Lattice-Based Cryptography

Most modern cryptography, and public-key crypto in particular, is based on mathematical problems that are conjectured to be infeasible (e.g., factoring large integers). Unfortunately, standard public-key techniques are often too inefficient to be employed in many environments; moreover, all commonly used schemes can in principle be broken by quantum computers. This talk will review my recent work on developing new mathematical foundations for cryptography, using geometric objects called lattices. Compared to more conventional proposals, lattice-based schemes offer a host of potential advantages: they are simple and highly parallelizable, they can be proved secure under mild worst-case hardness assumptions, and they remain unbroken by quantum algorithms. Due to the entirely different underlying mathematics, however, realizing even the most basic cryptographic notions has been a major challenge. Surprisingly, I will show that lattice-based schemes are also remarkably flexible and expressive, and that many important cryptographic goals can be achieved --- sometimes even more simply and efficiently than with conventional approaches. Some of our schemes provide interesting twists on old and cherished cryptographic notions, while others introduce entirely new concepts altogether.