# Invited Paper Session Abstracts - Cryptography and the Mathematics Behind It

Thursday, August 1, 1:30 p.m. - 4:00 p.m., Duke Energy Convention Center, Room 205

The security of our voting, banking, and military systems, the infrastructure of modern day society, and democracy itself rely on cryptography, to ensure privacy and allow secure and authenticated communication. Public key cryptography is based on mathematics, especially number theory and algebraic geometry. Recent proposals to solve long-standing important problems in the field of computer security use the geometry of numbers and the theory of lattices. Other mathematical ingredients found in modern day cryptography include factorization of numbers, discrete logarithms, and the theory of elliptic curves. Current research includes the area of fully homomorphic encryption and the search for cryptographically useful multilinear maps.

This session will have expository talks aimed at a general mathematical audience and will be suitable for both students and faculty.

Organizer:
Alice Silverberg, University of California, Irvine

### Language, Probability, and Cryptography

1:30 p.m. - 1:50 p.m.

#### Abstract

What rules define a language? Can we use this information to decrypt secret messages? In this talk, we look at the application of Markov Chain Monte Carlo methods to decrypt different types of substitution ciphers. By studying the state changes in consecutive letters of a particular language, one can program a computer to make exceptionally ‘smart’ guesses as to what an encrypted message actually says. We will discuss some famous examples, and how this can be made into a particularly interesting undergraduate research project.

### Inrtoduwtion to Erorr Dwtetcion and Czorrectmon

2:00 p.m. - 2:20 p.m.
Steven J Miller, Williams College

#### Abstract

In today's world it is essential for numerous applications, from cryptography to e-commerce to streaming videos, to be able to efficiently send information and have the recipient receive what we intend. We'll discuss some of the challenges in doing so, explore some systems that can not only detect but also correct transmission errors in real-time, and highlight some applications in cryptography. For example, due to how RSA encodes messages, a one bit error can lead to a completely different text being transmitted than intended. For another, by deliberately changing some bits we can hide a message in an image.

### Post-quantum Key Exchange Based on "Learning with Errors" Problems

2:30 p.m. - 2:50 p.m.
Jintai Ding, University of Cincinnati

#### Abstract

Public key cryptosystems are critical part of the foundation of modern communication systems, in particular, the Internet. However Shor's algorithm shows that the existing systems, such as Diffie-Hellmann key exchange, RSA, and Elliptic Curve Cryptography, can be broken by a quantum computer. To prepare for the coming age of quantum computing, we need to build new public key cryptosystems that could resist quantum computer attacks. In this lecture, we present first the basic mathematical idea and history of key exchange, and explain why it is hard to build new key exchanges. Then we will present a practical and provably secure (authenticated) key exchange protocol based on "learning with errors" problems and explain the fundamental mathematical idea behind such a construction.

### Public-key Cryptography from Supersingular Elliptic Curve Isogenies

3:00 p.m. - 3:20 p.m.
David Jao, University of Waterloo

#### Abstract

Isogeny-based cryptography represents one of the few number-theoretic approaches to cryptography that remains relevant in the context of post-quantum cryptosystems, schemes that are intended to be secure against adversaries with efficient universal quantum computers. Compared to other post-quantum cryptosystems, isogeny-based cryptography has smaller keys, and simple, straightforward parameter selection involving only one tunable security-sensitive parameter. In this talk, we survey one existing proposal for isogeny-based cryptography, the Supersingular Isogeny Diffie-Hellman (SIDH) proposal of Jao and De Feo, whose underlying hard problem is a variant of the Charles-Goren-Lauter cycle finding problem on Pizer’s family of expander graphs. We demonstrate some explicit examples of SIDH computations, and discuss the current status of the security of the cryptosystem against classical and quantum adversaries.

### $x^n+x+a$

3:30 p.m. - 3:50 p.m.
Kumar Murty, University of Toronto

#### Abstract

These polynomials are not only important computationally and in cryptographic applications, but they also have many beautiful mathematical properties. There are also many open problems.

Year:
2019