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On solving L P N using B K W and variants

Implementation and analysis

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Abstract

The Learning Parity with Noise problem (L P N) is appealing in cryptography as it is considered to remain hard in the post-quantum world. It is also a good candidate for lightweight devices due to its simplicity. In this paper we provide a comprehensive analysis of the existing L P N solving algorithms, both for the general case and for the sparse secret scenario. In practice, the L P N-based cryptographic constructions use as a reference the security parameters proposed by Levieil and Fouque. But, for these parameters, there remains a gap between the theoretical analysis and the practical complexities of the algorithms we consider. The new theoretical analysis in this paper provides tighter bounds on the complexity of L P N solving algorithms and narrows this gap between theory and practice. We show that for a sparse secret there is another algorithm that outperforms B K W and its variants. Following from our results, we further propose practical parameters for different security levels.

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Notes

  1. Definition 2 of [33] assumes independence of samples but Lemma 2 of [33] shows the reduction without proving independence.

  2. The term (a−1)2b is not included in Theorem 1 from [33]. This factor represents the number of queries lost during the reduction phase and it is the dominant one for all the algorithms except B K W ∗.

  3. The term b2b in the time complexity is missing in [33]. While in general kan is the dominant term, in the special case where a=1 (thus we apply no reduction step) a complexity of \(\mathcal {O}(kan)\) would be wrong since, in this case, we apply the Walsh-Hadamard transform on the whole secret and the term k2k dominates the final complexity.

  4. For the computation of n the authors of [24] use the term \(4 \ln \left (2^{\ell }\right )\) instead of \(8 \ln \left (\frac {2^{\ell }}{\theta }\right )\). If we use our formula, we obtain that we need more than 3⋅2b queries and obtain a complexity of t = 280.08.

  5. This n corresponds to covering code reduction using L F 1. For L F 2 reduction steps we need to have \(n = 3 \cdot 2^{b} + k \geq 8 \ln \left (\frac {2^{\ell }}{\theta }\right )\frac {1}{\delta ^{2^{a+1}} \varepsilon _{\mathsf {set}}^{2}}\).

  6. Given that we receive uniformly distributed vectors from the L P N oracle, from n+2 vectors v we expect to have n linearly independent ones.

  7. http://openmp.org/wp

  8. http://m4ri.sagemath.org/

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Acknowledgments

We would like to thank Thomas Johansson and all the authors of [24] for their help in providing us with their paper and for their useful discussions. We further congratulate them for receiving the Best Paper Award of Asiacrypt 2014.

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  1. EPFL, CH-1015, Lausanne, Switzerland

    Sonia Bogos, Florian Tramèr & Serge Vaudenay

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  1. Sonia Bogos
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Correspondence to Sonia Bogos.

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Supported by a grant of the Swiss National Science Foundation, 200021_143899/1.

Appendices

Appendix A: Hoeffding’s bounds

Theorem 12

[26] Let X 1 ,X 2 ,…,X n be n independent variables. We are given that \(\Pr [X_{i} \in [\alpha _{i},\beta _{i} ]] = 1\) for 1≤i≤n. We define X=X 1 +…+X n and E[X] is the expected value of X. We have that

$$\Pr[X-E[X] \geq \lambda] \leq e^{- \frac{2\lambda^{2}}{{\sum}_{i=1}^{n}(\beta_{i}-\alpha_{i})^{2}}} $$

and

$$\Pr[X-E[X] \leq - \lambda] \leq e^{- \frac{2\lambda^{2}}{{\sum}_{i=1}^{n}(\beta_{i}-\alpha_{i})^{2}}}, $$

for any λ>0.

Appendix B: L F 1 - full recovery of the secret

We provide here an example of the L F 1 algorithm, for the L P N 512,0.125 instance, where we recover the full secret. We provide the values of a, b, n and time complexity to show that indeed the number of queries for the first iteration, dominates the number of queries needed later on. Also, this shows that the time complexity of recovering the first block dominates the total time complexity. For L P N 512,0.125, we obtain the following values (See Table 24).

Table 24 Full secret recovery for the instance L P N 512,0.125
Full size table

The way one can interpret this table is the following: L F 1 recovers first 74 bits by taking a = 7 and requiring 276.59 queries. The total complexity of this step, i.e. the reduction, solving and updating operation, is of 288.43 bit operations. Next, L F 1 solves L P N 438,0.125 and continues this process until it recovers the whole secret.

We can easily see that indeed the number of queries and the time complexity of the first block dominate the other values.

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Bogos, S., Tramèr, F. & Vaudenay, S. On solving L P N using B K W and variants. Cryptogr. Commun. 8, 331–369 (2016). https://doi.org/10.1007/s12095-015-0149-2

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