Mathematics of Post-Quantum Cryptography

Cryptography keeps our everyday information communications secure.
Cryptography based on key sharing have been used mainly for military purposes since ancient times in human history, but with the advent of the Internet, cryptography that does not require key sharing has become necessary.
In 1976, Diffie and Hellman proposed the concept of public key cryptography, which does not require key sharing among communicators. Since then, research on public key cryptography has progressed, involving not only computer science but also mathematics, and has become an essential technology for the society we live in.

The security of public key cryptography is supported by computational hardness of problems derived from mathematics. For example, the integer factoring problem is a basis for the security of RSA cryptography, and the discrete logarithm problem is for elliptic curve cryptography.
However, in 1994, Shor proposed an efficient quantum algorithm that solves these problems. This means that emergence of large-scale quantum computers will break RSA and elliptic curve cryptography we use today.

For this reason, research on next-generation cryptography, so-called Post-Quantum Cryptography (PQC for short), is currently underway to prepare for a future in which quantum computers will emerge. In this talk, without assuming any knowledge of cryptography, I will give a brief overview of cryptography and the progress of PQC. The first half of the talk will mainly outline the relationship between mathematics and cryptography, while the second half will discuss isogeny-based cryptography, one of the promising PQC, with our recent results.