Learning Parity with Noise (LPN) is used as the underlying hardness problem in many cryptographic protocols. It is equivalent to decoding random linear codes, and thus offers strong security guarantees. The authors of this paper propose a memory-efficient algorithm to the LPN problem. They also propose the first quantum algorithm for LPN.

Let's first recall the definition of the LPN problem. Let a secret $s \in \mathbb{F}_2^k$ and samples $(\mathbb{a}_i, b_i)$, where $b_i$ equals $\langle \mathbb{a}_i,s \rangle + e_i$ for some $e \in \{0,1\}$ with $Pr(e_i=1)= \tau < 1/2$. From the samples $(\mathbb{a}_i,b_i)$, recover $s$. We see that the two LPN parameters are $k$ and $\tau$. Notice that this can be seen as a sub-case of the Ring Learning with Errors problems; in fact, LWR originated as an extension of LPN.

If $\tau$ is zero, we can draw $k$ independent samples and solve for $s$ by Gaussian elimination. This can also be extended to an algorithm for $\tau < 1/2$, by computing a guess $s'$ and testing whether $s'=s$. This works well for small $\tau$, for example $\tau = 1/\sqrt{k}$, used in some public key encryption schemes. Call this approach Gauss.

For larger constant $\tau$, the best current algorithm is BKW. However, although BKW has the best running time, it cannot be implemented for even medium size LPN parameters because of its memory consumption. Further, BKW has a bad running time dependency on $\tau$. Both algorithms also require many LPN oracle calls.

For larger constant $\tau$, the best current algorithm is BKW. However, although BKW has the best running time, it cannot be implemented for even medium size LPN parameters because of its memory consumption. Further, BKW has a bad running time dependency on $\tau$. Both algorithms also require many LPN oracle calls.

The authors take these two as a starting point. They describe a Pooled Gauss algorithm, which only requires a polynomial number of samples. From those, they look for error-free samples, similar to Gauss. The resulting algorithm has the same space and running time complexity, but requires significantly less oracle calls. It has the additional advantage of giving rise to a quantum version, where Grover search can be applied to save a square root faction in the running time.

They then describe a second hybrid algorithm, where a dimension reduction step is added. Thus Well-Pooled Gauss proceeds in two steps. First, reduce the dimension $k$ to $k'$ (for example, using methods such as BKW). Then, decode the smaller instance via Gaussian elimination. The latter step is improved upon by using the MMT algorithm.

For full results, see the original paper. Their conclusion is that Decoding remains the best strategy for small $\tau$. It also has quantum optimisations and is memory-efficient. The Hybrid approach has in fact no advantage over this for small values of $\tau$. For larger values however, they manage to solve for the first time an LPN instance of what they call medium parameters - $k = 243$, $\tau = 1/8$ - in 15 days.

For full results, see the original paper. Their conclusion is that Decoding remains the best strategy for small $\tau$. It also has quantum optimisations and is memory-efficient. The Hybrid approach has in fact no advantage over this for small values of $\tau$. For larger values however, they manage to solve for the first time an LPN instance of what they call medium parameters - $k = 243$, $\tau = 1/8$ - in 15 days.

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