Smoothing Out Binary Linear Codes and Worst-case Sub-exponential Hardness for LPN
Learning parity with noise (LPN) is a notorious (average-case) hard problem that has been well studied in learning theory, coding theory and cryptography since the early 90’s. It further inspires the Learning with Errors (LWE) problem [Regev, STOC 2005], which has become one of the central building blocks for post-quantum cryptography and advanced cryptographic primitives such as fully homomorphic encryption [Gentry, STOC 2009]. Unlike LWE whose hardness can be reducible from worst-case lattice problems, no corresponding worst-case hardness results were known for LPN until very recently. At Eurocrypt 2019, Brakerski et al. [BLVW19] established the first feasibility result that the worst-case hardness of nearest codeword problem (NCP) (on balanced linear code) at the extremely low noise rate log 2 n n implies the quasi-polynomial hardness of LPN at the extremely high noise rate 1/2−1/poly(n). It remained open whether a worst-case to average-case reduction can be established for standard (constant-noise) LPN, ideally with sub-exponential hardness. In this paper, we carry on the work of [BLVW19]. We first expand the underlying binary linear codes (of the NCP) to not only the balanced code considered in [BLVW19] but also to another code (in some sense dual to balanced code). At the core of our reduction is a new variant of smoothing lemma (for both binary codes) that circumvents the barriers (inherent in the underlying worst-case randomness extraction) and admits tradeoffs for a wider spectrum of parameter choices. In addition to the worst-case hardness result obtained in [BLVW19], we show that for any constant 0 < c < 1 the constant-noise LPN problem is (T = 2 ), = 2−Ω(n ), q = 2 ))-hard assuming that the NCP at the low-noise rate τ = n−c is (T ′ = 2, ′ = 2−Ω(τn),m = 2)-hard in the worst case, where T , , q and m are time complexity, success rate, sample complexity, and codeword length respectively. Moreover, refuting the worst-case hardness assumption would imply arbitrary polynomial speedups over the current state-of-the-art algorithms for solving the NCP (and LPN), which is a win-win result. Unfortunately, public-key encryptions and collision resistant hash functions need constant-noise LPN with (T = 2 √ , ′ = 2−ω( √ ,q = 2 √ )-hardness (Yu et al., CRYPTO 2016 & ASIACRYPT 2019), which is almost (up to an arbitrary ω(1) factor in the exponent) what is reducible from the worst-case NCP when c = 0.5. We leave it as an open problem whether the gap can be closed or there is a separation in place.