201563: [AtCoder]ARC156 D - Xor Sum 5
Memory Limit：1024 MB
Time Limit：2 S
Judge Style：Text Compare
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Solved：0
Description
Score : $700$ points
Problem Statement
You are given a sequence of $N$ non-negative integers $A=(A_1,A_2,\dots,A_N)$ and a positive integer $K$.
Find the bitwise $\mathrm{XOR}$ of $\displaystyle \sum_{i=1}^{K} A_{X_i}$ over all $N^K$ sequences of $K$ positive integer sequences $X=(X_1,X_2,\dots,X_K)$ such that $1 \leq X_i \leq N\ (1\leq i \leq K)$.
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 the digits in that place 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
- $1 \leq N \leq 1000$
- $1 \leq K \leq 10^{12}$
- $0 \leq A_i \leq 1000$
- All values in the input are integers.
Input
The input is given from Standard Input in the following format:
$N$ $K$ $A_1$ $A_2$ $\dots$ $A_N$
Output
Print the answer.
Sample Input 1
2 2 10 30
Sample Output 1
40
There are four sequences to consider: $(X_1,X_2)=(1,1),(1,2),(2,1),(2,2)$, for which $A_{X_1}+A_{X_2}$ is $20,40,40,60$, respectively. Thus, the answer is $20 \oplus 40 \oplus 40 \oplus 60=40$.
Sample Input 2
4 10 0 0 0 0
Sample Output 2
0
Sample Input 3
11 998244353 314 159 265 358 979 323 846 264 338 327 950
Sample Output 3
236500026047