201411: [AtCoder]ARC141 B - Increasing Prefix XOR
Memory Limit：1024 MB
Time Limit：2 S
Judge Style：Text Compare
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Solved：0
Description
Score : $500$ points
Problem Statement
You are given positive integers $N$ and $M$.
Find the number of sequences $A=(A_1,\ A_2,\ \dots,\ A_N)$ of $N$ positive integers that satisfy the following conditions, modulo $998244353$.
- $1 \leq A_1 < A_2 < \dots < A_N \leq M$.
- $B_1 < B_2 < \dots < B_N$, where $B_i = A_1 \oplus A_2 \oplus \dots \oplus A_i$.
Here, $\oplus$ denotes bitwise $\mathrm{XOR}$.
What is bitwise $\mathrm{XOR}$?
The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows:
- When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.
Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$.
Constraints
- $1 \leq N \leq M < 2^{60}$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$ $M$
Output
Print the answer.
Sample Input 1
2 4
Sample Output 1
5
For example, $(A_1,\ A_2)=(1,\ 3)$ counts, since $A_1 < A_2$ and $B_1 < B_2$ where $B_1=A_1=1,\ B_2=A_1\oplus A_2=2$.
The other pairs that count are $(A_1,\ A_2)=(1,\ 2),\ (1,\ 4),\ (2,\ 4),\ (3,\ 4)$.
Sample Input 2
4 4
Sample Output 2
0
Sample Input 3
10 123456789
Sample Output 3
205695670