Recovering Short Generators of Principal Ideals in Cyclotomic Rings

Explore a cryptographic presentation on recovering short generators of principal ideals in cyclotomic rings. Delve into the intricacies of principal ideals in cryptography, short generator recovery, and associated costs. Examine the problem, unit group, and log-unit lattice concepts. Study the logarithmic embedding and reduction modulo 1 using Z[√2] as an example. Learn about round-off decoding and the proof plan for recovering short generators, including folklore strategies and recent developments. Investigate geometric statements from analytic number theory, consider worst-case scenarios, and ponder open questions in this field.

Intro
Principal ideals in cryptography
Short generator recovery
Cost of those two steps
The Problem
The Unit Group and the log-unit lattice
Example: Logarithmic Embedding Log Z[V2]
Reduction modulo 1 - Log Z[V2]
Round-Off Decoding
Recovering Short Generator: Proof Plan Folklore strategy Bernstein 2014 Campbell et al. 2014 to recover a short generator
Geometric statement from Analytic Number Theory
What about the worst case?
Open questions