IdealSVP
Cyclotomic IdealSVP is a lattice problem of foundational importance in the security analysis of latticebased cryptography. There are "very strong hardness guarantees" stating, roughly, that RingLWE is secure if cyclotomic IdealSVP with polynomial approximation factor is secure. This was the explicit source of highprofile proposals of RingLWE/ModuleLWE cryptosystems for standardization and deployment.
Saying that one lattice problem is secure if another lattice problem is secure begs the question of whether the problems are in fact secure. These proposals assume that the best lattice attacks take exponential time, specifically time 2^((0.292...+o(1)) beta) where beta is the required "BKZ block size". (This is the time without quantum computers; otherwise 0.292 is replaced by something smaller.)
If RingLWE is in fact breakable much more efficiently than this, then it will not be surprising for the public development of a break to follow a path that includes a break of IdealSVP. After all, the "very strong hardness guarantees" say that a RingLWE attack implies an IdealSVP attack. Similar comments apply to ModuleLWE.
Sunit attacks
Traditional attacks against lattice problems, including cyclotomic IdealSVP, rely solely on the additive structure of lattices to search for short lattice vectors. Sunit attacks exploit the multiplicative structure of the lattices used in these specific lattice problems. This multiplicative structure is reflected in an auxiliary lattice, a standard numbertheoretic lattice called the "Sunit lattice".
Unit attacks are an early special case of Sunit attacks. A quantum polynomialtime unit attack broke the "h^+=1" cyclotomic case of Gentry's original STOC 2009 system for fully homomorphic encryption using ideal lattices. Various "barriers" were claimed for this line of attacks and then broken by subsequent developments, illustrating the importance of looking more closely at this area.
ShortSunit attacks
The attack against Gentry's system starts by very efficiently writing down a short basis for the unit lattice. The introduction of Sunit attacks in 2016 included asking whether one could find "a short enough basis for the Sunit lattice". A massive new research project on Sunit attacks began in early 2020, centered around the idea that one can quickly find short Sunits beyond short units.
This project led to a variety of advances in Sunit attacks against cyclotomic IdealSVP. Highlights were presented in a talk "Sunit attacks" (60 minutes, 2021.08.20), and extensive further resources are now available regarding five different aspects of the talk:

The special analytic features of Sunit lattices. First paper released by the project: "Nonrandomness of Sunit lattices" (58 pages, 2021.10.23). This paper shows that standard heuristics in the literature on latticebased cryptography, when applied to Sunit lattices as in 2019 PelletMary–Hanrot–Stehlé (presented more concisely in 2021 Ducas–PelletMary), are highly inaccurate. Trusting the standard heuristics led the 2019 paper to see only a small corner of the power of Sunit attacks.

Experimental evidence regarding the performance of Sunit attacks. First software package released by the project:
filteredsunit
(6883 lines, 2021.12.17). This software collects data regarding the tradeoffs between database size and output length in filteredSunit attacks for a range of cyclotomic fields. FilteredSunit attacks are Sunit attacks that obtain Sunits by filtering small ring elements, as in traditional classgroup computations and the numberfield sieve for integer factorization. 
Conjecturally subexponential scalability for Sunit attacks against polynomialapproximationfactor cyclotomic IdealSVP. The
filteredsunit
documentation reviews the rationale for the standard subexponentialtime conjectures for previous Sunit algorithms, quantifies this rationale for filteredSunit attacks against cyclotomic fields, and, from this perspective, evaluates the quantitative phase transitions visible in the data collected byfilteredsunit
. 
Faster norm computations for smoothdegree cyclotomic fields. Second paper released by the project: "Fast norm computation in smoothdegree Abelian number fields" (59 pages, 2022.07.31); accompanying software packages:
abelianfields
(3725 lines, 2022.07.11) andcyclo2power
(487 lines, 2022.05.30); also overview material. Computing norms of many small ring elements is one of the central bottlenecks in traditional classgroup computations and in filteredSunit attacks; this paper shows that algorithms for smoothdegree cyclotomic fields are much faster than the best techniques known for general number fields. 
Faster punit constructions for the case of cyclotomics. The talk's detailed construction of the full punit group via Jacobi sums and square roots (punits are a useful intermediate step between units and Sunits) was, after the talk, also described in two papers from other people: "A short basis of the Stickelberger ideal of a cyclotomic field" (20 pages, 2021.09.27) and "LogSunit lattices using explicit Stickelberger generators to solve Approx IdealSVP" (54 pages, 2021.10.13, updated 2021.11.29). Beware that those papers take limited smoothness bounds (following the standard lattice heuristics) and omit filtering; the approximation factors reached in those papers are a step backwards from the concrete examples given in the talk and from the conjecture of polynomial approximation factors in subexponential time.
Structurally, Sunit attacks are applicable to general IdealSVP, not just cyclotomic IdealSVP. However, many speedups in Sunit attacks rely on automorphisms, subfields, and specific structures of cyclotomic fields.
Version: This is version 2022.07.31 of the "Intro" web page.