Near-Optimal Time-Sparsity Trade-Offs for Solving Noisy Linear Equations
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3717823.3718284.pdf
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Author(s)
Bangachev, Kiril
Bresler, Guy
Tiegel, Stefan
Vaikuntanathan, Vinod
Date Issued
June 15, 2025
Publisher
ACM|Proceedings of the 57th Annual ACM Symposium on Theory of Computing
Citation
Kiril Bangachev, Guy Bresler, Stefan Tiegel, and Vinod Vaikuntanathan. 2025. Near-Optimal Time-Sparsity Trade-Offs for Solving Noisy Linear Equations. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC '25). Association for Computing Machinery, New York, NY, USA, 1910–1920.
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Final published version
Abstract
We present a polynomial-time reduction from solving noisy linear equations over in dimension Θ(klogn/(logk,logq,loglogn)) with a uniformly random coefficient matrix to noisy linear equations over in dimension n where each row of the coefficient matrix has uniformly random support of size k. This allows us to deduce the hardness of sparse problems from their dense counterparts. In particular, we derive hardness results in the following canonical settings:
• Assuming the ℓ-dimensional (dense) learning with errors () problem over a polynomial-size field takes time 2Ω(ℓ), k-sparse in dimension n takes time nΩ(k/(logk · (logk + loglogn))) .
• Assuming the ℓ-dimensional (dense) learning parity with noise () problem over ℤ/2ℤ takes time 2Ω(ℓ/logℓ), k-sparse in dimension n takes time nΩ(k/(logk · (logk + loglogn)2)) .
These running time lower bounds are nearly tight as both sparse problems can be solved in time nO(k), given sufficiently many samples.
Our reduction allows us to derive several consequences in cryptography and the computational complexity of statistical problems. In addition, as a new application, we give a reduction from k-sparse LWE to noisy tensor completion. Concretely, composing the two reductions implies that order-k rank-2k−1 noisy tensor completion in ℝn⊗ k takes time nΩ(k/ logk · (logk + loglogn)), assuming the exponential hardness of standard worst-case lattice problems.
Description
STOC ’25, Prague, Czechia
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3717823.3718284
