#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define REP(i,s,n) for(int i=(int)(s);i<(int)(n);i++) #define DEBUGP(val) cerr << #val << "=" << val << "\n" using namespace std; typedef long long int ll; typedef unsigned long long ull; typedef vector VI; typedef vector VL; typedef pair PI; typedef pair PL; const int DEBUG = 1; ll p; ll powmod(ll a, ll e) { ll prod = 1; ll cur = a % p; while (e > 0) { if (e % 2) prod = prod * cur % p; cur = cur * cur % p; e /= 2; } return prod; } typedef vector mat; VL factors(ll v); // Find min {x | a^x = 1 (mod p) }. ll period(ll a) { assert (a >= 1); assert (a < p); ll v = p - 1; VL fs = factors(v); for (ll i: fs) { while (v % i == 0) { if (powmod(a, v / i) == 1) { v /= i; } else { break; } } } return v; } const ll N = 80000; ll det(const mat &a) { ll tmp = a[0][0] * a[1][1] % p; tmp -= a[0][1] * a[1][0] % p; return (tmp + p) % p; } mat mul(const mat &a, const mat &b) { mat c(2, VL(2)); REP(i, 0, 2) REP(j, 0, 2) REP(k, 0, 2) c[i][k] = (c[i][k] + a[i][j] * b[j][k]) % p; return c; } mat powmat(const mat &a, ll e) { mat prod(2, VL(2)); mat cur(a); REP(i, 0, 2) prod[i][i] = 1; while (e > 0) { if (e % 2) prod = mul(prod, cur); cur = mul(cur, cur); e /= 2; } return prod; } mat inv(const mat &a) { ll d = det(a); assert (d != 0); d = powmod(d, p - 2); mat b(2, VL(2)); b[0][0] = a[1][1] * d % p; b[0][1] = a[0][1] * (p - d) % p; b[1][0] = a[1][0] * (p - d) % p; b[1][1] = a[0][0] * d % p; return b; } // Find min {x > 0 | a^x = b (mod p)}. ll disc_log(ll a, ll b) { if (a <= 1) { return a == b ? 1 : -1; } ll per = period(a); map big; REP(i, 0, N) { big[powmod(a, p - 1 - (N * (ll)i) % (p - 1)) * b % p] = N * (ll)i; } ll nxt = -1; REP(i, 0, N) { ll v = powmod(a, i); if (big.count(v)) { nxt = big[v] + i; break; } } return nxt < 0 ? -1 : (nxt + per - 1) % per + 1; } VL factors(ll v) { VL a; ll i = 2; while (i * i <= v) { int e = 0; while (v % i == 0) { e++; v /= i; } if (e) a.push_back(i); i++; } if (v > 1) a.push_back(v); return a; } ll mat_period(mat a) { assert (det(a) == 1); mat unit(2, VL(2)); REP(i, 0, 2) unit[i][i] = 1; if (a == unit) return 1; if (mul(a, a) == unit) return 2; if (powmat(a, p) == unit) return p; if (powmat(a, 2 * p) == unit) return 2 * p; ll v = p * p - 1; assert (powmat(a, v) == unit); VL f1 = factors(p - 1), f2 = factors(p + 1); for (int f: f2) f1.push_back(f); for (int f: f1) { while (v % f == 0) { if (powmat(a, v / f) == unit) { v /= f; } else { break; } } } assert (powmat(a, v) == unit); return v; } ll disc_log_mat(mat a, mat b) { assert (det(a) == 1); ll per = mat_period(a); assert (per <= 2 * p + 2); map big; mat step = powmat(inv(a), N); mat cur(b); REP(i, 0, N) { big[cur] = N * (ll) i; cur = mul(cur, step); } cur = mat(2, VL(2)); REP(i, 0, 2) cur[i][i] = 1; ll nxt = -1; REP(i, 0, N) { if (big.count(cur)) { nxt = big[cur] + i; nxt %= per; break; } cur = mul(cur, a); } return nxt < 0 ? -1 : nxt; } int main(void) { ios::sync_with_stdio(false); cin.tie(0); mat a(2, VL(2)), b(2, VL(2)); cin >> p; REP(i, 0, 2) { REP(j, 0, 2) { cin >> a[i][j]; } } REP(i, 0, 2) { REP(j, 0, 2) { cin >> b[i][j]; } } // A = kI? if (a[0][1] == 0 && a[1][0] == 0 && a[0][0] == a[1][1]) { if (b[0][1] == 0 && b[1][0] == 0 && b[0][0] == b[1][1]) { cout << disc_log(a[0][0], b[0][0]) << endl; } else { cout << -1 << endl; } return 0; } if (DEBUG) { cerr << det(a) << endl; cerr << det(b) << endl; } // B = xA + yI VL cons; if (a[0][1] != 0) { cons.push_back(b[0][1] * powmod(a[0][1], p - 2) % p); } if (a[1][0] != 0) { cons.push_back(b[1][0] * powmod(a[1][0], p - 2) % p); } if (a[0][0] != a[1][1]) { ll diff = (a[0][0] - a[1][1] + p) % p; cons.push_back((b[0][0] - b[1][1] + p) % p * powmod(diff, p - 2) % p); } assert (cons.size() > 0); cons.erase(unique(cons.begin(), cons.end()), cons.end()); if (cons.size() != 1) { cout << -1 << endl; return 0; } ll x = cons[0]; ll y = (b[0][0] - x * a[0][0] % p + p) % p; if (DEBUG) { cerr << "B = " << x << " A + " << y << " I" << endl; } if (det(a) == 0) { // A^2 = z A. Find z = tr(A). ll z = (a[0][0] + a[1][1]) % p; if (y != 0) { cout << -1 << endl; return 0; } if (z == 0) { ll ans = x == 1 ? 1 : x == 0 ? 2 : -1; cout << ans << endl; return 0; } cout << disc_log(z, z * x % p) << endl; return 0; } assert (det(a) != 0); ll e = disc_log(det(a), det(b)); if (e < 0) { cout << -1 << endl; return 0; } if (DEBUG) { cerr << "|A|^" << e << " = |B|" << endl; } ull per = period(det(a)); mat ap = powmat(a, per); b = mul(b, powmat(inv(a), e)); ull tmp = disc_log_mat(ap, b); if (DEBUG) { cerr << "tmp = " << tmp << endl; cerr << "per = " << per << endl; } if (tmp >= 0) { cout << tmp * per + (ull) e << endl; } else { cout << -1 << endl; } }