Lattice-based cryptography is the use of conjectured hard
problems on point lattices in $\R^{n}$ as the foundation for secure
cryptographic constructions. 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 work 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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