Unverified Commit 9b8441b2 authored by shaochenheng's avatar shaochenheng Committed by GitHub
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Update and rename pollar-rho.md to pollard-rho.md 

规范了一些命名
parent 5a33cc2b

docs/math/pollar-rho.md→docs/math/pollard-rho.md

+13 −13
Original line number Diff line number Diff line 
Pollar-Rho 算法是一种用于快速分解质因数的算法。

Pollard-Rho 算法是一种用于快速分解质因数的算法。



## 问题引入
@@ -63,9 +63,9 @@ $$ 


我们每次对 $d=\gcd(|x_i-x_j|,n)$ ，判断 d 是否满足 $1< d< n$ ，若满足则可直接返回 $d$ 。由于 $x_i$ 是一个伪随机数列，必定会形成环，在形成环时就不能再继续操作了，直接返回 n 本身，并且在后续操作里调整随机常数 $c$ ，重新分解。



??? note "基于Floyd判环的Pollar-Rho算法"

??? note "基于Floyd判环的Pollard-Rho算法"

    ```c++

    ll PR(ll N) {

    ll Pollard_Rho(ll N) {

      ll c = rand() % (N - 1) + 1;

      ll t = f(0, c, N);

      ll r = f(f(0, c, N), c, N);
@@ -87,7 +87,7 @@ $$ 


??? note "参考实现"

    ```c++

    ll PR(ll x) {

    ll Pollard_Rho(ll x) {

      ll s = 0, t = 0;

      ll c = rand() % (x - 1) + 1;

      int step = 0, goal = 1;
@@ -123,12 +123,12 @@ $$ 
    int t;

    ll max_factor, n;

    

    inline ll gcd(ll a, ll b) {

    ll gcd(ll a, ll b) {

      if (b == 0) return a;

      return gcd(b, a % b);

    }

    

    inline ll qp(ll x, ll p, ll mod) {

    ll quick_pow(ll x, ll p, ll mod) {

      ll ans = 1;

      while (p) {

        if (p & 1) ans = (lll)ans * x % mod;
@@ -138,7 +138,7 @@ $$ 
      return ans;

    }

    

    inline bool Miller_rabin(ll p) {

    bool Miller_Rabin(ll p) {

      if (p < 2) return 0;

      if (p == 2) return 1;

      if (p == 3) return 1;
@@ -146,7 +146,7 @@ $$ 
      while (!(d & 1)) ++r, d >>= 1;

      for (ll k = 0; k < 10; ++k) {

        ll a = rand() % (p - 2) + 2;

        ll x = qp(a, d, p);

        ll x = quick_pow(a, d, p);

        if (x == 1 || x == p - 1) continue;

        for (int i = 0; i < r - 1; ++i) {

          x = (lll)x * x % p;
@@ -157,9 +157,9 @@ $$ 
      return 1;

    }

    

    inline ll f(ll x, ll c, ll n) { return ((lll)x * x + c) % n; }

    ll f(ll x, ll c, ll n) { return ((lll)x * x + c) % n; }

    

    inline ll PR(ll x) {

    ll Pollard_Rho(ll x) {

      ll s = 0, t = 0;

      ll c = (ll)rand() % (x - 1) + 1;

      int step = 0, goal = 1;
@@ -178,14 +178,14 @@ $$ 
      }

    }

    

    inline void fac(ll x) {

    void fac(ll x) {

      if (x <= max_factor || x < 2) return;

      if (Miller_rabin(x)) {

      if (Miller_Rabin(x)) {

        max_factor = max(max_factor, x);

        return;

      }

      ll p = x;

      while (p >= x) p = PR(x);

      while (p >= x) p = Pollard_Rho(x);

      while ((x % p) == 0) x /= p;

      fac(x), fac(p);

    }