Robustness of Learning Parities with Dependent Noise

Kernkonzepte

The learning parities with noise (LPN) assumption is robust to weak dependencies in the noise distribution of small batches of samples.

Zusammenfassung

This expository note shows that the learning parities with noise (LPN) assumption is robust to weak dependencies in the noise distribution of small batches of samples. This provides a partial converse to the linearization technique of [AG11].

The key insights are:

1. Adding correlated noise to the LPN samples is necessary, as simply combining independent LPN samples does not work.
2. This can be achieved by adding an affine function of the secret key to the noise term, which can be simulated by appropriately adjusting the LPN samples.
3. A perturbation argument is used to show that the system of linear constraints on the distribution of the affine noise is well-conditioned, allowing a small perturbation to the noise bias to be tolerated.

The main result, Theorem 1.4, shows that for any constant batch size k and any δ-Santha-Vazirani source p over the batch noise, the standard LPN problem with noise level 1/2 - O(kδ) is polynomial-time reducible to learning parities with the batch noise distribution p. This provides a robustness guarantee for the LPN assumption in the face of small dependencies in the noise.

Statistiken

None.

Zitate

None.

Wichtige Erkenntnisse aus

On Learning Parities with Dependent Noise

by Noah Golowic... um arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11325.pdf

Tiefere Fragen

Can the dependence on the batch size k in the noise level be removed, so that batch LPN with a δ-Santha-Vazirani source is as hard as standard LPN with noise level 1/2 - δ

In the context of batch LPN with a δ-Santha-Vazirani source, the dependence on the batch size k in the noise level can potentially be removed to make it as hard as standard LPN with noise level 1/2 - δ. This adjustment would be a significant advancement in the field, as it would allow for a more seamless transition between different batch sizes without compromising the hardness of the problem. By refining the reduction process and potentially exploring alternative approaches, it may be feasible to achieve this goal. However, it would require a thorough analysis and potentially novel techniques to address this challenge effectively.

Is there a natural and general model where the joint noise distribution p is succinctly described, allowing larger batch sizes to be handled efficiently

Finding a natural and general model where the joint noise distribution p can be succinctly described is a crucial aspect for efficiently handling larger batch sizes in batch LPN. Such a model would streamline the process of working with diverse noise distributions, enabling more scalable and practical implementations. One potential direction could involve exploring structured noise distributions that can be efficiently represented and manipulated algorithmically. By identifying key characteristics and properties of these distributions, it may be possible to develop a framework that accommodates larger batch sizes while maintaining computational hardness.

Can the robustness of LPN be extended to other forms of dependent noise, beyond the Santha-Vazirani model

Extending the robustness of LPN to other forms of dependent noise beyond the Santha-Vazirani model presents an intriguing research opportunity. By investigating different types of noise dependencies and their impact on the hardness of LPN, researchers can broaden the understanding of the problem's resilience to various noise structures. This exploration could lead to valuable insights into the behavior of LPN under diverse noise conditions, potentially uncovering new cryptographic implications and applications. By delving into the complexities of dependent noise models, researchers can enhance the robustness analysis of LPN and its relevance in practical cryptographic scenarios.