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Seen here, at the bottom of page 5, $\operatorname{Flatten}(\vec{a})$ is defined as:

$\operatorname{Flatten}(\vec{a})=\operatorname{BitDecomp}(\operatorname{BitDecomp}^{-1}(\vec{a}))$

For an n-dimensional vector $\vec{a} = (a_{1,0},\ldots,a_{1,l}, \ldots, a_{k,0},\ldots,a_{k,l-1})$. Where $a_{i,j}$ is the $j$-th bit in $a_i$'s bit representation.

But if we're computing the function through its own inverse, would this not just result in the original vector a?

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When all the entries of $a$ are binaries, then yes, $Flatten(a) = a$, because $BitDecomp^{-1}(a)$ will give a $k$-dimensional vector whose entries are decomposed back to $\ell$ consecutive entries of $a$ (by $BitDecomp$).

However, when $a$ has non-binaries entries, this is not the case.

For instance, for $\ell = 3$ and $k = 2$, we would have $N = 6$. Then, consider $a = (0, 3, 0, 0, 0, 1)$.

Then, $BitDecomp^{-1}(a) = (3\cdot 2, 1\cdot 4) = (6, 4)$, and $BitDecomp(6, 4) = (0, 1, 1, 0, 0, 1)$.

Notice that for the last block of $a$ (last $\ell$ entries), $Flatten$ worked as the identity, because this block was binary.

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  • $\begingroup$ Thank you for your response. Would this still hold, when the entries are non-binary, if $\sum a_{1,j}> 7$, therefore couldn't be represented in $l$-bits? For example, if $a=(5,3,1,0,0,1)$ and the values of $l,k$ remain, would $Flatten(a)$ still be valid? $\endgroup$
    – Stilton
    Apr 3, 2019 at 14:56
  • $\begingroup$ @Stilton yes, it would work modulo $q$. Actually, you perform the operations modulo $q$ and $\ell = \lceil \log q \rceil$, thus, all the produced values can be represented with $\ell$ bits. $\endgroup$ Feb 7, 2020 at 7:34
  • $\begingroup$ For clarity, you might note that bits ordered least significant to most significant. $\endgroup$
    – kelalaka
    Feb 7, 2020 at 7:57

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