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Tight Estimate of the Local Leakage Resilience of the Additive Secret-Sharing Scheme & Its Consequences

Authors Hemanta K. Maji, Hai H. Nguyen, Anat Paskin-Cherniavsky, Tom Suad, Mingyuan Wang, Xiuyu Ye, Albert Yu



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Author Details

Hemanta K. Maji
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Hai H. Nguyen
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Anat Paskin-Cherniavsky
  • Department of Computer Science, Ariel University, Israel
Tom Suad
  • Department of Computer Science, Ariel University, Israel
Mingyuan Wang
  • Department of EECS, University of California Berkeley, CA, USA
Xiuyu Ye
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Albert Yu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

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Hemanta K. Maji, Hai H. Nguyen, Anat Paskin-Cherniavsky, Tom Suad, Mingyuan Wang, Xiuyu Ye, and Albert Yu. Tight Estimate of the Local Leakage Resilience of the Additive Secret-Sharing Scheme & Its Consequences. In 3rd Conference on Information-Theoretic Cryptography (ITC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 230, pp. 16:1-16:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITC.2022.16

Abstract

Innovative side-channel attacks have repeatedly exposed the secrets of cryptosystems. Benhamouda, Degwekar, Ishai, and Rabin (CRYPTO-2018) introduced local leakage resilience of secret-sharing schemes to study some of these vulnerabilities. In this framework, the objective is to characterize the unintended information revelation about the secret by obtaining independent leakage from each secret share. This work accurately quantifies the vulnerability of the additive secret-sharing scheme to local leakage attacks and its consequences for other secret-sharing schemes. Consider the additive secret-sharing scheme over a prime field among k parties, where the secret shares are stored in their natural binary representation, requiring λ bits - the security parameter. We prove that the reconstruction threshold k = ω(log λ) is necessary to protect against local physical-bit probing attacks, improving the previous ω(log λ/log log λ) lower bound. This result is a consequence of accurately determining the distinguishing advantage of the "parity-of-parity" physical-bit local leakage attack proposed by Maji, Nguyen, Paskin-Cherniavsky, Suad, and Wang (EUROCRYPT-2021). Our lower bound is optimal because the additive secret-sharing scheme is perfectly secure against any (k-1)-bit (global) leakage and (statistically) secure against (arbitrary) one-bit local leakage attacks when k = ω(log λ). Any physical-bit local leakage attack extends to (1) physical-bit local leakage attacks on the Shamir secret-sharing scheme with adversarially-chosen evaluation places, and (2) local leakage attacks on the Massey secret-sharing scheme corresponding to any linear code. In particular, for Shamir’s secret-sharing scheme, the reconstruction threshold k = ω(log λ) is necessary when the number of parties is n = O(λ log λ). Our analysis of the "parity-of-parity" attack’s distinguishing advantage establishes it as the best-known local leakage attack in these scenarios. Our work employs Fourier-analytic techniques to analyze the "parity-of-parity" attack on the additive secret-sharing scheme. We accurately estimate an exponential sum that captures the vulnerability of this secret-sharing scheme to the parity-of-parity attack, a quantity that is also closely related to the "discrepancy" of the Irwin-Hall probability distribution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Security and privacy → Cryptanalysis and other attacks
Keywords
  • leakage resilience
  • additive secret-sharing
  • Shamir’s secret-sharing
  • physical-bit probing leakage attacks
  • Fourier analysis

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