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FTX Raises $420M in Series B-1 Raise

Reducing a lattice basis with too many basis vectors

Suppose I have a basis $B$ of an $n$-dimensional lattice $Lsubseteqmathbb{Z}^n$ and $B$ has $n$ vectors. Now I take another $vin mathbb{Z}^nsetminus L$ and I define a new lattice $L’=L+mathbb{Z}v$. The set of vectors $B’:=Bcup{v}$ generates $L’$, but since $L’$ is $n$-dimensional, it’s rank is at most $n$, so $B’$ is too big. So there::Listen

Suppose I have a basis $B$ of an $n$-dimensional lattice $Lsubseteqmathbb{Z}^n$ and $B$ has $n$ vectors. Now I take another $vin mathbb{Z}^nsetminus L$ and I define a new lattice $L’=L+mathbb{Z}v$. The set of vectors $B’:=Bcup{v}$ generates $L’$, but since $L’$ is $n$-dimensional, it’s rank is at most $n$, so $B’$ is too big. So there must be some other basis that generates $L’$. How do we produce that from $B’$?

I vaguely recall reading that LLL could do this, but I have no idea how. Can anyone point to reference or give a quick argument/proof?

FTX Raises $420M in Series B-1 Raise

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