Sure, I can help you write a test case generator for the given problem statement. Here is the test case generator in Python:

```python
import random

def tcgen():
    # Define the constraints
    K = random.randint(1, 10**9)
    A = random.randint(1, 10**9)
    B = random.randint(1, 10**9)
    
    return K, A, B

# Example usage:
for _ in range(5):  # Generate 5 test cases
    print(tcgen())
```

This function will generate random values for \( K \), \( A \), and \( B \) within their specified ranges and return them as a tuple. You can run this function multiple times to get different test cases.

Let's briefly discuss what each component does:
- `random.randint(1, 10**9)`: Generates a random integer between 1 and \(10^9\) inclusive. This is used for \( K \), \( A \), and \( B \) as per the problem constraints.
- `for _ in range(5)`: This loop generates 5 test cases as an example usage of the `tcgen` function.

You can adjust the number of test cases generated or modify the function as needed to suit your specific requirements.