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feat: add lll_gram[_with_transform] for semidefinite matrices #2011

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Feb 2, 2025
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feat: add lll_gram[_with_transform] for semidefinite matrices #2011

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121 changes: 108 additions & 13 deletions src/flint/fmpz_mat.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -957,10 +957,7 @@
"""
function lll_with_transform(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51))
z = deepcopy(x)
u = similar(x, nrows(x), nrows(x))
for i in 1:nrows(u)
u[i, i] = 1
end
u = identity_matrix(ZZ, nrows(x))
@ccall libflint.fmpz_lll(z::Ref{ZZMatrix}, u::Ref{ZZMatrix}, ctx::Ref{LLLContext})::Nothing
return z, u
end
Expand Down Expand Up @@ -1003,6 +1000,30 @@
return x
end

# for lll_gram, we use the following helpers to deal with the degenerate case.

function _is_definitely_full_rank(x::ZZMatrix)
if Int == Int64
p = 1099511627791 % Int64
else
p = 1048583 % Int32

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end
F = Nemo.Native.GF(p; cached = false)
xmodp = F.(x)
return rank(xmodp) == nrows(x)
end

function _complete_to_basis(x::ZZMatrix)
@assert nrows(x) <= ncols(x)
# compute column HNF and take the inverse
h, t = hnf_with_transform(transpose(x))
for i in 1:nrows(x)
@assert is_one(h[i, i])
end
transpose!(t)
return inv!(t)
end

@doc raw"""
lll_gram_with_transform(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))

Expand All @@ -1015,43 +1036,117 @@
supplying an optional context object.
"""
function lll_gram_with_transform(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
z = deepcopy(x)
u = similar(x, nrows(x), nrows(x))
for i in 1:nrows(u)
u[i, i] = 1
@req is_square(x) && is_symmetric(x) "The matrix must be a symmetric square matrix"
# try to recognize the definite case
if nrows(x) == 0
return x, x
end
if _is_definitely_full_rank(x)
u = identity_matrix(ZZ, nrows(x))
if x[1, 1] < 0
y = neg(x)
_lll_gram_with_transform!(y, u, ctx)
return neg!(y), u

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else
y = deepcopy(x)
_lll_gram_with_transform!(y, u, ctx)
return y, u
end
end
@ccall libflint.fmpz_lll(z::Ref{ZZMatrix}, u::Ref{ZZMatrix}, ctx::Ref{LLLContext})::Nothing
return z, u
# remove the radical using a unimodular transformation
L = kernel(x; side = :left)
n = nrows(L)
r = nrows(x) - n
C = _complete_to_basis(L)
# move kernel to bottom, so that
# C * x * transpose(C) = [ * | 0 ]
# [-------]
# [ 0 | 0 ]
uu = deepcopy(reverse_rows!(C))
mul!(C, C, x * transpose(C))
u = identity_matrix(ZZ, nrows(x))
if C[1, 1] < 0
_lll_gram_with_transform!(neg!(@view(C[1:r, 1:r])), @view(u[1:r, 1:r]), ctx)
neg!(C)
else
_lll_gram_with_transform!(@view(C[1:r, 1:r]), @view(u[1:r, 1:r]), ctx)
end
return C, u * uu
end

function _lll_gram_with_transform!(x::ZZMatrix, u::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
# modifies x and u
@ccall libflint.fmpz_lll(x::Ref{ZZMatrix}, u::Ref{ZZMatrix}, ctx::Ref{LLLContext})::Nothing
return x, u
end

function _lll_gram_with_transform!(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
u = identity_matrix(ZZ, nrows(x))
@ccall libflint.fmpz_lll(x::Ref{ZZMatrix}, u::Ref{ZZMatrix}, ctx::Ref{LLLContext})::Nothing
return x, u

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end

@doc raw"""
lll_gram(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))

Return the Gram matrix $L$ of an LLL-reduced basis of the lattice given by the
Gram matrix $x$.
The matrix $x$ must be symmetric and non-singular.
The matrix $x$ must be symmetric and semidefinite. If an indefinite matrix is
provided, the output is undefined.

By default, the LLL is performed with reduction parameters $\delta = 0.99$ and
$\eta = 0.51$. These defaults can be overridden by specifying an optional context
object.
"""
function lll_gram(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
@req is_square(x) && is_symmetric(x) "The matrix must be a symmetric square matrix"
z = deepcopy(x)
return lll_gram!(z)
end

@doc raw"""
lll_gram!(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))

Compute the Gram matrix of an LLL-reduced basis of the lattice given by the
Gram matrix $x$ inplace.
The matrix $x$ must be symmetric and non-singular.
The matrix $x$ must be symmetric and semidefinite.

By default, the LLL is performed with reduction parameters $\delta = 0.99$ and
$\eta = 0.51$. These defaults can be overridden by specifying an optional context
object.
"""
function lll_gram!(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
if nrows(x) == 0
return x
end
# try to recognize the definite case
if _is_definitely_full_rank(x)
if x[1, 1] < 0
return neg!(_lll_gram!(!neg(x), ctx))

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else
return _lll_gram!(x, ctx)
end
end
# remove the radical using a unimodular transformation
L = kernel(x; side = :left)
n = nrows(L)
r = nrows(x) - n
C = _complete_to_basis(L)
# move kernel to bottom, so that
# C * x * transpose(C) = [ * | 0 ]
# [-------]
# [ 0 | 0 ]
reverse_rows!(C)
mul!(x, x, transpose(C))
C = mul!(x, C, x)
if C[1, 1] < 0
neg!(_lll_gram!(neg!(@view(x[1:r, 1:r])), ctx))
else
_lll_gram!(@view(x[1:r, 1:r]), ctx)
end
return x
end

function _lll_gram!(x::ZZMatrix, ctx::LLLContext = LLLContext(0.99, 0.51, :gram))
@ccall libflint.fmpz_lll(x::Ref{ZZMatrix}, C_NULL::Ptr{Nothing}, ctx::Ref{LLLContext})::Nothing
return x
end
Expand Down
31 changes: 31 additions & 0 deletions test/flint/fmpz_mat-test.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -612,6 +612,37 @@ end
B = gram(A)
lll_gram!(B)
@test B == lll_gram(gram(A))

# Semidefinite cases
A = ZZ[17070 17380 -4930 5840 28160;
17380 17975 -5135 6015 28835;
-4930 -5135 1485 -1715 -8155;
5840 6015 -1715 2015 9675;
28160 28835 -8155 9675 46705]
G = lll_gram(A)
GG = ZZ[5 0 0 0 0;
0 10 0 0 0;
0 0 10 0 0;
0 0 0 0 0;
0 0 0 0 0]
@test G == GG
@test lll_gram(-A) == -GG

G, T = lll_gram_with_transform(A)
@test T * A * transpose(T) == G
G, T = lll_gram_with_transform(-A)
@test T * (-A) * transpose(T) == G

A = zero_matrix(ZZ, 0, 0)
G, T = lll_gram_with_transform(A)
@test T * A * transpose(T) == G
G = lll_gram(A)
@test G == A

@test_throws ArgumentError lll_gram(ZZ[1 0])
@test_throws ArgumentError lll_gram_with_transform(ZZ[1 0])
@test_throws ArgumentError lll_gram(ZZ[1 0; 1 1])
@test_throws ArgumentError lll_gram_with_transform(ZZ[1 0; 1 1])
end

@testset "ZZMatrix.nullspace" begin
Expand Down
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