BAers Explore Modern Cryptography with Columbia's Tenured Professor

Beijing Academy's elective, Global Academician General Course: Mathematics Frontiers and Big Data Applications, recently concluded. Over the course of the semester, Prof. Andrew J. Blumberg, a tenured full professor at Columbia University, guided students from classical substitution ciphers to the frontier world of zero-knowledge proofs and secure multi-party protocols. Students brought their exploration to a satisfying close with individual English-language research reports.

I. Learning from a World-Class Scholar



Prof. Andrew J. Blumberg

Prof. Blumberg is a tenured full professor of Mathematics and Computer Science at Columbia University, where he also holds an endowed chair in cancer data research. His research spans multiple frontier domains across mathematics and computer science, and he has received numerous honors, including the NSF CAREER Award and the DARPA Young Faculty Award.

II. Course Content: From Classical Ciphers to Modern Cryptographic Frontiers

Themed An Introduction to Modern Cryptography and Privacy, each session consisted of a core concept lecture and an interactive seminar, with teaching assistants providing support throughout. The course progressed at a steady, well-balanced pace.

The curriculum began with the historical evolution and mathematical foundations of cryptography, moved through public-key encryption and digital signature schemes such as RSA and ElGamal, and extended to frontier applications including e-cash, anonymous communication, blockchain, zero-knowledge proofs, and secure multi-party protocols. It also delved into randomness, pseudorandom functions, vulnerability exploitation, and cryptanalysis.





From the substitution ciphers of Caesar's era to the modern cryptographic systems that safeguard the privacy of hundreds of millions of users today, the course painted a sweeping panorama of the revolutionary advances in cryptography and privacy protection over the past four decades.

III. Becoming "Cipher Attackers": Understanding Security by Breaking It

What left the deepest impression on students was the hands-on research component. Focusing on encryption and information security challenges, students stepped into the shoes of "cipher attackers" in realistic communication scenarios. They selected research tasks across a spectrum of cryptographic applications — substitution ciphers, one-time pads, RSA encryption, ElGamal encryption, digital signatures, secret sharing, and e-cash — and launched attacks using methods such as frequency analysis, brute force, and integer factorization. Throughout the process, they meticulously documented and analyzed their experimental procedures and results.

Through repeated role reversals between attacker and defender, students truly grasped the mathematical principles behind cryptography:

• Why can frequency analysis easily crack classical substitution ciphers?
• Why is the one-time pad theoretically unbreakable?
• Why does the security of RSA rest on the computational complexity of factoring large integers?

With these questions in mind, students went on to evaluate the security levels, potential vulnerabilities, and mathematical underpinnings of different encryption methods. In the end, each student produced a one-to-two-page English-language research report, presenting their research questions, experimental methods, and analytical conclusions in proper academic language — not only the culmination of their coursework, but also a first taste of English academic writing aligned with the standards of top global universities.

Students who successfully completed the course also received a certificate of completion personally signed by Prof. Blumberg — a meaningful testament to their academic growth.

IV. Takeaways and Looking Ahead



Over the course of a semester, this elective gave students far more than knowledge of cryptography. It offered them a window into the frontiers of academic research, the confidence to express themselves in English in an academic context, and the ability to "think like a researcher."

From listening in lectures to pursuing independent research, from receiving knowledge to tackling difficult problems head-on — this is precisely the spirit behind the school's commitment to bringing world-class academic resources to its students. Moving forward, the school will continue to build high-caliber academic platforms, enabling more students to ignite their passion for science and broaden their global perspectives through dialogue with world-class scholars, and ultimately step into a broader academic world.