The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors

Mårtensson, Erik

doi:10.1109/isit.2019.8849218

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…The idea of combining coded-BKW or LMS with techniques from lattice sieving [19] leads to the next improvement [20]. This combined approach was slightly improved in [9,21]. The distinguishing part of the BKW algorithm for solving LWE was improved by using the fast Fourier transform (FFT) in [22].…”

Section: Related Workmentioning

confidence: 99%

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Improvements on Making BKW Practical for Solving LWE

et al. 2021

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The learning with errors (LWE) problem is one of the main mathematical foundations of post-quantum cryptography. One of the main groups of algorithms for solving LWE is the Blum–Kalai–Wasserman (BKW) algorithm. This paper presents new improvements of BKW-style algorithms for solving LWE instances. We target minimum concrete complexity, and we introduce a new reduction step where we partially reduce the last position in an iteration and finish the reduction in the next iteration, allowing non-integer step sizes. We also introduce a new procedure in the secret recovery by mapping the problem to binary problems and applying the fast Walsh Hadamard transform. The complexity of the resulting algorithm compares favorably with all other previous approaches, including lattice sieving. We additionally show the steps of implementing the approach for large LWE problem instances. We provide two implementations of the algorithm, one RAM-based approach that is optimized for speed, and one file-based approach which overcomes RAM limitations by using file-based storage.

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Section: Related Workmentioning

confidence: 99%

“…Instead of testing all possible pairs of samples within the categories, this procedure was sped-up using lattice sieving techniques of [19]. This approach was slightly improved in [9,21].…”

Section: Coded-bkw With Sievingmentioning

confidence: 99%

Improvements on Making BKW Practical for Solving LWE

et al. 2021

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“…In [22] coded-BKW was introduced. Coded-BKW with sieving was introduced in [24] and improved in [9], [28].…”

Section: ) Improved Reductionmentioning

confidence: 99%

On the Sample Complexity of solving LWE using BKW-Style Algorithms

Guo¹,

Mårtensson²,

Wagner³

2021

Preprint

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The Learning with Errors (LWE) problem receives much attention in cryptography, mainly due to its fundamental significance in post-quantum cryptography. Among its solving algorithms, the Blum-Kalai-Wasserman (BKW) algorithm, originally proposed for solving the Learning Parity with Noise (LPN) problem, performs well, especially for certain parameter settings with cryptographic importance. The BKW algorithm consists of two phases, the reduction phase and the solving phase.In this work, we study the performance of distinguishers used in the solving phase. We show that the Fast Fourier Transform (FFT) distinguisher from Eurocrypt'15 has the same sample complexity as the optimal distinguisher, when making the same number of hypotheses. We also show that it performs much better than theory predicts and introduce an improvement of it called the pruned FFT distinguisher. Finally, we indicate, via extensive experiments, that the sample dependency due to both LF2 and sample amplification is limited.

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“…The idea of combining coded-BKW or LMS with techniques from lattice sieving [11] lead to the next improvement [21]. This combined approach was slightly improved in [22,30]. The distinguishing part of the BKW algorithm for solving LWE was improved by using the Fast Fourier Transform (FFT) in [16].…”

Section: Related Workmentioning

confidence: 99%

“…Instead of testing all possible pairs of samples within the categories, this procedure was sped-up using lattice sieving techniques of [11]. This approach was slightly improved in [22,30].…”

Section: Coded-bkw With Sievingmentioning

confidence: 99%

Making the BKW Algorithm Practical for LWE

Budroni

Guo

Johansson

et al. 2020

Progress in Cryptology – INDOCRYPT 2020

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The Learning with Errors (LWE) problem is one of the main mathematical foundations of post-quantum cryptography. One of the main groups of algorithms for solving LWE is the Blum-Kalai-Wasserman (BKW) algorithm. This paper presents new improvements for BKW-style algorithms for solving LWE instances. We target minimum concrete complexity and we introduce a new reduction step where we partially reduce the last position in an iteration and finish the reduction in the next iteration, allowing non-integer step sizes. We also introduce a new procedure in the secret recovery by mapping the problem to binary problems and applying the Fast Walsh Hadamard Transform. The complexity of the resulting algorithm compares favourably to all other previous approaches, including lattice sieving. We additionally show the steps of implementing the approach for large LWE problem instances. The core idea here is to overcome RAM limitations by using large file-based memory.

show abstract

The Asymptotic Complexity of Coded-BKW with Sieving Using Increasing Reduction Factors

Cited by 7 publications

References 18 publications

Improvements on Making BKW Practical for Solving LWE

Improvements on Making BKW Practical for Solving LWE

On the Sample Complexity of solving LWE using BKW-Style Algorithms

Making the BKW Algorithm Practical for LWE

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