Custom authentication tokens in Firebase - Time & Space Complexity

Time Complexity: Custom authentication tokens

O(n)

Understanding Time Complexity

When using custom authentication tokens in Firebase, it's important to understand how the time to create and verify these tokens changes as you handle more users.

We want to know: how does the work grow when we generate tokens for many users?

Scenario Under Consideration

Analyze the time complexity of the following operation sequence.

const admin = require('firebase-admin');

async function createCustomTokens(userIds) {
  const tokens = [];
  for (const uid of userIds) {
    const token = await admin.auth().createCustomToken(uid);
    tokens.push(token);
  }
  return tokens;
}

This code creates a custom authentication token for each user ID in a list.

Identify Repeating Operations

Identify the API calls, resource provisioning, data transfers that repeat.

• Primary operation: Calling createCustomToken(uid) for each user.
• How many times: Once per user ID in the input list.

How Execution Grows With Input

Each user requires one token creation call, so the total work grows directly with the number of users.

Input Size (n)	Approx. API Calls/Operations
10	10 token creations
100	100 token creations
1000	1000 token creations

Pattern observation: The number of operations increases evenly as the input size grows.

Final Time Complexity

Time Complexity: O(n)

This means the time to create tokens grows in direct proportion to the number of users.

Common Mistake

[X] Wrong: "Creating one token is slow, so creating many tokens will take the same time as one."

[OK] Correct: Each token creation is a separate operation, so more users mean more work and more time.

Interview Connect

Understanding how your code scales with more users shows you can build systems that handle growth smoothly and predict performance.

Self-Check

"What if we batch user IDs and create tokens in parallel? How would the time complexity change?"