Lower Bounds for Leakage-Resilient Secret-Sharing Schemes against Probing Attacks
Donald Q. Adams, Hemanta K. Maji, Hai H. Nguyen, Le-Minh Nguyen, Anat Paskin-Cherniavsky, Tom Suad, Mingyuan Wang · 2021
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.