Leaky LWE: Learning with Errors with Semi-Adaptive Secret- and Error-Leakage

The Learning with Errors (LWE) problem asks to distinguish noisy samples sT A + eT mod q from uniformly random values given the random matrix A. In this work, we show that a variant called Leaky LWE, where the distinguisher receives additionally noisy leakages (sT, eT) L + fT of the LWE secret s and error e for low-norm matrix L chosen adaptively by the distinguisher after seeing A, is not easier than the standard LWE of the same dimensions up to polynomial losses in the noise level and the modulus. More generally, we show that the Leaky LWE problem is hard even if the public matrix A is structured and/or hinted and if the non-leaky parts of the secret and error do not follow Gaussian distributions, as long as the corresponding LWE problem without leakage is hard. Our reduction from LWE to Leaky LWE unifies and extends prior results on the Error-Leakage LWE problem Döttling-Kolonelos-Lai-Lin-Malavolta-Rahimi, EUROCRYPT'23, where L only acts on the error e and the Hint-MLWE problem Kim-Lee-Seo-Song, CRYPTO'23, where L is restricted to concatenations of random Gaussian scalar matrices not controlled by the distinguisher. Previously, the Hint-MLWE and Error-Leakage LWE assumptions were used as computational replacements of the statistical noise flooding technique in security proofs which led to improved parameters in lattice-based cryptographic constructions such as zero-knowledge proofs, threshold signatures and registration-based encryption. We provide lemmas which abstract out such computational arguments based on Leaky LWE.