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The secret key of BFV, BGV schemes is generated as a random ternary polynomial from R2 ( R2 is the key distribution used to sample polynomials with integer coefficients in {−1,0,1}) Is there any specific reason for it to be a ternary polynomial? can we have it as polynomial from Rq i.e integer coefficients from {0, 1, 2, 3, ..q-1} and still have all the guarantees of being post quantum secure?

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Actually, if you sample the secret key (sk) from larger sets, you increase the security of the underlying problem (RLWE).

You can test it by yourself on the Lattice estimator, by keeping N and q fixed, then trying several distributions for sk.

So, it would be better to have an FHE scheme that uses sk sampled uniformly from $\mathcal{R}_q$. However, for schemes like BGV, CKKS, FV, in some operations, we have to divide the ciphertexts by some integer and round, and these operations increase the noise in a way that depends on the norm of sk.

For example, in BGV, there is the modulus switching. You can see that the noise after this operation depends on $||sk||$.

So, to reduce the noise growth due to these operations, we just set $||sk|| = 1$, which we can do by having ternary sk.

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  • $\begingroup$ Thank you. just realized in original FV paper, they were not using ternary sk but just {0.. q-1}. Seems this development came later. $\endgroup$
    – Kaneez sk
    Dec 22, 2022 at 18:54
  • $\begingroup$ @Kaneezsk Where did you see that FV used uniform secret keys? I will be very surprised if this is correct... $\endgroup$ Dec 23, 2022 at 21:41
  • $\begingroup$ not exactly {0,..q-1} but eprint.iacr.org/2012/144.pdf 2.2 and 3.2 indicate that it is uniform in [-B, B] $\endgroup$
    – Kaneez sk
    Dec 24, 2022 at 22:02
  • $\begingroup$ @Kaneezsk yes, that is fine. They are just saying that they assume the secret key to have small norm, bounded by B (for the reasons I explained here, since the noise depends on the norm of the key). But they even say that as an optimization, one can set $|||sk|| = 1$ $\endgroup$ Dec 26, 2022 at 7:35

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