/** \brief 整系数多项式的容度.
\param f 整系数多项式.
\param r 整数容度,包含正负号,使得本原部分lc为正整数.
*/
void UniContZ(mpz_ptr r,const poly_z & f)
{
if(f.size()==0)
{
mpz_set_ui(r,1);
return ;
}
mpz_set(r,f[f.size()-1]);
if(f.size()==1)
{
return ;
}
int sign;
sign=mpz_sgn(r);
mpz_gcd(r,f[0],f[1]);
if(mpz_cmp_ui(r,1)==0)
{
if(sign<0)mpz_neg(r,r);
return ;
}
for(uint i=2;i<f.size();i++)
{
mpz_gcd(r,r,f[i]);
if(mpz_cmp_ui(r,1)==0)
{
if(sign<0)mpz_neg(r,r);
return ;
}
}
if(sign<0)mpz_neg(r,r);
return ;
}
bool factor_compare(const poly_z & f,const poly_z & g)
{
if(f.size()<g.size())return 1;
if(f.size()>g.size())return 0;
for(size_t i=f.size()-1;i>=0;i--)
{
if(mpz_cmp(f[i],g[i])<0)return 1;
if(mpz_cmp(f[i],g[i])>0)return 0;
}
return 0;
}
/** \brief 整系数多项式本原部分.
\param f 整系数多项式.
\param r 本原部分.
\return 本原部分,其lc为正整数.
*/
void UniPPZ(poly_z & r,const poly_z & f)
{
mpz_t a;
mpz_init(a);
UniContZ(a,f);
r.resize(f.size());
for(size_t i=0;i<f.size();i++)
{
mpz_divexact(r[i],f[i],a);
}
mpz_clear(a);
}
/** \brief Fp域上多项式GCD.
\param f,g 域上多项式.
\param r 最大公因子.
\note Euclid算法.
*/
void UniGcdZp(poly_z & r,const poly_z & f,const poly_z & g,mpz_ptr p)
{
poly_z u,v;
poly_z q;
copy_poly_z(u,f);
copy_poly_z(v,g);
while(v.size()> 0){
UniDivModZp(q,r,u,v,p);
UniPolynomialMod(r,r,p);
u.resize(0);
u=v;v=r;
r=poly_z();
}
static mpz_t lc;
mpz_init_set(lc,u[u.size()-1]);
mpz_invert(lc,lc,p);
r.resize(u.size());
for(size_t i=0;i<u.size();i++)
{
mpz_mul(r[i],u[i],lc);
}
UniPolynomialMod(r,r,p);
mpz_clear(lc);
u.resize(0);v.resize(0);q.resize(0);
}
void poly_z_to_poly_fc(poly_fc & r, poly_z & f)
{
r.resize(f.size());
for(uint i=0;i<r.size();++i)
{
mpfc_set_z(r[i],f[i]);
}
return ;
}
/** \brief Fp域上多项式GCD.
\param f,g 域上多项式.
\param r,s,t 其中r为最大公因子,s,t为Bezout系数多项式.
*/
void UniGcdZp_Ext(poly_z & r,poly_z & s,poly_z & t,const poly_z & f,const poly_z & g,mpz_ptr p)
{
poly_z u,v;
poly_z q;
poly_z s1,s2,t1,t2,l1,l2;
copy_poly_z(u,f);
copy_poly_z(v,g);
s1.resize(1);mpz_set_si(s1[0],1);
s2.resize(0);
t1.resize(0);
t2.resize(1);mpz_set_si(t2[0],1);
while(v.size()> 0){
UniDivModZp(q,r,u,v,p);
UniPolynomialMod(r,r,p);
UniMulZ(l1,q,t1);UniSubZ(l1,s1,l1);
UniMulZ(l2,q,t2);UniSubZ(l2,s2,l2);
u.resize(0);u=v;v=r;r=poly_z();
s1.resize(0);s1=t1;t1=l1;l1=poly_z();
s2.resize(0);s2=t2;t2=l2;l2=poly_z();
}
static mpz_t lc;
mpz_init_set(lc,u[u.size()-1]);
mpz_invert(lc,lc,p);
r.resize(u.size());
for(size_t i=0;i<u.size();i++)
{
mpz_mul(r[i],u[i],lc);
}
UniPolynomialMod(r,r,p);
s.resize(s1.size());
for(size_t i=0;i<s1.size();i++)
{
mpz_mul(s[i],s1[i],lc);
}
UniPolynomialMod(s,s,p);
t.resize(s2.size());
for(size_t i=0;i<s2.size();i++)
{
mpz_mul(t[i],s2[i],lc);
}
UniPolynomialMod(t,t,p);
mpz_clear(lc);
u.resize(0);v.resize(0);q.resize(0);s1.resize(0);s2.resize(0);t1.resize(0);t2.resize(0);l1.resize(0);l2.resize(0);
}
/** \brief 整系数多项式1-范数.
\param f 整系数多项式.
\return 系数向量的1-范数.
*/
void UniOneNormZ(mpz_ptr r,const poly_z & f)
{
mpz_set_ui(r,0);
static mpz_t temp;
mpz_init(temp);
for(int i=0;i<f.size();i++)
{
mpz_abs(temp,f[i]);
mpz_add(r,r,temp);
}
mpz_clear(temp);
}
/** \brief 整系数多项式无穷范数.
\param f 整系数多项式.
\param r 系数向量的无穷范数.
*/
void UniMaxNormZ(mpz_ptr r,const poly_z & f)
{
mpz_set_ui(r,0);
mpz_t tempz;
mpz_init(tempz);
for(int i=0;i<f.size();i++)
{
mpz_abs(tempz,f[i]);
if(mpz_cmp(r,tempz)<0)mpz_set(r,tempz);
}
mpz_clear(tempz);
return ;
}
/**
\brief 整系数多项式最大公因子.
\param f,g 整系数多项式.
\return 最大公因子.
\note 调用大素数算法或小素数算法.
*/
void UniGcdZ(poly_z & r,const poly_z & f,const poly_z & g)
{
uint option=1;
if(f.size()==0)
{
copy_poly_z(r,g);
return ;
}
if(g.size()==0)
{
copy_poly_z(r,f);
return ;
}
mpz_t contf,contg,b;
mpz_init(contf);mpz_init(contg);mpz_init(b);
UniContZ(contf,f);UniContZ(contg,g);
mpz_gcd(b,contf,contg);
poly_z _f,_g,h;
UniPPZ(_f,f);UniPPZ(_g,g);
int n=_f.deg(),m=_g.deg();
if(n==0||m==0)
{
h.resize(1);
mpz_set_ui(h[0],1);
}
else
{
if(n<m)UniGcdZ_SmallPrime1(h,_g,_f);
else UniGcdZ_SmallPrime1(h,_f,_g);
}
r.resize(h.size());
for(size_t i=0;i<h.size();i++)
{
mpz_mul(r[i],h[i],b);
}
_f.resize(0);_g.resize(0);h.resize(0);
mpz_clear(contf);mpz_clear(contg);mpz_clear(b);
return ;
}
/**
\brief 整系数多项式的整数根(不含重数).
\param f 整系数多项式.
\return 整根的list.
\note 利用Zp中的根.
*/
void UniZRootZ_ByZp(std::vector<mpz_ptr> & rootlist,const poly_z & f)
{
static mpz_t A,B,p,z_temp,nA;
mpz_init(A);
mpz_init(B);
mpz_init(p);
mpz_init(z_temp);
mpz_init(nA);
UniMaxNormZ(A,f);
uint n=f.size()-1;
mpz_mul(B,A,A);
mpz_add(B,B,A);
mpz_mul_ui(B,B,2*n);
mpz_add_ui(p,B,1);
mpz_nextprime(p,p);
poly_z fp,vi,ui;
std::vector<mpz_ptr> ulist;
uint totalroot=0;
UniPolynomialMod(fp,f,p);
UniZpRootZp(ulist,fp,p);
mpz_mul_ui(nA,A,n);
ui.resize(2);
mpz_set_ui(ui[1],1);
for(uint i=0; i<ulist.size(); i++)
{
if(mpz_cmpabs(ulist[i],A)<=0)
{
mpz_neg(ui[0],ulist[i]);
UniDivZp(vi,fp,ui,p);
UniMaxNormZ(z_temp,vi);
if(mpz_cmp(z_temp,nA)<=0)
{
totalroot++;
resize_z_list(rootlist,totalroot);
mpz_set(rootlist[totalroot-1],ulist[i]);
}
}
}
mpz_clear(A);
mpz_clear(B);
mpz_clear(p);
mpz_clear(z_temp);
mpz_clear(nA);
fp.resize(0);
vi.resize(0);
ui.resize(0);
resize_z_list(ulist,0);
std::sort(rootlist.begin(),rootlist.end(),mpz_ptr_less);
return ;
}
void UniEqlDegFacZp(std::vector<poly_z> & eqlfactorlist,const poly_z & f,mpz_ptr p,uint d,uint & current)
{
uint n=f.size()-1;
if(n==d)
{
copy_poly_z(eqlfactorlist[current],f);
current++;
return ;
}
poly_z g,h;
while(1)
{
UniEqlDegFacZp_Odd(g,f,p,d);
if(g.size()>=d)break;
}
UniEqlDegFacZp(eqlfactorlist,g,p,d,current);
UniDivZp(h,f,g,p);
UniEqlDegFacZp(eqlfactorlist,h,p,d,current);
g.resize(0);h.resize(0);
return ;
}
//todo using iterated frobenius alg. to improve it?
void UniEqlDegFacZp_Odd(poly_z & factor,const poly_z & f,mpz_ptr p,uint d)
{
uint deg_bound=f.size()-2;
poly_z a;
while(1)
{
UniRandomZp(a,deg_bound,p);
if(a.size()>1)break;
}
UniGcdZp(factor,a,f,p);
if(factor.size()>1&&factor.size()<f.size())
{
a.resize(0);
return ;
}
power_specially_for_eqldegfactorization(a,f,p,d);
//static mpz_t exp;
//mpz_init(exp);
//mpz_pow_ui(exp,p,d);
//mpz_sub_ui(exp,exp,1);
//mpz_divexact_ui(exp,exp,2);
//PowerMod(a,exp,f,p);
//mpz_clear(exp);
mpz_sub_ui(a[0],a[0],1);
UniGcdZp(factor,a,f,p);
if(factor.size()>1&&factor.size()<f.size())
{
a.resize(0);
return;
}
a.resize(0);
factor.resize(0);
return ;
}
/**
\brief 整系数多项式最大公因子.
\param f,g 整系数本原多项式,且\f$\deg f=n\ge\deg g\ge 1\f$.
\param r 最大公因子.
\note 小素数模方法.
\todo 理论文档此处有误.
*/
void UniGcdZ_SmallPrime1(poly_z & r,const poly_z & f,const poly_z & g)
{
poly_z fp,gp,vp,v1;
poly_z fstar,gstar;
mpz_t c,s,t;
mpz_init(c);mpz_init(s);mpz_init(t);
mpz_t A,b,B,z_temp,k,p_bound,p,p_low_bound,p1,B2;
mpf_t float_temp;
mpz_init(A);mpz_init(b);mpz_init(B);mpz_init(z_temp);mpz_init(k);mpz_init(p_bound);mpz_init(p);mpz_init(p1);mpz_init(p_low_bound);mpf_init(float_temp);mpz_init(B2);
UniMaxNormZ(A,f);UniMaxNormZ(b,g);
int n=f.size()-1,m=g.size()-1;
if(mpz_cmp(A,b)<0)mpz_set(A,b);
mpz_gcd(b,f[n],g[m]);
mpz_ui_pow_ui(B,2,n);
mpz_mul(B,B,A);
mpz_mul(B,B,b);
mpf_sqrt_ui(float_temp,n+1);
mpz_set_f(z_temp,float_temp);
mpz_mul(B,B,z_temp);
mpz_mul_ui(B2,B,2);
mpz_set_si(z_temp,n);
int tempint;
tempint=2*n*mpz_sizeinbase(z_temp,2)+2*mpz_sizeinbase(b,2)+4*n*mpz_sizeinbase(A,2);
//tempint=2*n*Modules::NumberTheory::IntegerLength(Z(n),2)+2*Modules::NumberTheory::IntegerLength(b,2)+4*n*Modules::NumberTheory::IntegerLength(A,2);
mpz_set_si(k,tempint);
mpz_mul_ui(p_bound,k,2);
mpz_mul_ui(p_bound,p_bound,mpz_sizeinbase(k,2));
mpz_set_ui(p_low_bound,3);
fstar.resize(n+1);
for(size_t i=0;i<=n;i++)
{
mpz_mul(fstar[i],f[i],b);
}
gstar.resize(m+1);
for(size_t i=0;i<=m;i++)
{
mpz_mul(gstar[i],g[i],b);
}
while(1)
{
while(1)
{
random::randominteger(p,p_low_bound,p_bound);
if(mpz_probab_prime_p(p,10)>0&&mpz_divisible_p(b,p)==0)break;
}
UniPolynomialMod(fp,f,p);
UniPolynomialMod(gp,g,p);
UniGcdZp(vp,fp,gp,p);
if(vp.size()==1)
{
r.resize(1);
mpz_set_si(r[0],1);
fp.resize(0);gp.resize(0);vp.resize(0);v1.resize(0);
fstar.resize(0);gstar.resize(0);
mpz_clear(c);mpz_clear(s);mpz_clear(t);
mpz_clear(A);mpz_clear(b);mpz_clear(B);mpz_clear(z_temp);mpz_clear(k);mpz_clear(p_bound);mpz_clear(p);mpz_clear(p1);mpz_clear(p_low_bound);mpf_clear(float_temp);
return ;
}
mpz_set(p1,p);
copy_poly_z(v1,vp);
while(mpz_cmp(p1,B2)<0)
{
while(1)
{
random::randominteger(p,p_low_bound,p_bound);
if(mpz_probab_prime_p(p,10)>0&&mpz_divisible_p(b,p)==0)break;
}
UniPolynomialMod(fp,f,p);
UniPolynomialMod(gp,g,p);
UniGcdZp(vp,fp,gp,p);
if(vp.size()==1)
{
r.resize(1);
mpz_set_si(r[0],1);
fp.resize(0);gp.resize(0);vp.resize(0);v1.resize(0);
fstar.resize(0);gstar.resize(0);
mpz_clear(c);mpz_clear(s);mpz_clear(t);
mpz_clear(A);mpz_clear(b);mpz_clear(B);mpz_clear(z_temp);mpz_clear(k);mpz_clear(p_bound);mpz_clear(p);mpz_clear(p1);mpz_clear(p_low_bound);mpf_clear(float_temp);
return ;
}
if(vp.size()<v1.size())
{
mpz_set(p1,p);
copy_poly_z(v1,vp);
continue;
}
if(vp.size()==v1.size())
{
mpz_gcdext(c,s,t,p1,p);
mpz_mul(t,p,t);
mpz_mul(s,p1,s);
mpz_mul(p1,p1,p);
for(size_t i=0;i<vp.size();i++)
{
mpz_mul(c,v1[i],t);
mpz_addmul(c,vp[i],s);
mpz_set(v1[i],c);
}
UniPolynomialMod(v1,v1,p1);
}
}
for(size_t i=0;i<v1.size();i++)mpz_mul(v1[i],v1[i],b);
UniPolynomialMod(v1,v1,p1);
if(poly_z_divisible(fstar,v1)&&poly_z_divisible(gstar,v1))
{
UniPPZ(r,v1);break;
}
}
fp.resize(0);gp.resize(0);vp.resize(0);v1.resize(0);
fstar.resize(0);gstar.resize(0);
mpz_clear(c);mpz_clear(s);mpz_clear(t);
mpz_clear(A);mpz_clear(b);mpz_clear(B);mpz_clear(z_temp);mpz_clear(k);mpz_clear(p_bound);mpz_clear(p);mpz_clear(p1);mpz_clear(p_low_bound);mpf_clear(float_temp);
return ;
}
/**
\brief Berlekamp算法.
\param f 有限域上多项式.monic
\param p characteristic. small prime
\param faclist 返回因子list.
*/
void BerlekampZp_SmallPrime(const poly_z & f,mpz_ptr p,std::vector<poly_z> & faclist)
{
int n=f.size()-1;
clear_poly_z_list(faclist);
std::vector<poly_int> basis_int;
int k,r;
mat_int matq;
int p_int=mpz_get_si(p);
poly_int f_int;
poly_z_to_poly_int(f_int,f);
//str("matrix begin").print();
UniPetriMatrixZp_SmallPrime(matq,f_int,p_int);
//for(int i=0;i<n;i++)
//{
// for(int j=0;j<n;j++)
// {
// std::cout<<matq[i][j]<<" ";
// }
// std::cout<<"\n";
//}
//str("matrix end").print();
for(int i=0;i<n;i++)matq[i][i]=matq[i][i]-1;
NullSpaceZp_SmallPrime(basis_int,matq,n,p_int);
//str("nullspace end").print();
r=basis_int.size();
//for(size_t i=0;i<r;i++)
//{
// poly_z test;
// poly_int_to_poly_z(test,basis_int[i]);
// test.print();
//}
//
mat_int result_int;
poly_int bk_int,g_int;
int a_int;
int r_temp;
result_int.push_back(f_int);
while(result_int.size()<r)
{
random::randominteger(k,0,r-1);
bk_int=basis_int[k];
random::randominteger(a_int,0,p_int-1);
if(bk_int.size()<=1)continue;
bk_int[0]-=a_int;
r_temp=result_int.size();
for(int i=0;i<r_temp;i++)
{
UniGcdZp_SmallPrime(g_int,bk_int,result_int[i],p_int);
if(g_int.size()>1&&result_int[i].size()>g_int.size())
{
UniDivZp_SmallPrime(result_int[i],result_int[i],g_int,p_int);
result_int.push_back(g_int);
}
}
}
faclist.resize(r);
for(size_t i=0;i<r;i++)
{
poly_int_to_poly_z(faclist[i],result_int[i]);
}
//str("berlekamp end").print();
}