## Abstract

We study the local leakage resilience of Shamir’s secret sharing scheme. In Shamir’s scheme, a random polynomial f of degree t is sampled over a field of size p> n, conditioned on f(0 ) = s for a secret s. Any t shares (i, f(i)) can be used to fully recover f and thereby f(0). But, any t- 1 evaluations of f at non-zero coordinates are completely independent of f(0). Recent works ask whether the secret remains hidden even if say only 1 bit of information is leaked from each share, independently. This question is well motivated due to the wide range of applications of Shamir’s scheme. For instance, it is known that if Shamir’s scheme is leakage resilient in some range of parameters, then known secure computation protocols are secure in a local leakage model. Over characteristic-2 fields, the answer is known to be negative (e.g., Guruswami and Wootters, STOC ’16). Benhamouda, Degwekar, Ishai, and Rabin (CRYPTO ’18) were the first to give a positive answer assuming computation is done over prime-order fields. They showed that if t≥ 0.907 n, then Shamir’s scheme is leakage resilient. Since then, there has been extensive efforts to improve the above threshold and after a series of works, the current record shows leakage resilience for t≥ 0.78 n (Maji et al., ISIT ’22). All existing analyses of Shamir’s leakage resilience for general leakage functions follow a single framework for which there is a known barrier for any t≤ 0.5 n. In this work, we a develop a new analytical framework that allows us to significantly improve upon the previous record and obtain additional new results. Specifically, we show: 1.Shamir’s scheme is leakage resilient for any t≥ 0.69 n.2.If the leakage functions are guaranteed to be “balanced” (i.e., splitting the domain of possible shares into 2 roughly equal-size parts), then Shamir’s scheme is leakage resilient for any t≥ 0.58 n.3.If the leakage functions are guaranteed to be “unbalanced” (i.e., splitting the domain of possible shares into 2 parts of very different sizes), then Shamir’s scheme is leakage resilient as long as t≥ 0.01 n. Such a result is provably impossible to obtain using the previously known technique. All of the above apply more generally to any MDS codes-based secret sharing scheme. Confirming leakage resilience is most important in the range t≤ n/ 2, as in many applications, Shamir’s scheme is used with thresholds t≤ n/ 2. As opposed to the previous approach, ours does not seem to have a barrier at t= n/ 2, as demonstrated by our third contribution.

Original language | English |
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Title of host publication | Advances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings |

Editors | Helena Handschuh, Anna Lysyanskaya |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 139-170 |

Number of pages | 32 |

ISBN (Print) | 9783031385568 |

DOIs | |

State | Published - 2023 |

Externally published | Yes |

Event | Advances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings - Santa Barbara, United States Duration: 20 Aug 2023 → 24 Aug 2023 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 14081 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | Advances in Cryptology – CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Proceedings |
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Country/Territory | United States |

City | Santa Barbara |

Period | 20/08/23 → 24/08/23 |

### Bibliographical note

Publisher Copyright:© 2023, International Association for Cryptologic Research.

## Keywords

- Secret sharing
- Shamir’s scheme
- local leakage resilience