Phone number authentication in Firebase - Time & Space Complexity

Time Complexity: Phone number authentication

O(n)

Understanding Time Complexity

We want to understand how the time needed to verify phone numbers changes as more users try to sign in.

How does the system handle more requests and how does that affect speed?

Scenario Under Consideration

Analyze the time complexity of the following operation sequence.

const phoneNumber = "[phone]";
const appVerifier = window.recaptchaVerifier;

firebase.auth().signInWithPhoneNumber(phoneNumber, appVerifier)
  .then(confirmationResult => {
    // User receives SMS code
    const code = prompt('Enter the verification code');
    return confirmationResult.confirm(code);
  })
  .then(userCredential => {
    // User signed in
  })
  .catch(error => {
    // Handle errors
  });

This code sends a verification SMS to the user and then confirms the code they enter to sign them in.

Identify Repeating Operations

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

• Primary operation: Sending SMS verification via Firebase API.
• How many times: Once per user sign-in attempt.

How Execution Grows With Input

Each new user triggers one SMS send and one verification check.

Input Size (n)	Approx. Api Calls/Operations
10	10 SMS sends + 10 code verifications
100	100 SMS sends + 100 code verifications
1000	1000 SMS sends + 1000 code verifications

Pattern observation: The number of operations grows directly with the number of users trying to sign in.

Final Time Complexity

Time Complexity: O(n)

This means the time needed grows linearly as more users try to authenticate with their phone numbers.

Common Mistake

[X] Wrong: "Sending one SMS code will handle multiple users at once, so time stays the same."

[OK] Correct: Each user needs their own SMS and verification, so the system must do separate work for each user.

Interview Connect

Understanding how authentication scales helps you design systems that stay fast and reliable as more people use them.

Self-Check

"What if we batch phone number verifications instead of sending SMS one by one? How would the time complexity change?"