The LUMS Mathematics Department hosted another captivating session of the LUMS Math Circle titled “Locking & Unlocking Secrets: How Mathematics Keeps Your Messages Safe.” The event aimed to introduce students to the world of cryptography—an area where mathematics protects information, ensures privacy, and keeps digital communication secure.

The session was skillfully conducted by Dr. Aeysha Khalique and Dr. Adnan Khan, whose engaging teaching styles, real-life examples, and hands-on mathematical tasks kept the students completely immersed throughout the event.

Session Highlights

1. Beginning with the Caesar Cipher – Cracking the First Code

The session opened with Dr. Aeysha Khalique introducing the classical Caesar Cipher, one of history’s earliest encryption techniques. Students received two overlapping alphabet circles and were asked to decode a secret phrase. By matching letters from the inner circle to the outer circle, they successfully deciphered:

➡ “MATH IS FUN.”

To celebrate their effort, everyone received chocolates—an excellent motivational touch that made the activity more memorable.

Dr. Aeysha also demonstrated how shifting the alphabet wheel by 6, 7, or 8 places changes the coded message. She explained that this “shift number” is the key, and anyone with the key can reverse the encryption.

However, she pointed out one major limitation:

A single-key cipher is easy to break because the key must be shared and there are only 25 possible shifts, making brute-force attacks simple.

2. Understanding Modular Arithmetic – The Engine Behind Cryptography

Next, Dr. Aeysha introduced modular arithmetic, the foundational idea behind most encryption techniques.

She began with the relatable example of a clock, where numbers reset after 12—an intuitive demonstration of working “modulo 12.” She explained that in modular arithmetic, the only information that matters is the quotient and the remainder.

Students participated in an interactive activity where they computed:

•  (2mod7 = 2)

•  (9mod7 = 2)

•  (50mod7 = 1)

•  (-9mod7 = 5)

•  (65mod4 = 1)

•  (-65mod3 = 1)

These exercises helped her gauge students’ understanding, while the explanation of the floor function for finding remainders strengthened their conceptual foundation.

This section connected beautifully to the mathematical heart of modern encryption.

3. Why One-Key Encryption Is Not Enough

Drawing from the limitations of the Caesar Cipher, Dr. Aeysha transitioned to the idea of secure communication systems. She emphasized why relying on a single shared key is unsafe—because if the key is intercepted, the entire system collapses.

This set the stage for the central topic of the session:
➡ Public-key cryptography, which uses two keys—a public one for encryption and a private one for decryption.

4. RSA Algorithm – Modern Cryptography Explained

At this point, Dr. Adnan Khan took over to introduce the famous RSA Algorithm, a cornerstone of secure communication since 1977.

He explained that RSA uses:

•  A public key ((n, e)) → used for encrypting

•  A private key ((n, d)) → used for decrypting

Dr. Adnan demonstrated the process using simple numbers, guiding students through:

•  Choosing two primes.

•  Finding (n = pq).

•  Computing the quotient function

•  Selecting (e)

•  Finding the modular inverse for the private key

•  Performing modular exponentiation

He showed an example using small values such as (n = 77) and (e = 17) to illustrate how encryption works before scaling the logic to large real-world values.

The slides helped visualize the flow of RSA, from key generation to message encryption. Students were fascinated to learn that the security of RSA depends on the difficulty of prime factorization, a problem computers struggle with for very large numbers.

5. Perfect Secrecy – The Ideal, If Not Always Practical

The session also touched upon perfect secrecy, including the famous One-Time Pad, where a message is mathematically guaranteed secure if:

•  The key is random

•  The key is as long as the message

•  The key is used only once

Though rarely practical, it demonstrated beautifully how mathematics can create unbreakable codes.

Activities and Student Engagement

Throughout the session, students participated in:

•  Deciphering Caesar-coded messages

•  Solving modular arithmetic puzzles

•  Computing remainders

•  Applying modular inverses

•  Working through simplified RSA steps

These activities created a lively environment where every student felt involved and challenged. 

Acknowledgment

We extend our heartfelt gratitude to Dr. Aeysha Khalique and Dr. Adnan Khan for delivering an engaging and intellectually rich session that brought the world of cryptography to life for our young participants. We would also like to sincerely appreciate the exceptional organizational support provided by Ms. Noreen Sohail, Mr. Qamar Abbas, and Ms. Shazia Zafar, whose dedication, planning, and coordination ensured that every aspect of the event ran smoothly and successfully. Finally, a special thank you to Mr. M. Javaid Qayoom for thoughtfully preparing and writing this report, capturing the essence of the session with clarity and care.

Closing Ceremony

The event concluded with a warm closing session where certificates of participation were distributed to all students. This recognition celebrated their enthusiasm, curiosity, and willingness to explore advanced mathematical ideas.

Here are some highlights from the event: