SSH key generation (ssh-keygen) in Linux CLI - Time & Space Complexity

Time Complexity: SSH key generation (ssh-keygen)

O(k^4)

Understanding Time Complexity

When generating SSH keys, it is helpful to understand how the time taken grows as the key size changes.

We want to know how the work done by the key generation command changes with input size.

Scenario Under Consideration

Analyze the time complexity of the following command.

ssh-keygen -t rsa -b 2048 -f ~/.ssh/id_rsa

This command generates an RSA SSH key with a 2048-bit size and saves it to a file.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Generating large prime numbers by testing many candidates.
• How many times: The number of tests grows with the size of the key in bits.

How Execution Grows With Input

As the key size increases, the number of operations to find suitable primes grows faster than the key size itself.

Input Size (bits)	Approx. Operations
1024	Moderate number of prime tests
2048	Several times more prime tests than 1024
4096	Much larger number of prime tests, noticeably slower

Pattern observation: The time grows more than linearly as key size doubles, because prime testing is more complex for bigger numbers.

Final Time Complexity

Time Complexity: O(k^4)

This means the time to generate keys grows roughly with the fourth power of the key size in bits, so doubling the key size makes it take much longer.

Common Mistake

[X] Wrong: "Generating a bigger key only takes twice as long as a smaller key."

[OK] Correct: Prime number testing and key calculations get much harder with bigger numbers, so time grows much faster than just doubling.

Interview Connect

Understanding how key size affects generation time shows you can think about how input size impacts work done, a useful skill for many scripting and automation tasks.

Self-Check

"What if we changed the key type from RSA to Ed25519? How would the time complexity change?"