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I recently studied how to solve the LWE (learning with errors) using algebraic relations proposed by Arora and Ge (paper : New Algorithms for Learning in Presence of Errors). As I understood, for LWE instances of the form $\vec{a}_i,b_i = <\vec{a}_i,\vec{s}>+e_i~mod~q$ for some integer $q$, adversaries generate algebraic polynomials which has a root $e_i = b_i - <\vec{a}_i,\vec{s}>~mod~q$. If adversaries obtain sufficiently many polynomials, then they can recover the secret vector $\vec{s}$ using linearization technique.

Thus, I think that the Arora-Ge algorithm is also possible to solve LPN (learning parity with noise) problem since the instances are of the form $\vec{a}_i,b_i = <\vec{a}_i,\vec{s}>+e_i~mod~2$. However, I cannot find any references about the result, so I guess that I have some mistakes.

Why can't I use Arora-Ge algorithm to solve the LPN?

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It is natural to wonder if Arora-Ge can break LPN, but as you suspect, it does not work. The essential problem is that, because the modulus is $q=2$, the method does not find the unique solution $s$, nor does it even narrow down the set of possible solutions.

The reason is that the first step of the algorithm converts each LPN sample into a quadratic (in $s$) equation that encodes the condition $\langle a_i, s \rangle = b_i \text{ or } b_i +1 \pmod{2}$. It then linearizes these equations and solves for $s$. However, observe that any $s’$ is a solution to these equations, because the right-hand side is always “0 or 1.” So, the system set up by the algorithm does not carry any information about $s$, from the very start.

Arora-Ge works for LWE because/when the modulus $q$ is larger than the number of possible error values. Then, the condition “$\langle a_i, s \rangle = b_i \text{ or } b_i+1 \text{ or } b_i-1 \ldots$” captures a nontrivial constraint on the secret $s$, and enough such equations can uniquely specify it.

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  • $\begingroup$ Thanks a lot. I fully understood what the problem is. $\endgroup$ – filter hash Nov 16 '20 at 14:58
  • $\begingroup$ oh, I have one more question. When we want to solve LWE problem using Arora-Ge algorithm, we easily generate new LWE samples by adding for small random error. (We can immediately generate new LWE samples from given LWE samplers.) Why we cannot use Arora-Ge algorithm in this case? Since we add small error, the modulus $q$ is still larger than the number of possible errors as you said, so we easily obtain $m(>n^2)$ LWE samples with a dimension $n$ that is a constraint of Arora-Ge algorithm. $\endgroup$ – filter hash Nov 16 '20 at 15:06
  • $\begingroup$ When you create new samples by combining existing ones, the size of the resulting error is larger. Arora-Ge needs roughly $n^d$ samples, where $d$ is the number of possible values of the error. So, it’s not clear that there’s any benefit from using this technique, though I haven’t worked out the tradeoff carefully. $\endgroup$ – Chris Peikert Nov 16 '20 at 22:26
  • $\begingroup$ Sorry to late reply. Thank you for your kindness. $\endgroup$ – filter hash Nov 19 '20 at 0:07

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