# LLLplus.jl

LLLplus provides lattice tools such as Lenstra-Lenstra-Lovász (LLL) lattice reduction which are of practical and theoretical use in cryptography, digital communication, integer programming, and more. This package is experimental and not a robust tool; use at your own risk :-)

LLLplus has functions for LLL,
Seysen, and
Hermite-Korkine-Zolotarev
lattice reduction
techniques. Brun
integer relations is included in the form of lattice
reduction. Solvers for the shortest
vector
and the closest
vector
problems are also included; for more see the help text for the `lll`

,
`seysen`

, `hkz`

, `brun`

, `svp`

, and `cvp`

functions. Several toy (demo)
functions are also included; see the `subsetsum`

,
`integerfeasibility`

, `rationalapprox`

, and `spigotBBP`

functions.

**Examples** (click for details)

Each function contains documentation and examples available via Julia's
built-in documentation system (try `?lll`

or `@doc(lll)`

). Documentation
for all functions is available. A tutorial notebook is
found in the `docs`

directory or on
nbviewer.

Here are a few examples of using the functions in the package on random lattices.

```
Pkg.add("LLLplus")
using LLLplus
# do lattice reduction on a matrix with randn entries
N = 40;
H = randn(N,N);
B,T = brun(H);
B,T = lll(H);
B,T = seysen(H);
B,T = hkz(H);
# check out the CVP solver
Q,R=qr(H);
u=Int.(rand(0:1e10,N));
y=H*u+rand(N)/100;
uhat=cvp(Q'*y,R);
sum(abs.(u-uhat))
```

**Execution Time results** (click for details)

In the first test we compare several LLL functions: the `lll`

function from LLLplus, the
`l2avx`

function in the `src\l2.jl`

file in LLLplus, the
`lll_with_transform`

function from
Nemo.jl (which uses FLINT), and
the `lll_reduction`

function from
fplll. Nemo is written by number
theorists, while fplll is written
by lattice cryptanalysis academics; they are good benchmarks against which to compare.
We first show how the execution time varies as the basis (matrix) size
varies over [4 8 16 32 64]. For each matrix size, 20 random bases are
generated using fplll's `gen_qary`

function with depth of 25 bits,
with the average execution time shown; the `eltype`

is `Int64`

except
for NEMO, which can only use GMP (its own `BigInt`

); in all cases the
`δ=.99`

. The vertical axis shows execution time on a logarithmic
scale; the x-axis is also logarithmic. The generally linear nature of
the LLL curves supports the polynomial-time nature of the
algorithm. The `lll`

function is slower, while `l2avx`

is similar to
fplll. Though not shown, using bases from `gen_qary`

with bit depth of
45 gives fplll a larger advantage. Though the LLLplus functions are
not always the fastest, they are in the same ballpark as the C and
C++ tools; if this package gets more users, we'll spend more time on
speed :-) This figure was generated using code in `test/timeLLLs.jl`

.

One additional question that could arise when looking at the plot above is what
the quality of the basis is. In the next plot we show execution time
vs the norm of the first vector in the reduced basis, this first
vector is typically the smallest; its norm is an rough indication of
the quality of the reduced basis. We show results averaged over 20
random bases from `gen_qary`

with depth `25`

bits, this time with the
dimension fixed at `32`

. The curve is created by varying the `δ`

parameter from `.29`

to `.99`

in steps of `.2`

; the larger times and
smaller norms correspond to the largest `δ`

values. Though the `l2avx`

function is competitive with fplll in this case, in most cases
the fplll code is faster.

Finally, we show execution time for several built-in
datatypes (Int32, Int64, Int128, Float32, Float64, BitInt, and
BigFloat) as well as type from external packages (Float128 from
Quadmath.jl and Double64
from DoubleFloat.jl)
which are used to
generate 60 16x16 matrices, over which execution time for the
lattice reduction techniques is averaged. The vertical axis is a
logarithmic representation of execution time as in the previous
figure. This figure was generated using code in `test/perftest.jl`

.

**Notes** (click for details)

The 2020 Simons Institute lattice workshop, a survey paper by Wuebben, and the monograph by Bremner were helpful in writing the tools in LLLplus and are a good resource for further study. If you are trying to break one of the Lattice Challenge records or are looking for robust, well-proven lattice tools, look at fplll. Also, for many number-theoretic problems the Nemo.jl package is appropriate; it uses the FLINT C library to do LLL reduction on Nemo-specific data types. Finally, no number theorists have worked on LLLplus; please treat the package as experimental.