Cryptography to me is the "black magic," of cryptographers, enabling tasks that often seem paradoxical or simply just impossible. Like the space explorers, we cryptographers often wonder, "what are the boundaries of this world of black magic?" This work lays one of the founding stones in furthering our understanding of these edges.
We describe plausible lattice-based constructions with properties that approximate the sought after multilinear maps in hard-discrete-logarithm groups. The security of our constructions relies on seemingly hard problems in ideal lattices, which can be viewed as extensions of the assumed hardness of the NTRU function. These new constructions radically enhance our tool set and open a floodgate of applications. We present a survey of these applications. This book is based on my PhD thesis which was an extended version of a paper titled "Candidate Multilinear Maps from Ideal Lattices" co-authored with Craig Gentry and Shai Halevi. This paper was originally published at EUROCRYPT 2013.
The aim of cryptography is to design primitives and protocols that withstand adversarial behavior. Information theoretic cryptography, how-so-ever desirable, is extremely restrictive and most non-trivial cryptographic tasks are known to be information theoretically impossible. In order to realize sophisticated cryptographic primitives, we forgo information theoretic security and assume limitations on what can be efficiently computed. In other words we attempt to build secure systems conditioned on some computational intractability assumption such as factoring, discrete log, decisional Diffie-Hellman, learning with errors, and many more.
In this work, based on the 2013 ACM Doctoral Dissertation Award-winning thesis, we put forth new plausible lattice-based constructions with properties that approximate the sought after multilinear maps. The multilinear analog of the decision Diffie-Hellman problem appears to be hard in our construction, and this allows for their use in cryptography. These constructions open doors to providing solutions to a number of important open problems.
Table of Contents
Survey of Applications
Multilinear Maps and Graded Encoding Systems
Preliminaries I: Lattices
Preliminaries II: Algebraic Number Theory Background
The New Encoding Schemes
Security of Our Constructions
Preliminaries III: Computation in a Number Field
Survey of Lattice Cryptanalysis
One-Round Key Exchange
About the Author(s)Sanjam Garg
, University of California, Berkeley
Sanjam Garg grew up in royal Patiala, a beautiful city in southeastern Punjab, India. Growing up, even though he wasn't good at it, he enjoyed playing cricket with his friends. He was also remarkably curious about everything and this curiosity led him to the Indian Institute of Technology Delhi where he started studying cryptography by mistake. One mistake led to another and ultimately leading to a disaster - he ended up getting a doctoral degree from University of California, Los Angeles. During his graduate study he pondered over the lesser important questions in cryptography and the more important questions about life, making some progress on the former but none whatsoever on the latter. Unfettered, he marches on!