結果
| 問題 |
No.950 行列累乗
|
| ユーザー |
 gew1fw
|
| 提出日時 | 2025-06-12 13:14:32 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 8,065 bytes |
| コンパイル時間 | 154 ms |
| コンパイル使用メモリ | 82,668 KB |
| 実行使用メモリ | 83,992 KB |
| 最終ジャッジ日時 | 2025-06-12 13:16:51 |
| 合計ジャッジ時間 | 5,780 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge4 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 4 |
| other | AC * 37 WA * 20 |
ソースコード
import math
import random
def multiply(a, b, p):
res = [[0]*2 for _ in range(2)]
res[0][0] = (a[0][0] * b[0][0] + a[0][1] * b[1][0]) % p
res[0][1] = (a[0][0] * b[0][1] + a[0][1] * b[1][1]) % p
res[1][0] = (a[1][0] * b[0][0] + a[1][1] * b[1][0]) % p
res[1][1] = (a[1][0] * b[0][1] + a[1][1] * b[1][1]) % p
return res
def matrix_pow(mat, power, p):
result = [[1, 0], [0, 1]] # Identity matrix
while power > 0:
if power % 2 == 1:
result = multiply(result, mat, p)
mat = multiply(mat, mat, p)
power //= 2
return result
def is_prime(n):
if n < 2:
return False
for p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]:
if n % p == 0:
return n == p
d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1
for a in [2, 3, 5, 7, 11]:
if a >= n:
continue
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(s - 1):
x = pow(x, 2, n)
if x == n - 1:
break
else:
return False
return True
def pollards_rho(n):
if n % 2 == 0:
return 2
if n % 3 == 0:
return 3
if n % 5 == 0:
return 5
while True:
c = random.randint(1, n - 1)
f = lambda x: (pow(x, 2, n) + c) % n
x, y, d = 2, 2, 1
while d == 1:
x = f(x)
y = f(f(y))
d = math.gcd((x - y) % n, n)
if d != n:
return d
def factor(n):
factors = []
def _factor(n):
if n == 1:
return
if is_prime(n):
factors.append(n)
return
d = pollards_rho(n)
_factor(d)
_factor(n // d)
_factor(n)
factors.sort()
return factors
def prime_factors(n):
factors = factor(n)
res = {}
for p in factors:
res[p] = res.get(p, 0) + 1
return res
def compute_order(a, p):
if a == 0 or p <= 1 or math.gcd(a, p) != 1:
return -1
m = p - 1
factors = prime_factors(m)
unique_factors = list(factors.keys())
for factor_p in unique_factors:
exponent = factors[factor_p]
for e in range(exponent, 0, -1):
if pow(a, m // (factor_p ** e), p) == 1:
m = m // (factor_p ** e)
break
return m
def baby_step_giant_step(a, b, p):
if b == 1:
return 0
if a == 0:
return -1
a = a % p
b = b % p
n = int(math.isqrt(p)) + 1
table = {}
current = 1
for i in range(n):
if current not in table:
table[current] = i
current = (current * a) % p
an = pow(a, n, p)
an = pow(an, p-2, p)
current = b
for i in range(n):
if current in table:
return i * n + table[current]
current = (current * an) % p
return -1
def discrete_log(a, b, p):
if p == 1:
return 0 if b == 1 else -1
if a == 0:
if b == 0:
return 1
else:
return -1
a = a % p
b = b % p
if b == 1:
return 0
if a == 0:
return -1
return baby_step_giant_step(a, b, p)
def main():
p = int(input())
A = []
for _ in range(2):
A.append(list(map(int, input().split())))
B = []
for _ in range(2):
B.append(list(map(int, input().split())))
# Compute determinants
det_A = (A[0][0] * A[1][1] - A[0][1] * A[1][0]) % p
det_B = (B[0][0] * B[1][1] - B[0][1] * B[1][0]) % p
if det_A != 0:
# Check if det_B is possible
log_val = discrete_log(det_A, det_B, p)
if log_val == -1:
print(-1)
return
# Check if A^log_val == B
a_pow = matrix_pow(A, log_val, p)
if a_pow == B:
print(log_val)
return
# Find the order of det_A
ord_det = compute_order(det_A, p)
if ord_det == -1:
print(-1)
return
current_x = log_val
min_x = None
for _ in range(100):
current_x += ord_det
a_pow = matrix_pow(A, current_x, p)
if a_pow == B:
if min_x is None or current_x < min_x:
min_x = current_x
if min_x is not None:
print(min_x)
return
else:
print(-1)
return
else:
if det_B != 0:
print(-1)
return
# Check if A^2 is zero matrix
A_sq = multiply(A, A, p)
is_zero = True
for i in range(2):
for j in range(2):
if A_sq[i][j] != 0:
is_zero = False
if is_zero:
# Check if B is zero matrix or A
is_B_zero = True
for i in range(2):
for j in range(2):
if B[i][j] != 0:
is_B_zero = False
if is_B_zero:
print(2)
return
if A == B:
print(1)
return
else:
print(-1)
return
# Check if A^2 = c*A
c = None
for i in range(2):
for j in range(2):
if A[i][j] == 0:
if A_sq[i][j] != 0:
c = None
break
else:
inv = pow(A[i][j], p-2, p)
current_c = (A_sq[i][j] * inv) % p
if c is None:
c = current_c
else:
if current_c != c:
c = None
break
if c is None:
break
if c is not None:
# Check if B is k*A
k = None
for i in range(2):
for j in range(2):
if A[i][j] == 0:
if B[i][j] != 0:
k = None
break
else:
inv = pow(A[i][j], p-2, p)
current_k = (B[i][j] * inv) % p
if k is None:
k = current_k
else:
if current_k != k:
k = None
break
if k is None:
break
if k is not None:
if c == 0:
if k == 0:
# Find minimal n >=2
a_pow = matrix_pow(A, 2, p)
if a_pow == B:
print(2)
return
else:
print(-1)
return
else:
print(-1)
return
else:
log_val = discrete_log(c, k, p)
if log_val == -1:
print(-1)
return
n_candidate = log_val + 1
if n_candidate < 1:
print(-1)
return
a_pow = matrix_pow(A, n_candidate, p)
if a_pow == B:
print(n_candidate)
return
else:
print(-1)
return
else:
print(-1)
return
else:
# Check small n
current = A
if current == B:
print(1)
return
for n in range(2, 10**5+1):
current = multiply(current, A, p)
if current == B:
print(n)
return
print(-1)
return
if __name__ == "__main__":
main()