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Yes, it appears that it can be solved in practical time in $GF(2^n)$, if the attacker gets $n+\epsilon$ random $a_i$ values, even if he gets a single bit of the $a_i \times k$ values. The chief observation is that the mapping from $a_i$ to bit $j$ of $a_i \times k$ (which I'll refer to as $bit_j(a_i \times k)$) is bitwise linear (for constant $j$, $k$). ...


Thanks @poncho for providing a correct answer. I investigated it deeply, viewing it as a linear algebra problem. Here's what I obtained: in $GF(2^n)$, the series of equations $r_i=a_i\times k$ can be written as $R = K \cdot A$ where: $A$ is a known $n \times n$ matrix, where each column is a bit representation of $n$, linearly independent, $a_i$ $R$ is a ...

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