Lattice-based cryptography is the use of conjectured hard problems
on point lattices in Rn as the foundation for secure cryptographic
systems. Attractive features of lattice cryptography include
apparent resistance to quantum attacks (in contrast with most
number-theoretic cryptography), high asymptotic efficiency and
parallelism, security under worst-case intractability assumptions,
and solutions to long-standing open problems in cryptography. This
monograph surveys most of the major developments in lattice
cryptography over the past ten years. The main focus is on the
foundational short integer solution (SIS) and learning with errors
(LWE) problems (and their more efficient ring-based variants),
their provable hardness assuming the worst-case intractability of
standard lattice problems, and their many cryptographic
applications.
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