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Erfan Hosseini avatar Crypto

composition of RLWE distributions

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Erfan Hosseini

Assume we have the polynomial space $R_q$ defined as $R_q = Z_q/(X^n + 1)$. Additionally, we define the error distribution $\chi$ as a discrete centred Gaussian bounded by $B$. Let $s,t \in R_q$ be randomly selected secrets. Let $r_0=as+e_0$ where $a \gets R_q$ is selected uniformly at random and $e_0 \gets \chi$ is sampled from the noise distribution. We know that given $a$, the distribution of $r_0$ is computationally indistinguishable from uniform. Let $r_1=tr_0+e_1$ where $e_1 \gets \chi$ is sampled from the noise distribution. Can we claim that given $a$ and $r_0$, the distribution of $r_1$ is also indistinguishable from uniform even though $r_0$ is not truly random?

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Andy Dienes avatar
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Andy Dienes
By decisional-RWLE, $r_0$ is (computationally) indistinguishable from $u$, so $(r_0, tr_0 + e_1) \cong (u, tu+e_1)$. The hardness of RWLE again implies $tu+e_1$ is indistinguishable from random.
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