Accumulate XOR

Accumulate XOR involves calculating the running bitwise XOR of all elements in a sequence.

For a sequence of integers a1, a2, …, an, the accumulate XOR is:

accum = 0 for i from 1 to n: accum ^= ai

This XORs each integer element into a running result.

Bitwise XOR finds the differing bits between values. Accumulate XOR is used in applications like encryption and checksum verification.

Example in Java:

1
2
3
4
5
6
7
8

int[] a = {3, 5, 2, 7}; 

int accumXOR = 0;
for(int num : a) {
  accumXOR ^= num;
}

// accumXOR = 7

Example in C++:

1
2
3
4
5
6
7
8

vector<int> a{4, 8, 2, 10};

int accumXOR = 0;
for(int num : a) {
  accumXOR ^= num; 
} 

// accumXOR = 14

Example in Python:

1
2
3
4
5
6
7

a = [1, 3, 5, 7]

accum_xor = 0
for num in a:
  accum_xor ^= num
  
# accum_xor = 4  

In summary, accumulate XOR bitwise XORs all integers in a sequence into a result. This is useful for checksums and encryption.

Accumulate XOR

Concept

Accumulate XOR is the process of performing the bitwise XOR operation between each successive pair of elements in an array, and then summing up these XOR results. XOR, or “exclusive or,” is a bitwise operation that returns 1 for each position where either of the corresponding bits of its operands is 1, but not both. This is particularly useful in cryptography, error detection, and data compression.

Why is it Important?

• Cryptography: Used in simple encryption and decryption algorithms.
• Error Detection: Can be useful for checksums in network communications.
• Data Transformation: Helps in masking or altering data.

Formula

The formula to calculate Accumulated XOR ( D ) for an array ( A ) of length ( n ) is:

[ D = A[0] \oplus A[1] \oplus \cdots \oplus A[n-1] ]

Example Code

Java

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15

public class Main {
    public static int accumulateXOR(int[] A) {
        int sum = 0;
        for (int num : A) {
            sum ^= num;
        }
        return sum;
    }

    public static void main(String[] args) {
        int[] A = {1, 2, 3, 4};
        int result = accumulateXOR(A);
        System.out.println("Accumulated XOR: " + result);
    }
}

C++

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18

#include <iostream>
using namespace std;

int accumulateXOR(int A[], int n) {
    int sum = 0;
    for (int i = 0; i < n; ++i) {
        sum ^= A[i];
    }
    return sum;
}

int main() {
    int A[] = {1, 2, 3, 4};
    int n = sizeof(A)/sizeof(A[0]);
    int result = accumulateXOR(A, n);
    cout << "Accumulated XOR: " << result << endl;
    return 0;
}

Python

1
2
3
4
5
6
7
8
9

def accumulate_xor(A):
    result = 0
    for num in A:
        result ^= num
    return result

A = [1, 2, 3, 4]
result = accumulate_xor(A)
print(f"Accumulated XOR: {result}")

Key Takeaways

• Accumulate XOR is a powerful tool for performing bitwise XOR operations across an array to obtain a single result.
• It is computationally efficient due to the simple nature of the XOR operation.
• This technique finds uses in various applications, such as cryptography for data encryption and error detection in data transmission.