Question

Suppose a password consisted of a string of nine characters from the English alphabet (twenty-six characters). If each possible password could be tested in a millisecond, how long would it take to test all possible passwords?

Short answer

It would take about 172 years.

Step-by-step solution

Understanding the Problem

We have to determine the total number of possible passwords made from nine letters where each letter can be any of the twenty-six English alphabet letters.

Calculate Total Possible Passwords

Each character in the password has 26 possible choices. The total number of passwords is calculated by multiplying the number of choices for each of the 9 positions, which gives us the equation: \[ 26^9 \]

Compute 26 to the Power of 9

Calculate \( 26^9 \), the power gives the total number of possible passwords. Compute this as: \[ 26^9 = 5,429,503,678,976 \]

Determine Time to Test All Passwords

If each password is tested in one millisecond, then the total time to test all passwords is \[ 5,429,503,678,976 \text{ milliseconds.} \]Convert this to more conventional units, such as seconds, minutes, or years.

Convert Milliseconds to Years

Convert the milliseconds to seconds by dividing by 1000, to minutes by dividing by 60, to hours by dividing by 60 again, and finally to days by dividing by 24, and to years by dividing by 365.\[\begin{align*}\text{Seconds} & = \frac{5,429,503,678,976}{1000} = 5,429,503,678.976 \\text{Minutes} & = \frac{5,429,503,678.976}{60} \approx 90,491,727.983 \\text{Hours} & = \frac{90,491,727.983}{60} \approx 1,508,195.466 \\text{Days} & = \frac{1,508,195.466}{24} \approx 62,841.478 \\text{Years} & = \frac{62,841.478}{365} \approx 172.147\end{align*}\]

Conclusion

It would take approximately 172.147 years to test all possible passwords at a rate of one password per millisecond.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing finite structures. It is used in calculating the total number of possible configurations in different situations. In the context of password security, combinatorics allows us to determine how many potential passwords exist.
In our problem, each of the 9 positions in the password can be filled with any of the 26 English alphabet letters. To find the total number of possible passwords, we multiply the number of choices for each position using the formula:

• 26 choices per position
• 9 positions in total
• Resulting in: \( 26^9 \)

After calculating, you find a massive number of possibilities: 5,429,503,678,976. Understanding combinatorics is essential as it shows the complexity and vastness of password combinations.

Time Complexity

Time complexity describes how the computation time of an algorithm scales with the size of the input. It's an essential concept in assessing the efficiency of algorithms. In password security, when estimating how long it takes to guess all possible passwords, we consider how the required time increases with password length and character set size.
In this task, testing each password takes 1 millisecond, hence testing all possibilities (5,429,503,678,976 of them) requires:

• Total time in milliseconds: same as the number of tests, 5,429,503,678,976
• Time in more understandable units after conversion:
• 172.147 years

Showing these conversions emphasizes the huge amount of time needed due to the exponential growth in possibilities.

Brute Force Attacks

Brute force attacks entail systematically guessing every possible key or password until the correct one is found. This method highlights the importance of password strength and complexity.
Our exercise demonstrates the impracticality of brute force for strong passwords, as such a large number of potential combinations lead to testing times that are not feasible within a human lifetime. Considering each password takes 1 millisecond to test, the total would be more than 172 years to explore all combinations.

• Brute force effectiveness decreases with longer passwords and larger character sets.
• Time required skyrockets as the number of possibilities grows exponentially.
• This serves as a reminder to always use complex, lengthy passwords, reducing vulnerability to brute force methods.

Understanding brute force attack limitations guides us in creating and maintaining secure passwords.