AtCoder Regular Contest 084 - XorShift

There are N non-negative integers written on a blackboard. The i-th integer is A_i.
Takahashi can perform the following two kinds of operations any number of times in any order:
 - Select one integer written on the board (let this integer be X). Write 2X on the board, without erasing the selected integer.
 - Select two integers, possibly the same, written on the board (let these integers be X and Y). Write X XOR Y (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.
How many different integers not exceeding X can be written on the blackboard? We will also count the integers that are initially written on the board.
Since the answer can be extremely large, find the count modulo 998244353.

-----Constraints-----
 - 1 \leq N \leq 6
 - 1 \leq X < 2^{4000}
 - 1 \leq A_i < 2^{4000}(1\leq i\leq N)
 - All input values are integers.
 - X and A_i(1\leq i\leq N) are given in binary notation, with the most significant digit in each of them being 1.

-----Input-----
Input is given from Standard Input in the following format:
N X
A_1
:
A_N

-----Output-----
Print the number of different integers not exceeding X that can be written on the blackboard.

-----Sample Input-----
3 111
1111
10111
10010

-----Sample Output-----
4

Initially, 15, 23 and 18 are written on the blackboard. Among the integers not exceeding 7, four integers, 0, 3, 5 and 6, can be written.
For example, 6 can be written as follows:
 - Double 15 to write 30.
 - Take XOR of 30 and 18 to write 12.
 - Double 12 to write 24.
 - Take XOR of 30 and 24 to write 6.

Source: atcoder | Difficulty: VERY_HARD


Solutions:
import sys
read = sys.stdin.read
readline = sys.stdin.readline
sys.setrecursionlimit(10 ** 5)
(n, x) = readline().split()
n = int(n)
x = int(x, 2)
(*a,) = [int(x, 2) for x in read().split()]

def gcd(a, b):
	c = 0
	while b:
		if a < b:
			(a, b) = (b, a)
			continue
		a ^= b << a.bit_length() - b.bit_length()
	return a
v = 0
for ai in a:
	v = gcd(ai, v)
MOD = 998244353
p2 = [1]
for _ in range(5000):
	p2.append(p2[-1] * 2 % MOD)
dv = v.bit_length()
dx = x.bit_length()

def dfs(x, y, d):
	if d < dv:
		return int(x >= y)
	else:
		bx = x >> d - 1 & 1
		by = y >> d - 1 & 1
		if bx and by:
			return (p2[d - dv] + dfs(x, y, d - 1)) % MOD
		elif not bx and by:
			return dfs(x, y ^ v << d - dv, d - 1)
		elif not by and bx:
			return (p2[d - dv] + dfs(x, y ^ v << d - dv, d - 1)) % MOD
		else:
			return dfs(x, y, d - 1)
print(dfs(x, 0, dx))


import sys
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
from functools import reduce
X = int(readline().split()[1], 2)
A = [int(x, 2) for x in readlines()]
MOD = 998244353

def gcd(a, b):
	if a < b:
		(a, b) = (b, a)
	while b:
		LA = a.bit_length()
		LB = b.bit_length()
		a ^= b << LA - LB
		if a < b:
			(a, b) = (b, a)
	return a
g = reduce(gcd, A)
LX = X.bit_length()
Lg = g.bit_length()
answer = X >> Lg - 1
prod = 0
x = X
Lx = LX
while Lx >= Lg:
	prod ^= g << Lx - Lg
	x ^= g << Lx - Lg
	Lx = x.bit_length()
if prod <= X:
	answer += 1
answer %= MOD
print(answer)


import sys
sys.setrecursionlimit(10 ** 6)
int1 = lambda x: int(x) - 1
p2D = lambda x: print(*x, sep='\n')

def MI():
	return map(int, sys.stdin.readline().split())

def LI():
	return list(map(int, sys.stdin.readline().split()))

def LLI(rows_number):
	return [LI() for _ in range(rows_number)]

def gcd(a, b):
	if b == 0:
		return a
	if a < b:
		(a, b) = (b, a)
	k = a.bit_length() - b.bit_length()
	return gcd(b, a ^ b << k)

def main():
	md = 998244353
	(n, x) = input().split()
	x = int(x, 2)
	aa = [input() for _ in range(int(n))]
	aa = [int(a, 2) for a in aa]
	g = aa[0]
	for a in aa[1:]:
		g = gcd(g, a)
	ans = x >> g.bit_length() - 1
	s = 0
	while 1:
		k = (x ^ s).bit_length() - g.bit_length()
		if k < 0:
			break
		s ^= g << k
	if s <= x:
		ans += 1
	print(ans % md)
main()


def divmod(f, g):
	assert g
	h = 0
	for i in reversed(range(f.bit_length() - g.bit_length() + 1)):
		if f & 1 << g.bit_length() + i - 1:
			f ^= g << i
			h ^= 1 << i
	return (h, f)

def gcd(f, g):
	while g:
		(q, r) = divmod(f, g)
		(f, g) = (g, r)
	return f
import functools

def solve(n, x, a):
	g = functools.reduce(gcd, a)
	cnt = 0
	h = 0
	for k in reversed(range(x.bit_length() - g.bit_length() + 1)):
		bit = 1 << g.bit_length() + k - 1
		if x & bit:
			cnt += 1 << k
		if x & bit != h & bit:
			h ^= g << k
	cnt += h <= x
	return cnt % 998244353

def main():
	(n, x) = input().split()
	n = int(n)
	x = int(x, 2)
	a = [int(input(), 2) for _ in range(n)]
	print(solve(n, x, a))
main()


import random
mod = 998244353
(N, X) = input().split()
N = int(N)
A = []
for i in range(N):
	A.append(int(input(), 2))
A.sort()
a = A[-1]
M = max(len(X) - 1, a.bit_length() - 1)
data = [0] * (M + 1)
n = a.bit_length() - 1
for i in range(M - n, -1, -1):
	data[i + n] = a << i
for i in range(0, N - 1):
	a = A[i]
	flag = True
	while flag:
		n = a.bit_length()
		for j in range(n - 1, -1, -1):
			a = min(a, a ^ data[j])
		if a != 0:
			data[a.bit_length() - 1] = a
			id = a.bit_length() - 1
			while data[id + 1] == 0:
				data[id + 1] = min(data[id] << 1 ^ a, data[id] << 1)
				if data[id + 1]:
					id += 1
				else:
					flag = False
			else:
				a = data[id] << 1
		else:
			break
data2 = [0] * (M + 1)
for i in range(M + 1):
	data2[i] = data[i] != 0
for i in range(1, M + 1):
	data2[i] += data2[i - 1]
data2 = [0] + data2
x = 0
ans = 0
n = len(X) - 1
for i in range(len(X)):
	if X[i] == '1':
		if x >> n - i & 1 == 1:
			if data[n - i]:
				ans += pow(2, data2[n - i], mod)
				ans %= mod
		else:
			ans += pow(2, data2[n - i], mod)
			ans %= mod
			if data[n - i]:
				x = x ^ data[n - i]
			else:
				break
	elif x >> n - i & 1 == 1:
		if data[n - i]:
			x = x ^ data[n - i]
		else:
			break
	else:
		continue
else:
	ans += 1
	ans %= mod
print(ans)


(N, X) = input().split()
N = int(N)
X = int(X, 2)
p = int(input(), 2)
for i in range(N - 1):
	v = int(input(), 2)
	if p > v:
		(p, v) = (v, p)
	while 0 < p:
		q = v ^ p << v.bit_length() - p.bit_length()
		if p < q:
			(p, v) = (p, q)
		else:
			(p, v) = (q, p)
	p = v
lx = X.bit_length()
lp = p.bit_length()
MOD = 998244353
ans = (X >> p.bit_length() - 1) % MOD
q = p << lx - lp
b = 1 << lx - 1
x = 0
y = 0
for k in range(lx - lp, -1, -1):
	if b & (X ^ y):
		y ^= q
	b >>= 1
	q >>= 1
if y <= X:
	ans = (ans + 1) % MOD
print(ans)


import random
mod = 998244353
(N, X) = input().split()
N = int(N)
A = []
for i in range(N):
	A.append(int(input(), 2))
A.sort()
a = A[-1]
M = max(len(X) - 1, a.bit_length() - 1)
base = []
n = a.bit_length() - 1
for i in range(M - n, -1, -1):
	base.append(a << i)
for i in range(0, N - 1):
	a = A[i]
	for j in range(M):
		for b in base:
			a = min(a, a ^ b)
		if a == 0:
			break
		else:
			base.append(a)
			a = a << 1
data = [0] * (M + 1)
data2 = [0] * (M + 1)
for b in base:
	data[b.bit_length() - 1] = b
	data2[b.bit_length() - 1] = 1
for i in range(1, M + 1):
	data2[i] += data2[i - 1]
data2 = [0] + data2
x = 0
ans = 0
n = len(X) - 1
for i in range(len(X)):
	if X[i] == '1':
		if x >> n - i & 1 == 1:
			if data[n - i]:
				ans += pow(2, data2[n - i], mod)
				ans %= mod
		else:
			ans += pow(2, data2[n - i], mod)
			ans %= mod
			if data[n - i]:
				x = x ^ data[n - i]
			else:
				break
	elif x >> n - i & 1 == 1:
		if data[n - i]:
			x = x ^ data[n - i]
		else:
			break
	else:
		continue
else:
	ans += 1
	ans %= mod
print(ans)