			}
+	if ( a.empty() ) return 0;
+	cl_modint_ring R = a[0].ring();
+	cl_I mod = R->modulus;
+	cl_I halfmod = (mod-1) >> 1;
+	ex e;
+	for ( int i=degree(a); i>=0; --i ) {
+		cl_I n = R->retract(a[i]);
+		if ( n > halfmod ) {
+			e += numeric(n-mod) * pow(x, i);
+		} else {
+			e += numeric(n) * pow(x, i);
 		}
 	}
-};
+	return e;
+}
 
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniFactorVec& ufv)
+static upoly umodpoly_to_upoly(const umodpoly& a)
 {
-	for ( size_t i=0; i<ufv.factors.size(); ++i ) {
-		if ( i != ufv.factors.size()-1 ) {
-			o << "*";
-		}
-		else {
-			o << " ";
+	upoly e(a.size());
+	if ( a.empty() ) return e;
+	cl_modint_ring R = a[0].ring();
+	cl_I mod = R->modulus;
+	cl_I halfmod = (mod-1) >> 1;
+	for ( int i=degree(a); i>=0; --i ) {
+		cl_I n = R->retract(a[i]);
+		if ( n > halfmod ) {
+			e[i] = n-mod;
+		} else {
+			e[i] = n;
 		}
-		o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
 	}
-	return o;
+	return e;
 }
-#endif // def DEBUGFACTOR
 
-static void rem(const UniPoly& a_, const UniPoly& b, UniPoly& c)
+static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
 {
-	if ( a_.degree() < b.degree() ) {
-		c = a_;
-		return;
-	}
+	umodpoly e;
+	if ( a.empty() ) return e;
+	cl_modint_ring oldR = a[0].ring();
+	size_t sa = a.size();
+	e.resize(sa+m, R->zero());
+	for ( int i=0; i<sa; ++i ) {
+		e[i+m] = R->canonhom(oldR->retract(a[i]));
+	}
+	canonicalize(e);
+	return e;
+}
 
-	unsigned int k, n;
-	n = b.degree();
-	k = a_.degree() - n;
+/** Divides all coefficients of the polynomial a by the integer x.
+ *  All coefficients are supposed to be divisible by x. If they are not, the
+ *  the<cl_I> cast will raise an exception.
+ *
+ *  @param[in,out] a  polynomial of which the coefficients will be reduced by x
+ *  @param[in]     x  integer that divides the coefficients
+ */
+static void reduce_coeff(umodpoly& a, const cl_I& x)
+{
+	if ( a.empty() ) return;
 
-	if ( n == 0 ) {
-		c.terms.clear();
-		return;
+	cl_modint_ring R = a[0].ring();
+	umodpoly::iterator i = a.begin(), end = a.end();
+	for ( ; i!=end; ++i ) {
+		// cln cannot perform this division in the modular field
+		cl_I c = R->retract(*i);
+		*i = cl_MI(R, the<cl_I>(c / x));
 	}
+}
 
-	c = a_;
-	Term termbuf;
-
-	while ( true ) {
-		cl_MI qk = div(c[n+k], b[n]);
+/** Calculates remainder of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] r  polynomial remainder
+ */
+static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
+{
+	int k, n;
+	n = degree(b);
+	k = degree(a) - n;
+	r = a;
+	if ( k < 0 ) return;
+
+	do {
+		cl_MI qk = div(r[n+k], b[n]);
 		if ( !zerop(qk) ) {
-			unsigned int j;
-			for ( unsigned int i=0; i<n; ++i ) {
-				j = n + k - 1 - i;
-				c.set(j, c[j] - qk*b[j-k]);
+			for ( int i=0; i<n; ++i ) {
+				unsigned int j = n + k - 1 - i;
+				r[j] = r[j] - qk * b[j-k];
 			}
 		}
-		if ( k == 0 ) break;
-		--k;
-	}
-	list<Term>::iterator i = c.terms.begin(), end = c.terms.end();
-	while ( i != end ) {
-		if ( i->exp <= n-1 ) {
-			break;
-		}
-		++i;
-	}
-	c.terms.erase(c.terms.begin(), i);
+	} while ( k-- );
+
+	fill(r.begin()+n, r.end(), a[0].ring()->zero());
+	canonicalize(r);
 }
 
-static void div(const UniPoly& a_, const UniPoly& b, UniPoly& q)
+/** Calculates quotient of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] q  polynomial quotient
+ */
+static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
 {
-	if ( a_.degree() < b.degree() ) {
-		q.terms.clear();
-		return;
-	}
-
-	unsigned int k, n;
-	n = b.degree();
-	k = a_.degree() - n;
+	int k, n;
+	n = degree(b);
+	k = degree(a) - n;
+	q.clear();
+	if ( k < 0 ) return;
+
+	umodpoly r = a;
+	q.resize(k+1, a[0].ring()->zero());
+	do {
+		cl_MI qk = div(r[n+k], b[n]);
+		if ( !zerop(qk) ) {
+			q[k] = qk;
+			for ( int i=0; i<n; ++i ) {
+				unsigned int j = n + k - 1 - i;
+				r[j] = r[j] - qk * b[j-k];
+			}
+		}
+	} while ( k-- );
 
-	UniPoly c = a_;
-	Term termbuf;
+	canonicalize(q);
+}
 
-	while ( true ) {
-		cl_MI qk = div(c[n+k], b[n]);
+/** Calculates quotient and remainder of a/b.
+ *  Assertion: a and b not empty.
+ *
+ *  @param[in]  a  polynomial dividend
+ *  @param[in]  b  polynomial divisor
+ *  @param[out] r  polynomial remainder
+ *  @param[out] q  polynomial quotient
+ */
+static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
+{
+	int k, n;
+	n = degree(b);
+	k = degree(a) - n;
+	q.clear();
+	r = a;
+	if ( k < 0 ) return;
+
+	q.resize(k+1, a[0].ring()->zero());
+	do {
+		cl_MI qk = div(r[n+k], b[n]);
 		if ( !zerop(qk) ) {
-			Term t;
-			t.c = qk;
-			t.exp = k;
-			q.terms.push_back(t);
-			unsigned int j;
-			for ( unsigned int i=0; i<n; ++i ) {
-				j = n + k - 1 - i;
-				c.set(j, c[j] - qk*b[j-k]);
+			q[k] = qk;
+			for ( int i=0; i<n; ++i ) {
+				unsigned int j = n + k - 1 - i;
+				r[j] = r[j] - qk * b[j-k];
 			}
 		}
-		if ( k == 0 ) break;
-		--k;
-	}
+	} while ( k-- );
+
+	fill(r.begin()+n, r.end(), a[0].ring()->zero());
+	canonicalize(r);
+	canonicalize(q);
 }
 
-static void gcd(const UniPoly& a, const UniPoly& b, UniPoly& c)
+/** Calculates the GCD of polynomial a and b.
+ *
+ *  @param[in]  a  polynomial
+ *  @param[in]  b  polynomial
+ *  @param[out] c  GCD
+ */
+static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
 {
-	c = a;
-	c.unit_normal();
-	UniPoly d = b;
-	d.unit_normal();
-
-	if ( c.degree() < d.degree() ) {
-		gcd(b, a, c);
-		return;
-	}
+	if ( degree(a) < degree(b) ) return gcd(b, a, c);
 
-	while ( !d.zero() ) {
-		UniPoly r(a.R);
+	c = a;
+	normalize_in_field(c);
+	umodpoly d = b;
+	normalize_in_field(d);
+	umodpoly r;
+	while ( !d.empty() ) {
 		rem(c, d, r);
 		c = d;
 		d = r;
 	}
-	c.unit_normal();
+	normalize_in_field(c);
 }
 
-static bool is_one(const UniPoly& w)
+/** Calculates the derivative of the polynomial a.
+ *  
+ *  @param[in]  a  polynomial of which to take the derivative
+ *  @param[out] d  result/derivative
+ */
+static void deriv(const umodpoly& a, umodpoly& d)
 {
-	if ( w.terms.size() == 1 && w[0] == w.R->one() ) {
-		return true;
+	d.clear();
+	if ( a.size() <= 1 ) return;
+
+	d.insert(d.begin(), a.begin()+1, a.end());
+	int max = d.size();
+	for ( int i=1; i<max; ++i ) {
+		d[i] = d[i] * (i+1);
 	}
-	return false;
+	canonicalize(d);
 }
 
-static void sqrfree_main(const UniPoly& a, UniFactorVec& fvec)
+static bool unequal_one(const umodpoly& a)
 {
-	unsigned int i = 1;
-	UniPoly b(a.R);
-	a.deriv(b);
-	if ( !b.zero() ) {
-		UniPoly c(a.R), w(a.R);
-		gcd(a, b, c);
-		div(a, c, w);
-		while ( !is_one(w) ) {
-			UniPoly y(a.R), z(a.R);
-			gcd(w, c, y);
-			div(w, y, z);
-			if ( !is_one(z) ) {
-				UniFactor uf(z, i++);
-				fvec.factors.push_back(uf);
-			}
-			w = y;
-			UniPoly cbuf(a.R);
-			div(c, y, cbuf);
-			c = cbuf;
-		}
-		if ( !is_one(c) ) {
-			unsigned int prime = cl_I_to_uint(c.R->modulus);
-			c.reduce_exponents(prime);
-			unsigned int pos = fvec.factors.size();
-			sqrfree_main(c, fvec);
-			for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
-				fvec.factors[p].exp *= prime;
-			}
-			return;
-		}
-	}
-	else {
-		unsigned int prime = cl_I_to_uint(a.R->modulus);
-		UniPoly amod = a;
-		amod.reduce_exponents(prime);
-		unsigned int pos = fvec.factors.size();
-		sqrfree_main(amod, fvec);
-		for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
-			fvec.factors[p].exp *= prime;
-		}
-		return;
-	}
+	if ( a.empty() ) return true;
+	return ( a.size() != 1 || a[0] != a[0].ring()->one() );
 }
 
-static void squarefree(const UniPoly& a, UniFactorVec& fvec)
+static bool equal_one(const umodpoly& a)
 {
-	sqrfree_main(a, fvec);
-	fvec.unique();
+	return ( a.size() == 1 && a[0] == a[0].ring()->one() );
 }
 
-class Matrix
+/** Returns true if polynomial a is square free.
+ *
+ *  @param[in] a  polynomial to check
+ *  @return       true if polynomial is square free, false otherwise
+ */
+static bool squarefree(const umodpoly& a)
 {
-	friend ostream& operator<<(ostream& o, const Matrix& m);
-public:
-	Matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
-	{
-		m.resize(c*r, init);
+	umodpoly b;
+	deriv(a, b);
+	if ( b.empty() ) {
+		return true;
+	}
+	umodpoly c;
+	gcd(a, b, c);
+	return equal_one(c);
+}
+
+// END modular univariate polynomial code
+////////////////////////////////////////////////////////////////////////////////
+
+////////////////////////////////////////////////////////////////////////////////
+// modular matrix
+
+class modular_matrix
+{
+	friend ostream& operator<<(ostream& o, const modular_matrix& m);
+public:
+	modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
+	{
+		m.resize(c*r, init);
 	}
 	size_t rowsize() const { return r; }
 	size_t colsize() const { return c; }
@@ -626,7 +633,7 @@ public:
 	cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
 	void mul_col(size_t col, const cl_MI x)
 	{
-		Vec::iterator i = m.begin() + col;
+		mvec::iterator i = m.begin() + col;
 		for ( size_t rc=0; rc<r; ++rc ) {
 			*i = *i * x;
 			i += c;
@@ -634,8 +641,8 @@ public:
 	}
 	void sub_col(size_t col1, size_t col2, const cl_MI fac)
 	{
-		Vec::iterator i1 = m.begin() + col1;
-		Vec::iterator i2 = m.begin() + col2;
+		mvec::iterator i1 = m.begin() + col1;
+		mvec::iterator i2 = m.begin() + col2;
 		for ( size_t rc=0; rc<r; ++rc ) {
 			*i1 = *i1 - *i2 * fac;
 			i1 += c;
@@ -645,17 +652,57 @@ public:
 	void switch_col(size_t col1, size_t col2)
 	{
 		cl_MI buf;
-		Vec::iterator i1 = m.begin() + col1;
-		Vec::iterator i2 = m.begin() + col2;
+		mvec::iterator i1 = m.begin() + col1;
+		mvec::iterator i2 = m.begin() + col2;
 		for ( size_t rc=0; rc<r; ++rc ) {
 			buf = *i1; *i1 = *i2; *i2 = buf;
 			i1 += c;
 			i2 += c;
 		}
 	}
+	void mul_row(size_t row, const cl_MI x)
+	{
+		vector<cl_MI>::iterator i = m.begin() + row*c;
+		for ( size_t cc=0; cc<c; ++cc ) {
+			*i = *i * x;
+			++i;
+		}
+	}
+	void sub_row(size_t row1, size_t row2, const cl_MI fac)
+	{
+		vector<cl_MI>::iterator i1 = m.begin() + row1*c;
+		vector<cl_MI>::iterator i2 = m.begin() + row2*c;
+		for ( size_t cc=0; cc<c; ++cc ) {
+			*i1 = *i1 - *i2 * fac;
+			++i1;
+			++i2;
+		}
+	}
+	void switch_row(size_t row1, size_t row2)
+	{
+		cl_MI buf;
+		vector<cl_MI>::iterator i1 = m.begin() + row1*c;
+		vector<cl_MI>::iterator i2 = m.begin() + row2*c;
+		for ( size_t cc=0; cc<c; ++cc ) {
+			buf = *i1; *i1 = *i2; *i2 = buf;
+			++i1;
+			++i2;
+		}
+	}
+	bool is_col_zero(size_t col) const
+	{
+		mvec::const_iterator i = m.begin() + col;
+		for ( size_t rr=0; rr<r; ++rr ) {
+			if ( !zerop(*i) ) {
+				return false;
+			}
+			i += c;
+		}
+		return true;
+	}
 	bool is_row_zero(size_t row) const
 	{
-		Vec::const_iterator i = m.begin() + row*c;
+		mvec::const_iterator i = m.begin() + row*c;
 		for ( size_t cc=0; cc<c; ++cc ) {
 			if ( !zerop(*i) ) {
 				return false;
@@ -666,47 +713,76 @@ public:
 	}
 	void set_row(size_t row, const vector<cl_MI>& newrow)
 	{
-		Vec::iterator i1 = m.begin() + row*c;
-		Vec::const_iterator i2 = newrow.begin(), end = newrow.end();
+		mvec::iterator i1 = m.begin() + row*c;
+		mvec::const_iterator i2 = newrow.begin(), end = newrow.end();
 		for ( ; i2 != end; ++i1, ++i2 ) {
 			*i1 = *i2;
 		}
 	}
-	Vec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
-	Vec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
+	mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
+	mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
 private:
 	size_t r, c;
-	Vec m;
+	mvec m;
 };
 
 #ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Matrix& m)
+modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
 {
-	vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
-	size_t wrap = 1;
-	for ( ; i != end; ++i ) {
-		o << *i << " ";
-		if ( !(wrap++ % m.c) ) {
-			o << endl;
+	const unsigned int r = m1.rowsize();
+	const unsigned int c = m2.colsize();
+	modular_matrix o(r,c,m1(0,0));
+
+	for ( size_t i=0; i<r; ++i ) {
+		for ( size_t j=0; j<c; ++j ) {
+			cl_MI buf;
+			buf = m1(i,0) * m2(0,j);
+			for ( size_t k=1; k<c; ++k ) {
+				buf = buf + m1(i,k)*m2(k,j);
+			}
+			o(i,j) = buf;
 		}
 	}
-	o << endl;
+	return o;
+}
+
+ostream& operator<<(ostream& o, const modular_matrix& m)
+{
+	cl_modint_ring R = m(0,0).ring();
+	o << "{";
+	for ( size_t i=0; i<m.rowsize(); ++i ) {
+		o << "{";
+		for ( size_t j=0; j<m.colsize()-1; ++j ) {
+			o << R->retract(m(i,j)) << ",";
+		}
+		o << R->retract(m(i,m.colsize()-1)) << "}";
+		if ( i != m.rowsize()-1 ) {
+			o << ",";
+		}
+	}
+	o << "}";
 	return o;
 }
 #endif // def DEBUGFACTOR
 
-static void q_matrix(const UniPoly& a, Matrix& Q)
+// END modular matrix
+////////////////////////////////////////////////////////////////////////////////
+
+static void q_matrix(const umodpoly& a_, modular_matrix& Q)
 {
-	unsigned int n = a.degree();
-	unsigned int q = cl_I_to_uint(a.R->modulus);
-	vector<cl_MI> r(n, a.R->zero());
-	r[0] = a.R->one();
+	umodpoly a = a_;
+	normalize_in_field(a);
+
+	int n = degree(a);
+	unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
+	umodpoly r(n, a[0].ring()->zero());
+	r[0] = a[0].ring()->one();
 	Q.set_row(0, r);
 	unsigned int max = (n-1) * q;
 	for ( size_t m=1; m<=max; ++m ) {
 		cl_MI rn_1 = r.back();
 		for ( size_t i=n-1; i>0; --i ) {
-			r[i] = r[i-1] - rn_1 * a[i];
+			r[i] = r[i-1] - (rn_1 * a[i]);
 		}
 		r[0] = -rn_1 * a[0];
 		if ( (m % q) == 0 ) {
@@ -715,7 +791,7 @@ static void q_matrix(const UniPoly& a, Matrix& Q)
 	}
 }
 
-static void nullspace(Matrix& M, vector<Vec>& basis)
+static void nullspace(modular_matrix& M, vector<mvec>& basis)
 {
 	const size_t n = M.rowsize();
 	const cl_MI one = M(0,0).ring()->one();
@@ -753,58 +829,65 @@ static void nullspace(Matrix& M, vector<Vec>& basis)
 	}
 	for ( size_t i=0; i<n; ++i ) {
 		if ( !M.is_row_zero(i) ) {
-			Vec nu(M.row_begin(i), M.row_end(i));
+			mvec nu(M.row_begin(i), M.row_end(i));
 			basis.push_back(nu);
 		}
 	}
 }
 
-static void berlekamp(const UniPoly& a, UniPolyVec& upv)
+static void berlekamp(const umodpoly& a, upvec& upv)
 {
-	Matrix Q(a.degree(), a.degree(), a.R->zero());
+	cl_modint_ring R = a[0].ring();
+	umodpoly one(1, R->one());
+
+	modular_matrix Q(degree(a), degree(a), R->zero());
 	q_matrix(a, Q);
-	VecVec nu;
+	vector<mvec> nu;
 	nullspace(Q, nu);
+
 	const unsigned int k = nu.size();
 	if ( k == 1 ) {
 		return;
 	}
 
-	list<UniPoly> factors;
+	list<umodpoly> factors;
 	factors.push_back(a);
 	unsigned int size = 1;
 	unsigned int r = 1;
-	unsigned int q = cl_I_to_uint(a.R->modulus);
+	unsigned int q = cl_I_to_uint(R->modulus);
 
-	list<UniPoly>::iterator u = factors.begin();
+	list<umodpoly>::iterator u = factors.begin();
 
 	while ( true ) {
 		for ( unsigned int s=0; s<q; ++s ) {
-			UniPoly g(a.R);
-			UniPoly nur(a.R, nu[r]);
-			nur.set(0, nur[0] - cl_MI(a.R, s));
+			umodpoly nur = nu[r];
+			nur[0] = nur[0] - cl_MI(R, s);
+			canonicalize(nur);
+			umodpoly g;
 			gcd(nur, *u, g);
-			if ( !is_one(g) && g != *u ) {
-				UniPoly uo(a.R);
+			if ( unequal_one(g) && g != *u ) {
+				umodpoly uo;
 				div(*u, g, uo);
-				if ( is_one(uo) ) {
+				if ( equal_one(uo) ) {
 					throw logic_error("berlekamp: unexpected divisor.");
 				}
 				else {
 					*u = uo;
 				}
 				factors.push_back(g);
-				++size;
+				size = 0;
+				list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
+				while ( i != end ) {
+					if ( degree(*i) ) ++size; 
+					++i;
+				}
 				if ( size == k ) {
-					list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+					list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
 					while ( i != end ) {
 						upv.push_back(*i++);
 					}
 					return;
 				}
-				if ( u->degree() < nur.degree() ) {
-					break;
-				}
 			}
 		}
 		if ( ++r == k ) {
@@ -814,121 +897,299 @@ static void berlekamp(const UniPoly& a, UniPolyVec& upv)
 	}
 }
 
-static void factor_modular(const UniPoly& p, UniPolyVec& upv)
+static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
 {
-	berlekamp(p, upv);
-	return;
+	size_t newdeg = degree(a)/prime;
+	ap.resize(newdeg+1);
+	ap[0] = a[0];
+	for ( size_t i=1; i<=newdeg; ++i ) {
+		ap[i] = a[i*prime];
+	}
 }
 
-static void exteuclid(const UniPoly& a, const UniPoly& b, UniPoly& g, UniPoly& s, UniPoly& t)
+static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
 {
-	if ( a.degree() < b.degree() ) {
-		exteuclid(b, a, g, t, s);
+	const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
+	int i = 1;
+	umodpoly b;
+	deriv(a, b);
+	if ( b.size() ) {
+		umodpoly c;
+		gcd(a, b, c);
+		umodpoly w;
+		div(a, c, w);
+		while ( unequal_one(w) ) {
+			umodpoly y;
+			gcd(w, c, y);
+			umodpoly z;
+			div(w, y, z);
+			factors.push_back(z);
+			mult.push_back(i);
+			++i;
+			w = y;
+			umodpoly buf;
+			div(c, y, buf);
+			c = buf;
+		}
+		if ( unequal_one(c) ) {
+			umodpoly cp;
+			expt_1_over_p(c, prime, cp);
+			size_t previ = mult.size();
+			modsqrfree(cp, factors, mult);
+			for ( size_t i=previ; i<mult.size(); ++i ) {
+				mult[i] *= prime;
+			}
+		}
+	}
+	else {
+		umodpoly ap;
+		expt_1_over_p(a, prime, ap);
+		size_t previ = mult.size();
+		modsqrfree(ap, factors, mult);
+		for ( size_t i=previ; i<mult.size(); ++i ) {
+			mult[i] *= prime;
+		}
+	}
+}
+
+static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
+{
+	umodpoly a = a_;
+
+	cl_modint_ring R = a[0].ring();
+	int q = cl_I_to_int(R->modulus);
+	int nhalf = degree(a)/2;
+
+	int i = 1;
+	umodpoly w(2);
+	w[0] = R->zero();
+	w[1] = R->one();
+	umodpoly x = w;
+
+	bool nontrivial = false;
+	while ( i <= nhalf ) {
+		expt_pos(w, q);
+		umodpoly buf;
+		rem(w, a, buf);
+		w = buf;
+		umodpoly wx = w - x;
+		gcd(a, wx, buf);
+		if ( unequal_one(buf) ) {
+			degrees.push_back(i);
+			ddfactors.push_back(buf);
+		}
+		if ( unequal_one(buf) ) {
+			umodpoly buf2;
+			div(a, buf, buf2);
+			a = buf2;
+			nhalf = degree(a)/2;
+			rem(w, a, buf);
+			w = buf;
+		}
+		++i;
+	}
+	if ( unequal_one(a) ) {
+		degrees.push_back(degree(a));
+		ddfactors.push_back(a);
+	}
+}
+
+static void same_degree_factor(const umodpoly& a, upvec& upv)
+{
+	cl_modint_ring R = a[0].ring();
+	int deg = degree(a);
+
+	vector<int> degrees;
+	upvec ddfactors;
+	distinct_degree_factor(a, degrees, ddfactors);
+
+	for ( size_t i=0; i<degrees.size(); ++i ) {
+		if ( degrees[i] == degree(ddfactors[i]) ) {
+			upv.push_back(ddfactors[i]);
+		}
+		else {
+			berlekamp(ddfactors[i], upv);
+		}
+	}
+}
+
+static void factor_modular(const umodpoly& p, upvec& upv)
+{
+	upvec factors;
+	vector<int> mult;
+	modsqrfree(p, factors, mult);
+
+#define USE_SAME_DEGREE_FACTOR
+#ifdef USE_SAME_DEGREE_FACTOR
+	for ( size_t i=0; i<factors.size(); ++i ) {
+		upvec upvbuf;
+		same_degree_factor(factors[i], upvbuf);
+		for ( int j=mult[i]; j>0; --j ) {
+			upv.insert(upv.end(), upvbuf.begin(), upvbuf.end());
+		}
+	}
+#else
+	for ( size_t i=0; i<factors.size(); ++i ) {
+		upvec upvbuf;
+		berlekamp(factors[i], upvbuf);
+		if ( upvbuf.size() ) {
+			for ( size_t j=0; j<upvbuf.size(); ++j ) {
+				upv.insert(upv.end(), mult[i], upvbuf[j]);
+			}
+		}
+		else {
+			for ( int j=mult[i]; j>0; --j ) {
+				upv.push_back(factors[i]);
+			}
+		}
+	}
+#endif
+}
+
+/** Calculates polynomials s and t such that a*s+b*t==1.
+ *  Assertion: a and b are relatively prime and not zero.
+ *
+ *  @param[in]  a  polynomial
+ *  @param[in]  b  polynomial
+ *  @param[out] s  polynomial
+ *  @param[out] t  polynomial
+ */
+static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
+{
+	if ( degree(a) < degree(b) ) {
+		exteuclid(b, a, t, s);
 		return;
 	}
-	UniPoly c1(a.R), c2(a.R), d1(a.R), d2(a.R), q(a.R), r(a.R), r1(a.R), r2(a.R);
-	UniPoly c = a; c.unit_normal();
-	UniPoly d = b; d.unit_normal();
-	c1.set(0, a.R->one());
-	d2.set(0, a.R->one());
-	while ( !d.zero() ) {
-		q.terms.clear();
+
+	umodpoly one(1, a[0].ring()->one());
+	umodpoly c = a; normalize_in_field(c);
+	umodpoly d = b; normalize_in_field(d);
+	s = one;
+	t.clear();
+	umodpoly d1;
+	umodpoly d2 = one;
+	umodpoly q;
+	while ( true ) {
 		div(c, d, q);
-		r = c - q * d;
-		r1 = c1 - q * d1;
-		r2 = c2 - q * d2;
+		umodpoly r = c - q * d;
+		umodpoly r1 = s - q * d1;
+		umodpoly r2 = t - q * d2;
 		c = d;
-		c1 = d1;
-		c2 = d2;
+		s = d1;
+		t = d2;
+		if ( r.empty() ) break;
 		d = r;
 		d1 = r1;
 		d2 = r2;
 	}
-	g = c; g.unit_normal();
-	s = c1;
-	s.divide(a.unit());
-	s.divide(c.unit());
-	t = c2;
-	t.divide(b.unit());
-	t.divide(c.unit());
+	cl_MI fac = recip(lcoeff(a) * lcoeff(c));
+	umodpoly::iterator i = s.begin(), end = s.end();
+	for ( ; i!=end; ++i ) {
+		*i = *i * fac;
+	}
+	canonicalize(s);
+	fac = recip(lcoeff(b) * lcoeff(c));
+	i = t.begin(), end = t.end();
+	for ( ; i!=end; ++i ) {
+		*i = *i * fac;
+	}
+	canonicalize(t);
 }
 
-static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
+static upoly replace_lc(const upoly& poly, const cl_I& lc)
 {
-	ex r = expand(poly + (lc - poly.lcoeff(x)) * pow(x, poly.degree(x)));
+	if ( poly.empty() ) return poly;
+	upoly r = poly;
+	r.back() = lc;
 	return r;
 }
 
-static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const UniPoly& u1_, const UniPoly& w1_, const ex& gamma_ = 0)
+static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
 {
-	ex a = a_;
-	const cl_modint_ring& R = u1_.R;
+	upoly a = a_;
+	const cl_modint_ring& R = u1_[0].ring();
 
 	// calc bound B
-	ex maxcoeff;
-	for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
-		maxcoeff += pow(abs(a.coeff(x, i)),2);
+	cl_R maxcoeff = 0;
+	for ( int i=degree(a); i>=0; --i ) {
+		maxcoeff = maxcoeff + square(abs(a[i]));
 	}
-	cl_I normmc = ceiling1(the<cl_F>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
-	unsigned int maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
-	unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+	cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(maxcoeff)));
+	cl_I maxdegree = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
+	cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
 
 	// step 1
-	ex alpha = a.lcoeff(x);
-	ex gamma = gamma_;
-	if ( gamma == 0 ) {
-		gamma = alpha;
-	}
-	unsigned int gamma_ui = ex_to<numeric>(abs(gamma)).to_int();
-	a = a * gamma;
-	UniPoly nu1 = u1_;
-	nu1.unit_normal();
-	UniPoly nw1 = w1_;
-	nw1.unit_normal();
-	ex phi;
-	phi = expand(gamma * nu1.to_ex(x));
-	UniPoly u1(R, phi, x);
-	phi = expand(alpha * nw1.to_ex(x));
-	UniPoly w1(R, phi, x);
+	cl_I alpha = lcoeff(a);
+	a = a * alpha;
+	umodpoly nu1 = u1_;
+	normalize_in_field(nu1);
+	umodpoly nw1 = w1_;
+	normalize_in_field(nw1);
+	upoly phi;
+	phi = umodpoly_to_upoly(nu1) * alpha;
+	umodpoly u1;
+	umodpoly_from_upoly(u1, phi, R);
+	phi = umodpoly_to_upoly(nw1) * alpha;
+	umodpoly w1;
+	umodpoly_from_upoly(w1, phi, R);
 
 	// step 2
-	UniPoly s(R), t(R), g(R);
-	exteuclid(u1, w1, g, s, t);
+	umodpoly s;
+	umodpoly t;
+	exteuclid(u1, w1, s, t);
 
 	// step 3
-	ex u = replace_lc(u1.to_ex(x), x, gamma);
-	ex w = replace_lc(w1.to_ex(x), x, alpha);
-	ex e = expand(a - u * w);
-	unsigned int modulus = p;
+	u = replace_lc(umodpoly_to_upoly(u1), alpha);
+	w = replace_lc(umodpoly_to_upoly(w1), alpha);
+	upoly e = a - u * w;
+	cl_I modulus = p;
+	const cl_I maxmodulus = 2*B*abs(alpha);
 
 	// step 4
-	while ( !e.is_zero() && modulus < 2*B*gamma_ui ) {
-		ex c = e / modulus;
-		phi = expand(s.to_ex(x)*c);
-		UniPoly sigmatilde(R, phi, x);
-		phi = expand(t.to_ex(x)*c);
-		UniPoly tautilde(R, phi, x);
-		UniPoly q(R), r(R);
-		div(sigmatilde, w1, q);
-		rem(sigmatilde, w1, r);
-		UniPoly sigma = r;
-		phi = expand(tautilde.to_ex(x) + q.to_ex(x) * u1.to_ex(x));
-		UniPoly tau(R, phi, x);
-		u = expand(u + tau.to_ex(x) * modulus);
-		w = expand(w + sigma.to_ex(x) * modulus);
-		e = expand(a - u * w);
+	while ( !e.empty() && modulus < maxmodulus ) {
+		// ad-hoc divisablity check
+		for ( size_t k=0; k<e.size(); ++k ) {
+			if ( !zerop(mod(e[k], modulus)) ) {
+				goto quickexit;
+			}
+		}
+		upoly c = e / modulus;
+		phi = umodpoly_to_upoly(s) * c;
+		umodpoly sigmatilde;
+		umodpoly_from_upoly(sigmatilde, phi, R);
+		phi = umodpoly_to_upoly(t) * c;
+		umodpoly tautilde;
+		umodpoly_from_upoly(tautilde, phi, R);
+		umodpoly r, q;
+		remdiv(sigmatilde, w1, r, q);
+		umodpoly sigma = r;
+		phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
+		umodpoly tau;
+		umodpoly_from_upoly(tau, phi, R);
+		u = u + umodpoly_to_upoly(tau) * modulus;
+		w = w + umodpoly_to_upoly(sigma) * modulus;
+		e = a - u * w;
 		modulus = modulus * p;
 	}
+quickexit: ;
 
 	// step 5
-	if ( e.is_zero() ) {
-		ex delta = u.content(x);
-		u = u / delta;
-		w = w / gamma * delta;
-		return lst(u, w);
+	if ( e.empty() ) {
+		cl_I g = u[0];
+		for ( size_t i=1; i<u.size(); ++i ) {
+			g = gcd(g, u[i]);
+			if ( g == 1 ) break;
+		}
+		if ( g != 1 ) {
+			u = u / g;
+			w = w * g;
+		}
+		if ( alpha != 1 ) {
+			w = w / alpha;
+		}
 	}
 	else {
-		return lst();
+		u.clear();
 	}
 }
 
@@ -962,84 +1223,117 @@ static unsigned int next_prime(unsigned int p)
 	throw logic_error("next_prime: should not reach this point!");
 }
 
-class Partition
+class factor_partition
 {
 public:
-	Partition(size_t n_) : n(n_)
+	factor_partition(const upvec& factors_) : factors(factors_)
 	{
+		n = factors.size();
 		k.resize(n, 1);
 		k[0] = 0;
 		sum = n-1;
+		one.resize(1, factors.front()[0].ring()->one());
+		split();
 	}
 	int operator[](size_t i) const { return k[i]; }
 	size_t size() const { return n; }
 	size_t size_first() const { return n-sum; }
 	size_t size_second() const { return sum; }
+#ifdef DEBUGFACTOR
+	void get() const { DCOUTVAR(k); }
+#endif
 	bool next()
 	{
 		for ( size_t i=n-1; i>=1; --i ) {
 			if ( k[i] ) {
 				--k[i];
 				--sum;
-				return sum > 0;
+				if ( sum > 0 ) {
+					split();
+					return true;
+				}
+				else {
+					return false;
+				}
 			}
 			++k[i];
 			++sum;
 		}
 		return false;
 	}
+	void split()
+	{
+		left = one;
+		right = one;
+		for ( size_t i=0; i<n; ++i ) {
+			if ( k[i] ) {
+				right = right * factors[i];
+			}
+			else {
+				left = left * factors[i];
+			}
+		}
+	}
+public:
+	umodpoly left, right;
 private:
+	upvec factors;
+	umodpoly one;
 	size_t n, sum;
 	vector<int> k;
 };
 
-static void split(const UniPolyVec& factors, const Partition& part, UniPoly& a, UniPoly& b)
-{
-	a.set(0, a.R->one());
-	b.set(0, a.R->one());
-	for ( size_t i=0; i<part.size(); ++i ) {
-		if ( part[i] ) {
-			b = b * factors[i];
-		}
-		else {
-			a = a * factors[i];
-		}
-	}
-}
-
 struct ModFactors
 {
-	ex poly;
-	UniPolyVec factors;
+	upoly poly;
+	upvec factors;
 };
 
 static ex factor_univariate(const ex& poly, const ex& x)
 {
-	ex unit, cont, prim;
-	poly.unitcontprim(x, unit, cont, prim);
+	ex unit, cont, prim_ex;
+	poly.unitcontprim(x, unit, cont, prim_ex);
+	upoly prim;
+	upoly_from_ex(prim, prim_ex, x);
 
-	// determine proper prime
-	unsigned int p = 3;
-	cl_modint_ring R = find_modint_ring(p);
-	while ( true ) {
-		if ( irem(ex_to<numeric>(prim.lcoeff(x)), p) != 0 ) {
-			UniPoly modpoly(R, prim, x);
-			UniFactorVec sqrfree_ufv;
-			squarefree(modpoly, sqrfree_ufv);
-			if ( sqrfree_ufv.factors.size() == 1 ) break;
-		}
-		p = next_prime(p);
-		R = find_modint_ring(p);
-	}
-
-	// do modular factorization
-	UniPoly modpoly(R, prim, x);
-	UniPolyVec factors;
-	factor_modular(modpoly, factors);
-	if ( factors.size() <= 1 ) {
-		// irreducible for sure
-		return poly;
+	// determine proper prime and minimize number of modular factors
+	unsigned int p = 3, lastp = 3;
+	cl_modint_ring R;
+	unsigned int trials = 0;
+	unsigned int minfactors = 0;
+	cl_I lc = lcoeff(prim);
+	upvec factors;
+	while ( trials < 2 ) {
+		umodpoly modpoly;
+		while ( true ) {
+			p = next_prime(p);
+			if ( !zerop(rem(lc, p)) ) {
+				R = find_modint_ring(p);
+				umodpoly_from_upoly(modpoly, prim, R);
+				if ( squarefree(modpoly) ) break;
+			}
+		}
+
+		// do modular factorization
+		upvec trialfactors;
+		factor_modular(modpoly, trialfactors);
+		if ( trialfactors.size() <= 1 ) {
+			// irreducible for sure
+			return poly;
+		}
+
+		if ( minfactors == 0 || trialfactors.size() < minfactors ) {
+			factors = trialfactors;
+			minfactors = factors.size();
+			lastp = p;
+			trials = 1;
+		}
+		else {
+			++trials;
+		}
 	}
+	p = lastp;
+	R = find_modint_ring(p);
 
 	// lift all factor combinations
 	stack<ModFactors> tocheck;
@@ -1047,24 +1341,22 @@ static ex factor_univariate(const ex& poly, const ex& x)
 	mf.poly = prim;
 	mf.factors = factors;
 	tocheck.push(mf);
+	upoly f1, f2;
 	ex result = 1;
 	while ( tocheck.size() ) {
 		const size_t n = tocheck.top().factors.size();
-		Partition part(n);
+		factor_partition part(tocheck.top().factors);
 		while ( true ) {
-			UniPoly a(R), b(R);
-			split(tocheck.top().factors, part, a, b);
-
-			ex answer = hensel_univar(tocheck.top().poly, x, p, a, b);
-			if ( answer != lst() ) {
+			hensel_univar(tocheck.top().poly, p, part.left, part.right, f1, f2);
+			if ( !f1.empty() ) {
 				if ( part.size_first() == 1 ) {
 					if ( part.size_second() == 1 ) {
-						result *= answer.op(0) * answer.op(1);
+						result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
 						tocheck.pop();
 						break;
 					}
-					result *= answer.op(0);
-					tocheck.top().poly = answer.op(1);
+					result *= upoly_to_ex(f1, x);
+					tocheck.top().poly = f2;
 					for ( size_t i=0; i<n; ++i ) {
 						if ( part[i] == 0 ) {
 							tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
@@ -1075,12 +1367,12 @@ static ex factor_univariate(const ex& poly, const ex& x)
 				}
 				else if ( part.size_second() == 1 ) {
 					if ( part.size_first() == 1 ) {
-						result *= answer.op(0) * answer.op(1);
+						result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
 						tocheck.pop();
 						break;
 					}
-					result *= answer.op(1);
-					tocheck.top().poly = answer.op(0);
+					result *= upoly_to_ex(f2, x);
+					tocheck.top().poly = f1;
 					for ( size_t i=0; i<n; ++i ) {
 						if ( part[i] == 1 ) {
 							tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
@@ -1090,8 +1382,8 @@ static ex factor_univariate(const ex& poly, const ex& x)
 					break;
 				}
 				else {
-					UniPolyVec newfactors1(part.size_first(), R), newfactors2(part.size_second(), R);
-					UniPolyVec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+					upvec newfactors1(part.size_first()), newfactors2(part.size_second());
+					upvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
 					for ( size_t i=0; i<n; ++i ) {
 						if ( part[i] ) {
 							*i2++ = tocheck.top().factors[i];
@@ -1101,16 +1393,17 @@ static ex factor_univariate(const ex& poly, const ex& x)
 						}
 					}
 					tocheck.top().factors = newfactors1;
-					tocheck.top().poly = answer.op(0);
+					tocheck.top().poly = f1;
 					ModFactors mf;
 					mf.factors = newfactors2;
-					mf.poly = answer.op(1);
+					mf.poly = f2;
 					tocheck.push(mf);
+					break;
 				}
 			}
 			else {
 				if ( !part.next() ) {
-					result *= tocheck.top().poly;
+					result *= upoly_to_ex(tocheck.top().poly, x);
 					tocheck.pop();
 					break;
 				}
@@ -1121,7 +1414,780 @@ static ex factor_univariate(const ex& poly, const ex& x)
 	return unit * cont * result;
 }
 
-struct FindSymbolsMap : public map_function {
+struct EvalPoint
+{
+	ex x;
+	int evalpoint;
+};
+
+// forward declaration
+vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
+
+upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
+{
+	const size_t r = a.size();
+	cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+	upvec q(r-1);
+	q[r-2] = a[r-1];
+	for ( size_t j=r-2; j>=1; --j ) {
+		q[j-1] = a[j] * q[j];
+	}
+	umodpoly beta(1, R->one());
+	upvec s;
+	for ( size_t j=1; j<r; ++j ) {
+		vector<ex> mdarg(2);
+		mdarg[0] = umodpoly_to_ex(q[j-1], x);
+		mdarg[1] = umodpoly_to_ex(a[j-1], x);
+		vector<EvalPoint> empty;
+		vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
+		umodpoly sigma1;
+		umodpoly_from_ex(sigma1, exsigma[0], x, R);
+		umodpoly sigma2;
+		umodpoly_from_ex(sigma2, exsigma[1], x, R);
+		beta = sigma1;
+		s.push_back(sigma2);
+	}
+	s.push_back(beta);
+	return s;
+}
+
+/**
+ *  Assert: a not empty.
+ */
+void change_modulus(const cl_modint_ring& R, umodpoly& a)
+{
+	if ( a.empty() ) return;
+	cl_modint_ring oldR = a[0].ring();
+	umodpoly::iterator i = a.begin(), end = a.end();
+	for ( ; i!=end; ++i ) {
+		*i = R->canonhom(oldR->retract(*i));
+	}
+	canonicalize(a);
+}
+
+void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
+{
+	cl_modint_ring R = find_modint_ring(p);
+	umodpoly amod = a;
+	change_modulus(R, amod);
+	umodpoly bmod = b;
+	change_modulus(R, bmod);
+
+	umodpoly smod;
+	umodpoly tmod;
+	exteuclid(amod, bmod, smod, tmod);
+
+	cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
+	umodpoly s = smod;
+	change_modulus(Rpk, s);
+	umodpoly t = tmod;
+	change_modulus(Rpk, t);
+
+	cl_I modulus(p);
+	umodpoly one(1, Rpk->one());
+	for ( size_t j=1; j<k; ++j ) {
+		umodpoly e = one - a * s - b * t;
+		reduce_coeff(e, modulus);
+		umodpoly c = e;
+		change_modulus(R, c);
+		umodpoly sigmabar = smod * c;
+		umodpoly taubar = tmod * c;
+		umodpoly sigma, q;
+		remdiv(sigmabar, bmod, sigma, q);
+		umodpoly tau = taubar + q * amod;
+		umodpoly sadd = sigma;
+		change_modulus(Rpk, sadd);
+		cl_MI modmodulus(Rpk, modulus);
+		s = s + sadd * modmodulus;
+		umodpoly tadd = tau;
+		change_modulus(Rpk, tadd);
+		t = t + tadd * modmodulus;
+		modulus = modulus * p;
+	}
+
+	s_ = s; t_ = t;
+}
+
+upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+{
+	cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+
+	const size_t r = a.size();
+	upvec result;
+	if ( r > 2 ) {
+		upvec s = multiterm_eea_lift(a, x, p, k);
+		for ( size_t j=0; j<r; ++j ) {
+			umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
+			umodpoly buf;
+			rem(bmod, a[j], buf);
+			result.push_back(buf);
+		}
+	}
+	else {
+		umodpoly s, t;
+		eea_lift(a[1], a[0], x, p, k, s, t);
+		umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
+		umodpoly buf, q;
+		remdiv(bmod, a[0], buf, q);
+		result.push_back(buf);
+		umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
+		buf = t1mod + q * a[1];
+		result.push_back(buf);
+	}
+
+	return result;
+}
+
+struct make_modular_map : public map_function {
+	cl_modint_ring R;
+	make_modular_map(const cl_modint_ring& R_) : R(R_) { }
+	ex operator()(const ex& e)
+	{
+		if ( is_a<add>(e) || is_a<mul>(e) ) {
+			return e.map(*this);
+		}
+		else if ( is_a<numeric>(e) ) {
+			numeric mod(R->modulus);
+			numeric halfmod = (mod-1)/2;
+			cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
+			numeric n(R->retract(emod));
+			if ( n > halfmod ) {
+				return n-mod;
+			}
+			else {
+				return n;
+			}
+		}
+		return e;
+	}
+};
+
+static ex make_modular(const ex& e, const cl_modint_ring& R)
+{
+	make_modular_map map(R);
+	return map(e.expand());
+}
+
+vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k)
+{
+	vector<ex> a = a_;
+
+	const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+	const size_t r = a.size();
+	const size_t nu = I.size() + 1;
+
+	vector<ex> sigma;
+	if ( nu > 1 ) {
+		ex xnu = I.back().x;
+		int alphanu = I.back().evalpoint;
+
+		ex A = 1;
+		for ( size_t i=0; i<r; ++i ) {
+			A *= a[i];
+		}
+		vector<ex> b(r);
+		for ( size_t i=0; i<r; ++i ) {
+			b[i] = normal(A / a[i]);
+		}
+
+		vector<ex> anew = a;
+		for ( size_t i=0; i<r; ++i ) {
+			anew[i] = anew[i].subs(xnu == alphanu);
+		}
+		ex cnew = c.subs(xnu == alphanu);
+		vector<EvalPoint> Inew = I;
+		Inew.pop_back();
+		sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+
+		ex buf = c;
+		for ( size_t i=0; i<r; ++i ) {
+			buf -= sigma[i] * b[i];
+		}
+		ex e = make_modular(buf, R);
+
+		ex monomial = 1;
+		for ( size_t m=1; m<=d; ++m ) {
+			while ( !e.is_zero() && e.has(xnu) ) {
+				monomial *= (xnu - alphanu);
+				monomial = expand(monomial);
+				ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+				cm = make_modular(cm, R);
+				if ( !cm.is_zero() ) {
+					vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+					ex buf = e;
+					for ( size_t j=0; j<delta_s.size(); ++j ) {
+						delta_s[j] *= monomial;
+						sigma[j] += delta_s[j];
+						buf -= delta_s[j] * b[j];
+					}
+					e = make_modular(buf, R);
+				}
+			}
+		}
+	}
+	else {
+		upvec amod;
+		for ( size_t i=0; i<a.size(); ++i ) {
+			umodpoly up;
+			umodpoly_from_ex(up, a[i], x, R);
+			amod.push_back(up);
+		}
+
+		sigma.insert(sigma.begin(), r, 0);
+		size_t nterms;
+		ex z;
+		if ( is_a<add>(c) ) {
+			nterms = c.nops();
+			z = c.op(0);
+		}
+		else {
+			nterms = 1;
+			z = c;
+		}
+		for ( size_t i=0; i<nterms; ++i ) {
+			int m = z.degree(x);
+			cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
+			upvec delta_s = univar_diophant(amod, x, m, p, k);
+			cl_MI modcm;
+			cl_I poscm = cm;
+			while ( poscm < 0 ) {
+				poscm = poscm + expt_pos(cl_I(p),k);
+			}
+			modcm = cl_MI(R, poscm);
+			for ( size_t j=0; j<delta_s.size(); ++j ) {
+				delta_s[j] = delta_s[j] * modcm;
+				sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
+			}
+			if ( nterms > 1 ) {
+				z = c.op(i+1);
+			}
+		}
+	}
+
+	for ( size_t i=0; i<sigma.size(); ++i ) {
+		sigma[i] = make_modular(sigma[i], R);
+	}
+
+	return sigma;
+}
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
+{
+	for ( size_t i=0; i<v.size(); ++i ) {
+		o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
+	}
+	return o;
+}
+#endif // def DEBUGFACTOR
+
+ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
+{
+	const size_t nu = I.size() + 1;
+	const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
+
+	vector<ex> A(nu);
+	A[nu-1] = a;
+
+	for ( size_t j=nu; j>=2; --j ) {
+		ex x = I[j-2].x;
+		int alpha = I[j-2].evalpoint;
+		A[j-2] = A[j-1].subs(x==alpha);
+		A[j-2] = make_modular(A[j-2], R);
+	}
+
+	int maxdeg = a.degree(I.front().x);
+	for ( size_t i=1; i<I.size(); ++i ) {
+		int maxdeg2 = a.degree(I[i].x);
+		if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
+	}
+
+	const size_t n = u.size();
+	vector<ex> U(n);
+	for ( size_t i=0; i<n; ++i ) {
+		U[i] = umodpoly_to_ex(u[i], x);
+	}
+
+	for ( size_t j=2; j<=nu; ++j ) {
+		vector<ex> U1 = U;
+		ex monomial = 1;
+		for ( size_t m=0; m<n; ++m) {
+			if ( lcU[m] != 1 ) {
+				ex coef = lcU[m];
+				for ( size_t i=j-1; i<nu-1; ++i ) {
+					coef = coef.subs(I[i].x == I[i].evalpoint);
+				}
+				coef = make_modular(coef, R);
+				int deg = U[m].degree(x);
+				U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
+			}
+		}
+		ex Uprod = 1;
+		for ( size_t i=0; i<n; ++i ) {
+			Uprod *= U[i];
+		}
+		ex e = expand(A[j-1] - Uprod);
+
+		vector<EvalPoint> newI;
+		for ( size_t i=1; i<=j-2; ++i ) {
+			newI.push_back(I[i-1]);
+		}
+
+		ex xj = I[j-2].x;
+		int alphaj = I[j-2].evalpoint;
+		size_t deg = A[j-1].degree(xj);
+		for ( size_t k=1; k<=deg; ++k ) {
+			if ( !e.is_zero() ) {
+				monomial *= (xj - alphaj);
+				monomial = expand(monomial);
+				ex dif = e.diff(ex_to<symbol>(xj), k);
+				ex c = dif.subs(xj==alphaj) / factorial(k);
+				if ( !c.is_zero() ) {
+					vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
+					for ( size_t i=0; i<n; ++i ) {
+						deltaU[i] *= monomial;
+						U[i] += deltaU[i];
+						U[i] = make_modular(U[i], R);
+					}
+					ex Uprod = 1;
+					for ( size_t i=0; i<n; ++i ) {
+						Uprod *= U[i];
+					}
+					e = A[j-1] - Uprod;
+					e = make_modular(e, R);
+				}
+			}
+		}
+	}
+
+	ex acand = 1;
+	for ( size_t i=0; i<U.size(); ++i ) {
+		acand *= U[i];
+	}
+	if ( expand(a-acand).is_zero() ) {
+		lst res;
+		for ( size_t i=0; i<U.size(); ++i ) {
+			res.append(U[i]);
+		}
+		return res;
+	}
+	else {
+		lst res;
+		return lst();
+	}
+}
+
+static ex put_factors_into_lst(const ex& e)
+{
+	lst result;
+
+	if ( is_a<numeric>(e) ) {
+		result.append(e);
+		return result;
+	}
+	if ( is_a<power>(e) ) {
+		result.append(1);
+		result.append(e.op(0));
+		result.append(e.op(1));
+		return result;
+	}
+	if ( is_a<symbol>(e) || is_a<add>(e) ) {
+		result.append(1);
+		result.append(e);
+		result.append(1);
+		return result;
+	}
+	if ( is_a<mul>(e) ) {
+		ex nfac = 1;
+		for ( size_t i=0; i<e.nops(); ++i ) {
+			ex op = e.op(i);
+			if ( is_a<numeric>(op) ) {
+				nfac = op;
+			}
+			if ( is_a<power>(op) ) {
+				result.append(op.op(0));
+				result.append(op.op(1));
+			}
+			if ( is_a<symbol>(op) || is_a<add>(op) ) {
+				result.append(op);
+				result.append(1);
+			}
+		}
+		result.prepend(nfac);
+		return result;
+	}
+	throw runtime_error("put_factors_into_lst: bad term.");
+}
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+	for ( size_t i=0; i<v.size(); ++i ) {
+		o << v[i] << " ";
+	}
+	return o;
+}
+#endif // def DEBUGFACTOR
+
+static bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+	const int k = f.nops()-2;
+	numeric q, r;
+	d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
+	if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
+		return false;
+	}
+	for ( int i=1; i<=k; ++i ) {
+		q = ex_to<numeric>(abs(f.op(i)));
+		for ( int j=i-1; j>=0; --j ) {
+			r = d[j];
+			do {
+				r = gcd(r, q);
+				q = q/r;
+			} while ( r != 1 );
+			if ( q == 1 ) {
+				return true;
+			}
+		}
+		d[i] = q;
+	}
+	return false;
+}
+
+static bool generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+{
+	// computation of d is actually not necessary
+	const ex& x = *syms.begin();
+	bool trying = true;
+	do {
+		ex u0 = u;
+		ex vna = vn;
+		ex vnatry;
+		exset::const_iterator s = syms.begin();
+		++s;
+		for ( size_t i=0; i<a.size(); ++i ) {
+			do {
+				a[i] = mod(numeric(rand()), 2*modulus) - modulus;
+				vnatry = vna.subs(*s == a[i]);
+			} while ( vnatry == 0 );
+			vna = vnatry;
+			u0 = u0.subs(*s == a[i]);
+			++s;
+		}
+		if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+			continue;
+		}
+		if ( is_a<numeric>(vn) ) {
+			trying = false;
+		}
+		else {
+			lst fnum;
+			lst::const_iterator i = ex_to<lst>(f).begin();
+			fnum.append(*i++);
+			bool problem = false;
+			while ( i!=ex_to<lst>(f).end() ) {
+				ex fs = *i;
+				if ( !is_a<numeric>(fs) ) {
+					s = syms.begin();
+					++s;
+					for ( size_t j=0; j<a.size(); ++j ) {
+						fs = fs.subs(*s == a[j]);
+						++s;
+					}
+					if ( abs(fs) == 1 ) {
+						problem = true;
+						break;
+					}
+				}
+				fnum.append(fs);
+				++i; ++i;
+			}
+			if ( problem ) {
+				return true;
+			}
+			ex con = u0.content(x);
+			fnum.append(con);
+			trying = checkdivisors(fnum, d);
+		}
+	} while ( trying );
+	return false;
+}
+
+static ex factor_multivariate(const ex& poly, const exset& syms)
+{
+	exset::const_iterator s;
+	const ex& x = *syms.begin();
+
+	/* make polynomial primitive */
+	ex p = poly.expand().collect(x);
+	ex cont = p.lcoeff(x);
+	for ( numeric i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
+		cont = gcd(cont, p.coeff(x,ex_to<numeric>(i).to_int()));
+		if ( cont == 1 ) break;
+	}
+	ex pp = expand(normal(p / cont));
+	if ( !is_a<numeric>(cont) ) {
+		return factor(cont) * factor(pp);
+	}
+
+	/* factor leading coefficient */
+	pp = pp.collect(x);
+	ex vn = pp.lcoeff(x);
+	pp = pp.expand();
+	ex vnlst;
+	if ( is_a<numeric>(vn) ) {
+		vnlst = lst(vn);
+	}
+	else {
+		ex vnfactors = factor(vn);
+		vnlst = put_factors_into_lst(vnfactors);
+	}
+
+	const numeric maxtrials = 3;
+	numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
+	numeric minimalr = -1;
+	vector<numeric> a(syms.size()-1, 0);
+	vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+
+	while ( true ) {
+		numeric trialcount = 0;
+		ex u, delta;
+		unsigned int prime = 3;
+		size_t factor_count = 0;
+		ex ufac;
+		ex ufaclst;
+		while ( trialcount < maxtrials ) {
+			bool problem = generate_set(pp, vn, syms, vnlst, modulus, a, d);
+			if ( problem ) {
+				++modulus;
+				continue;
+			}
+			u = pp;
+			s = syms.begin();
+			++s;
+			for ( size_t i=0; i<a.size(); ++i ) {
+				u = u.subs(*s == a[i]);
+				++s;
+			}
+			delta = u.content(x);
+
+			// determine proper prime
+			prime = 3;
+			cl_modint_ring R = find_modint_ring(prime);
+			while ( true ) {
+				if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
+					umodpoly modpoly;
+					umodpoly_from_ex(modpoly, u, x, R);
+					if ( squarefree(modpoly) ) break;
+				}
+				prime = next_prime(prime);
+				R = find_modint_ring(prime);
+			}
+
+			ufac = factor(u);
+			ufaclst = put_factors_into_lst(ufac);
+			factor_count = (ufaclst.nops()-1)/2;
+
+			// veto factorization for which gcd(u_i, u_j) != 1 for all i,j
+			upvec tryu;
+			for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
+				umodpoly newu;
+				umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
+				tryu.push_back(newu);
+			}
+			bool veto = false;
+			for ( size_t i=0; i<tryu.size()-1; ++i ) {
+				for ( size_t j=i+1; j<tryu.size(); ++j ) {
+					umodpoly tryg;
+					gcd(tryu[i], tryu[j], tryg);
+					if ( unequal_one(tryg) ) {
+						veto = true;
+						goto escape_quickly;
+					}
+				}
+			}
+			escape_quickly: ;
+			if ( veto ) {
+				continue;
+			}
+
+			if ( factor_count <= 1 ) {
+				return poly;
+			}
+
+			if ( minimalr < 0 ) {
+				minimalr = factor_count;
+			}
+			else if ( minimalr == factor_count ) {
+				++trialcount;
+				++modulus;
+			}
+			else if ( minimalr > factor_count ) {
+				minimalr = factor_count;
+				trialcount = 0;
+			}
+			if ( minimalr <= 1 ) {
+				return poly;
+			}
+		}
+
+		vector<numeric> ftilde((vnlst.nops()-1)/2+1);
+		ftilde[0] = ex_to<numeric>(vnlst.op(0));
+		for ( size_t i=1; i<ftilde.size(); ++i ) {
+			ex ft = vnlst.op((i-1)*2+1);
+			s = syms.begin();
+			++s;
+			for ( size_t j=0; j<a.size(); ++j ) {
+				ft = ft.subs(*s == a[j]);
+				++s;
+			}
+			ftilde[i] = ex_to<numeric>(ft);
+		}
+
+		vector<bool> used_flag((vnlst.nops()-1)/2+1, false);
+		vector<ex> D(factor_count, 1);
+		for ( size_t i=0; i<=factor_count; ++i ) {
+			numeric prefac;
+			if ( i == 0 ) {
+				prefac = ex_to<numeric>(ufaclst.op(0));
+				ftilde[0] = ftilde[0] / prefac;
+				vnlst.let_op(0) = vnlst.op(0) / prefac;
+				continue;
+			}
+			else {
+				prefac = ex_to<numeric>(ufaclst.op(2*(i-1)+1).lcoeff(x));
+			}
+			for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
+				if ( abs(ftilde[j-1]) == 1 ) {
+					used_flag[j-1] = true;
+					continue;
+				}
+				numeric g = gcd(prefac, ftilde[j-1]);
+				if ( g != 1 ) {
+					prefac = prefac / g;
+					numeric count = abs(iquo(g, ftilde[j-1]));
+					used_flag[j-1] = true;
+					if ( i > 0 ) {
+						if ( j == 1 ) {
+							D[i-1] = D[i-1] * pow(vnlst.op(0), count);
+						}
+						else {
+							D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
+						}
+					}
+					else {
+						ftilde[j-1] = ftilde[j-1] / prefac;
+						break;
+					}
+					++j;
+				}
+			}
+		}
+
+		bool some_factor_unused = false;
+		for ( size_t i=0; i<used_flag.size(); ++i ) {
+			if ( !used_flag[i] ) {
+				some_factor_unused = true;
+				break;
+			}
+		}
+		if ( some_factor_unused ) {
+			continue;
+		}
+
+		vector<ex> C(factor_count);
+		if ( delta == 1 ) {
+			for ( size_t i=0; i<D.size(); ++i ) {
+				ex Dtilde = D[i];
+				s = syms.begin();
+				++s;
+				for ( size_t j=0; j<a.size(); ++j ) {
+					Dtilde = Dtilde.subs(*s == a[j]);
+					++s;
+				}
+				C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
+			}
+		}
+		else {
+			for ( size_t i=0; i<D.size(); ++i ) {
+				ex Dtilde = D[i];
+				s = syms.begin();
+				++s;
+				for ( size_t j=0; j<a.size(); ++j ) {
+					Dtilde = Dtilde.subs(*s == a[j]);
+					++s;
+				}
+				ex ui;
+				if ( i == 0 ) {
+					ui = ufaclst.op(0);
+				}
+				else {
+					ui = ufaclst.op(2*(i-1)+1);
+				}
+				while ( true ) {
+					ex d = gcd(ui.lcoeff(x), Dtilde);
+					C[i] = D[i] * ( ui.lcoeff(x) / d );
+					ui = ui * ( Dtilde[i] / d );
+					delta = delta / ( Dtilde[i] / d );
+					if ( delta == 1 ) break;
+					ui = delta * ui;
+					C[i] = delta * C[i];
+					pp = pp * pow(delta, D.size()-1);
+				}
+			}
+		}
+
+		EvalPoint ep;
+		vector<EvalPoint> epv;
+		s = syms.begin();
+		++s;
+		for ( size_t i=0; i<a.size(); ++i ) {
+			ep.x = *s++;
+			ep.evalpoint = a[i].to_int();
+			epv.push_back(ep);
+		}
+
+		// calc bound B
+		ex maxcoeff;
+		for ( int i=u.degree(x); i>=u.ldegree(x); --i ) {
+			maxcoeff += pow(abs(u.coeff(x, i)),2);
+		}
+		cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+		unsigned int maxdegree = 0;
+		for ( size_t i=0; i<factor_count; ++i ) {
+			if ( ufaclst[2*i+1].degree(x) > (int)maxdegree ) {
+				maxdegree = ufaclst[2*i+1].degree(x);
+			}
+		}
+		cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+		cl_I l = 1;
+		cl_I pl = prime;
+		while ( pl < B ) {
+			l = l + 1;
+			pl = pl * prime;
+		}
+
+		upvec uvec;
+		cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
+		for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
+			umodpoly newu;
+			umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
+			uvec.push_back(newu);
+		}
+
+		ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+		if ( res != lst() ) {
+			ex result = cont * ufaclst.op(0);
+			for ( size_t i=0; i<res.nops(); ++i ) {
+				result *= res.op(i).content(x) * res.op(i).unit(x);
+				result *= res.op(i).primpart(x);
+			}
+			return result;
+		}
+	}
+}
+
+struct find_symbols_map : public map_function {
 	exset syms;
 	ex operator()(const ex& e)
 	{
@@ -1136,13 +2202,14 @@ struct FindSymbolsMap : public map_function {
 static ex factor_sqrfree(const ex& poly)
 {
 	// determine all symbols in poly
-	FindSymbolsMap findsymbols;
+	find_symbols_map findsymbols;
 	findsymbols(poly);
 	if ( findsymbols.syms.size() == 0 ) {
 		return poly;
 	}
 
 	if ( findsymbols.syms.size() == 1 ) {
+		// univariate case
 		const ex& x = *(findsymbols.syms.begin());
 		if ( poly.ldegree(x) > 0 ) {
 			int ld = poly.ldegree(x);
@@ -1155,16 +2222,53 @@ static ex factor_sqrfree(const ex& poly)
 		}
 	}
 
-	// multivariate case not yet implemented!
-	throw runtime_error("multivariate case not yet implemented!");
+	// multivariate case
+	ex res = factor_multivariate(poly, findsymbols.syms);
+	return res;
 }
 
+struct apply_factor_map : public map_function {
+	unsigned options;
+	apply_factor_map(unsigned options_) : options(options_) { }
+	ex operator()(const ex& e)
+	{
+		if ( e.info(info_flags::polynomial) ) {
+			return factor(e, options);
+		}
+		if ( is_a<add>(e) ) {
+			ex s1, s2;
+			for ( size_t i=0; i<e.nops(); ++i ) {
+				if ( e.op(i).info(info_flags::polynomial) ) {
+					s1 += e.op(i);
+				}
+				else {
+					s2 += e.op(i);
+				}
+			}
+			s1 = s1.eval();
+			s2 = s2.eval();
+			return factor(s1, options) + s2.map(*this);
+		}
+		return e.map(*this);
+	}
+};
+
 } // anonymous namespace
 
-ex factor(const ex& poly)
+ex factor(const ex& poly, unsigned options)
 {
+	// check arguments
+	if ( !poly.info(info_flags::polynomial) ) {
+		if ( options & factor_options::all ) {
+			options &= ~factor_options::all;
+			apply_factor_map factor_map(options);
+			return factor_map(poly);
+		}
+		return poly;
+	}
+
 	// determine all symbols in poly
-	FindSymbolsMap findsymbols;
+	find_symbols_map findsymbols;
 	findsymbols(poly);
 	if ( findsymbols.syms.size() == 0 ) {
 		return poly;
@@ -1190,6 +2294,7 @@ ex factor(const ex& poly)
 		return pow(f, sfpoly.op(1));
 	}
 	if ( is_a<mul>(sfpoly) ) {
+		// case: multiple factors
 		ex res = 1;
 		for ( size_t i=0; i<sfpoly.nops(); ++i ) {
 			const ex& t = sfpoly.op(i);
@@ -1213,9 +2318,16 @@ ex factor(const ex& poly)
 		}
 		return res;
 	}
+	if ( is_a<symbol>(sfpoly) ) {
+		return poly;
+	}
 	// case: (polynomial)
 	ex f = factor_sqrfree(sfpoly);
 	return f;
 }
 
 } // namespace GiNaC
+
+#ifdef DEBUGFACTOR
+#include "test.h"
+#endif