51Nod-1038-X^A Mod P

描述

题解

代码

#include <cmath>
#include <vector>
#include <cstdio>
#include <iostream>
#include <algorithm>

using namespace std;

typedef long long ll;

const int MAXN = 100100;

ll qk_pow(ll a, ll b, ll mod)
{
ll ret = 1;
while (b)
{
if (b & 1)
{
ret = ret * a % mod;
}
b >>= 1;
a = a * a % mod;
}
return ret;
}

ll ex_gcd(ll a, ll b, ll &x, ll &y)
{
if (b == 0)
{
x = 1;
y = 0;
return a;
}
else
{
ll r = ex_gcd(b, a % b, y, x);
y -= x * (a / b);
return r;
}
}

vector<ll> a;

bool check(ll g, ll p)
{
for (int i = 0; i < a.size(); i++)
{
if (qk_pow(g, (p - 1) / a[i], p) == 1)
{
return 0;
}
}

return 1;
}

//  求解原根
ll primitive_root(ll p)
{
ll tmp = p - 1;
for (int i = 2; i <= tmp / i; i++)
{
if (tmp % i == 0)
{
a.push_back(i);
while (tmp % i == 0)
{
tmp /= i;
}
}
}
if (tmp != 1)
{
a.push_back(tmp);
}

ll g = 1;
while (true)
{
if (check(g, p))
{
return g;
}
++g;
}
}

struct sa
{
ll x;
int id;
bool operator < (const sa &b) const
{
if (x == b.x)
{
return id < b.id;
}
return x < b.x;
}
} rec[MAXN];

//  求解离散对数
ll discerte_log(ll x, ll n, ll m)
{
int s = (int)(sqrt((double)m + 0.5));
while ((ll)s * s <= m)
{
s++;
}

ll cur = 1;
sa tmp;
for (int i = 0; i < s; i++)
{
tmp.x = cur;
tmp.id = i;
rec[i] = tmp;
cur = cur * x % m;
}

sort(rec, rec + s);
ll mul = qk_pow(cur, m - 2, m) % m;
cur = 1;

for (int i = 0; i < s; i++)
{
ll more = n * cur % m;
tmp.x = more;
tmp.id = -1;
int j = (int)(lower_bound(rec, rec + s, tmp) - rec);
if (rec[j].x == more)
{
return i * s + rec[j].id;
}

cur = cur * mul % m;
}

return -1;
}

//  求解n次剩余
vector<ll> residue(ll p, ll n, ll a)
{
vector<ll> ret;
if (a == 0)
{
ret.push_back(0);
return ret;
}

ll g = primitive_root(p);
ll m = discerte_log(g, a, p);

if (m == -1)
{
return ret;
}

ll A = n, B = p - 1, C = m, x, y;
ll G = ex_gcd(A, B, x, y);
if (C % G != 0)
{
return ret;
}

x = x * (C / G) % B;
ll delta = B / G;
for (int i = 0; i < G; i++)
{
x = ((x + delta) % B + B) % B;
ret.push_back(qk_pow(g, x, p));
}
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());

return ret;
}

ll P, A, B;

int main()
{
int T;
scanf("%d", &T);

while (T--)
{
a.clear();
scanf("%lld%lld%lld", &P, &A, &B);

vector<ll> ans;
ans = residue(P, A, B);

if (ans.empty())
{
puts("No Solution");
}
else
{
for (int i = 0; i < ans.size(); i++)
{
printf("%lld ", ans[i]);
}
putchar(10);
}
}

return 0;
}