51nod 1162 质因子分解

https://www.51nod.com/onlineJudge/questionCode.html#!problemId=1162

数据范围大约是2^97，需要高精度计算

可以使用pollard-rho算法，期望需要$O((logn)^{1/4})$次整数操作

需要采用 蒙哥马利约化(Montgomery reduction)避免大量取模，并每隔一段时间检查一次gcd。

#include<bits/stdc++.h>
typedef __int128 num;
typedef long long i64;
const int B=,K=;
bool np[B+];
int ps[B],pp=,pps[B],fp=,qp=;
num fs[],P,qs[];
num read(){
    num x=;
    int c=getchar();
    while(!isdigit(c))c=getchar();
    while(isdigit(c))x=x*+c-,c=getchar();
    return x;
}
void pr(num x){
    if(x<)putchar('-'),x=-x;
    int ss[],sp=;
    do ss[++sp]=x%;while(x/=);
    while(sp)putchar(+ss[sp--]);
    putchar();
}
num rnd(){
    static std::mt19937 mt(time());
    num z=mt();
    z=z<<^mt();
    z=z<<^mt();
    return z%(P-)+;
}
const unsigned X=(<<)-;
inline num fix1(num a){return a>=P?a-P:a;}
inline num fix2(num a){return a<?a+P:a;}
inline num mm(num a,num b,num C=){
    unsigned b0=b&X;b>>=;
    unsigned b1=b&X;b>>=;
    unsigned b2=b&X;b>>=;
    unsigned b3=b&X;
    num c=a*b3%P;
    c=((c<<)+a*b2)%P;
    c=((c<<)+a*b1)%P;
    c=((c<<)+a*b0+C)%P;
    return c;
}
num iP,RR,iR;
#define HI(x) u64((x)>>52)
#define LO(x) u64(x&((1ll<<52)-1))
typedef unsigned __int128 u128;
typedef unsigned long long u64;
const num R=num()<<,R1=R-;
__attribute__((optimize("Ofast")))
inline u128 mul(u128 a,u128 b){
    u128 m=a*b*iP&R1;
    u128 t=(
        (u128)HI(a)*HI(b)+
        (u128)HI(m)*HI(P)
    );
    u128 t1=(
        (u128)HI(a)*LO(b)+
        (u128)LO(a)*HI(b)+
        (u128)HI(m)*LO(P)+
        (u128)LO(m)*HI(P)
    );
    u128 t0=(
        (u128)LO(a)*LO(b)+
        (u128)LO(m)*LO(P)
    );
    t+=HI(HI(t0)+LO(t1))+HI(t1);
    return fix1(t);
}
inline u128 mulRR(u128 a){return mul(a,RR);}
num exgcd(num a,num b,num&x,num&y){
    if(!b){x=,y=;return a;}
    num g=exgcd(b,a%b,y,x);
    y-=a/b*x;
    return g;
}
void setP(num x){
    P=x;
    exgcd(P,R,iP,iR);
    iP=R1&-iP;
    iR=fix2(iR);
    RR=mm(R%P,R%P);
}
num pw(num a,num n){
    num v=;
    for(;n;n>>=,a=mm(a,a))if(n&)v=mm(v,a);
    return v;
}
bool mr(){
    num s=P-;
    int r=;
    for(;~s&;s>>=,++r);
    num a=pw(rnd(),s);
    for(;;){
        if(a==||a==P-)return ;
        if(!--r)return ;
        a=mm(a,a);
    }
}
bool isp(){
    if(P<=B)return !np[P];
    if(~P&)return ;
    for(int i=;i<;++i)if(!mr())return ;
    return ;
}
num gcd(num a,num b){return b?gcd(b,a%b):a;}
num abs(num x){return x>?x:-x;}
void pollard_rho(num n){
    setP(n);
    if(isp()){
        fs[fp++]=n;
        return;
    }
    const int mod=;
    for(;;){
        num a,b,c,prod=mulRR();
        a=b=mulRR(rnd());
        c=mulRR(rnd());
        for(i64 ii=,k=;;){
            a=fix1(mul(a,a)+c);
            prod=mul(prod,abs(a-b));
            if(!prod){
                prod=abs(a-b);
                num g=gcd(P,mul(prod,));
                if(g==n)break;
                if(g!=){
                    qs[qp++]=g;
                    qs[qp++]=n/g;
                    return;
                }
            }
            if(++ii==k)k<<=,b=a;
            if(!(ii&mod)){
                num g=gcd(P,mul(prod,));
                if(g==n)break;
                if(g!=){
                    qs[qp++]=g;
                    qs[qp++]=n/g;
                    return;
                }
            }
        }
    }
}
void factor(num x){
    for(int i=;i<pp;++i){
        for(int p=ps[i];x%p==;fs[fp++]=p,x/=p);
    }
    if(x>)qs[qp++]=x;
    while(qp)pollard_rho(qs[--qp]);
    std::sort(fs,fs+fp);
    for(int i=;i<fp;++i)pr(fs[i]);
}
void init(){
    for(int i=;i<=B;++i){
        if(!np[i])ps[pp++]=i;
        for(int j=;j<pp&&i*ps[j]<=B;++j){
            np[i*ps[j]]=;
            if(i%ps[j]==)break;
        }
    }
    for(int i=;i<pp;++i){
        int p=ps[i],s=;
        while(s<=B/p)s*=p;
        pps[i]=s;
    }
}
int main(){
    init();
    factor(read());
    return ;
}