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muhammad haris avatar Crypto

How to understand noise growth in BFV?

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muhammad haris

I am trying to understand the noise growth due to multiplication in BFV encryption.

As explained in section 4 and equation 3 of this paper: https://eprint.iacr.org/2012/144.pdf.

I couldn't follow what is $r_a$ and $r_r$ is in their equations.

Also how they are bounding the values of all rounding errors?

Also is there are simpler explanation?

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lwe
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Stanley

$\boldsymbol{r}_a$ is the approximation error for Equation 3, since we are creating an approximation where we round the ciphertext values.

$\boldsymbol{r}_r$ are the non-integer values in the equality generated by scaling Equation 2 by $t/q$, i.e.: $$\textbf{r}_r=\frac{t}{q}\cdot[\textbf{v}_1\cdot\textbf{v}_2]_\Delta-\frac{r_t(q)}{q}\cdot (\Delta\cdot \textbf{m}_1\cdot\textbf{m}_2+(\textbf{m}_1\cdot\textbf{v}_2+\textbf{m}_2\cdot\textbf{v}_1)+\textbf{r}_v$$ Note that these will be the only values affected by the rounding. The error from this rounding is bounded by $\lVert r_r\rVert$ (since $[\boldsymbol{r}_r]_q=\sum_{i=0}{[r_{r_i}]_q\cdot x^i})$

As for a simpler explanation I've yet to find one, if you only need the implementation and not the correctness perhaps consider this site.

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