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Given the ring $R_2=\mathbb{Z}_{2}[x]/(x^8+1)$, initiate a public key encryption based on RLWE $(A,b=As+e)$ where all the parameters are selected as polynomials over the given ring. Under the assumption of brute force attack only and if the vector $s$ is 248-byte long, I have the following questions:

  • is this secure initiation?
  • What should be the dimension of the matrix $A$?
  • Can Gaussian elimination finds $s$ given $A$ and $b$?.
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1 Answer 1

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It is not clear how to create correct encryption from this, let alone secure encryption.

I'll show the easiest lattice-based scheme to make correct, namely secret key encryption. Here, we have

  1. Key Gen samples $s(x)$ somehow, generally with independent coordinates. The only reasonable choice here seems to be uniform.
  2. $\mathsf{Enc}_{s}(m) = [a(x), a(x)s(x) + e(x) + \mathsf{encode}(m(x))]$
  3. $\mathsf{Dec}_s([a(x), b(x)]) = \mathsf{decode}(b(x) - a(x)s(x))$

Here, $\mathsf{encode}$ and $\mathsf{decode}$ must correct errors, in the sense that

$$\mathsf{decode}(\mathsf{encode}(m(x)) + e(x)) = m(x)$$

Typically this is done by having $\mathsf{encode}(m(x)) = \Delta m(x)$ for some scaling factor $\Delta$, and then $\mathsf{decode}(c(x)) = \lfloor c(x)/\Delta\rceil$. The issue with your scheme is that working over $\mathbb{Z}_2$ leaves no room for any scaling factor $\Delta$, so it's really not clear what you can do to make the scheme correct, let alone secure.

Note that for this scheme, if you assume the coordinates of $e(x)$ are from a Gaussian of standard deviation $\sigma$, then one expects something like $\Pr[\lVert e(x)\rVert_\infty > k\sigma]\leq n\exp(-k^2)$. Under the assumption that $k\approx 10$, $\sigma \approx 3$ (which are common), this means that we really need to be working over $\mathbb{Z}_q$ for $q\geq 30$ just to get a basic notion of correctness for private-key encryption. Public-key encryption would require larger $q$ even though.

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