Lower Bounds for Leakage-Resilient Secret-Sharing Schemes against Probing Attacks

Historically, side-channel attacks have revealed partial information about the intermediate values and secrets of computations to compromise the security of cryptographic primitives. The objective of leakage-resilient cryptography is to model such avenues of information leakage and study techniques to realize them securely. This work studies the local leakage-resilience of prominent secret-sharing schemes like Shamir's secret-sharing scheme and the additive secret-sharing scheme against probing attacks that leak physical-bits from the memory hardware storing the secret shares. Consider the additive secret-sharing scheme among k parties over a prime field such that the prime needs \lambda-bits for its binary representation, where \lambda is the security parameter. We prove that k must be at least \omega(\log\lambda/\log\log\lambda) for the scheme to be secure against even one physical-bit leakage from each secret share. This result improves the previous state-of-the-art result where an identical lower bound was known for one-bit general leakage from each secret share (Benhamouda, Degwekar, Ishai, and Rabin, CRYPTO-2018). This lower bound on the reconstruction threshold extends to Shamir's secret-sharing scheme if one does not carefully choose the evaluation places for generating the secret shares. For this scheme, our result additionally improves another lower bound on the reconstruction threshold k of Shamir's secret-sharing scheme (Nielsen and Simkin, EUROCRYPT-2020) when the total number of parties is \mathcal{O}(\lambda\log\lambda/\log\log\lambda). Our work provides the analysis of the recently-proposed (explicit) physical-bit leakage attack of Maji, Nguyen, Paskin-Cherniavsky, Suad, and Wang (EUROCRYPT-2021), namely the 'parity of parity' attack. This analysis relies on lower-bounding the 'discrepancy' of the Irwin-Hall probability distribution.