number theory algorithms

nt.cpp

//gcd
int gcd(int a, int b) {
  return b!=0 : gcd(b, a%b), a;  
}

//extended gcd
int exgcd(int a, int b, int& x, int& y) {
  if (b == 0) {
    x = 1; y = 0;
    return a;
  }
  int x1, y1;
  int g = gcd(b, a%b, x1, y1);
  x = y1;
  y = x1 - (a/b) * y1;
  return g;
}

//inverse a (mod b)
int inv(int a, int b) {
  int x, y; 
  int g = exgcd(a, b, x, y);
  return g==1 ? (x%b + b) % b : -1;
}

// a + b (mod m)
int add(int a, int b, int m) {
  return (a + b) % m;
}

//a * b (mod m)
int mul(long long a, long long b, int m) {
  return a * b % m;  
}

//a * b (mod m) if m is long long
long long mul(long long a, long long b, long long m) {
  long long r = 0;  
  while (b > 0) {
    if (b&1) r = add(r, a);
    a = add(a, a);
    b >>= 1;
  }
  return b;
}


//a^b (mod m)
int ex(int a, int b, int m) {
  int r = 1;
  while (b > 0) {
    if (b&1) r = mul(r, a);
    a = mul(a, a);
    b >>= 1;
  }
  return r;
}

//chinese remainder theorem
/**
 * solve:
 *  ai = x (mod bi)
 */
int crt(vector<int> a, vector<int> b) {
  int A = a[0], B = b[0]; 
  int n = a.size();
  for (int i = 1; i < n; ++i) {
    int x, y;
    int g = exgcd(B, b[i], x, y);
    A -= (a[i] - A) / g * x * B;
    B *= b[i] / g;
    A = (A % B + B) % B;
  }
  return A;
}

//find generator
[emaxx-eng](https://cp-algorithms.com/algebra/primitive-root.html)