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Understanding Secret Sharing: Foundations and Applications in Cryptography

This document explores the concept of secret sharing, a method essential in cryptography for secure data storage and communication. It delves into various secret sharing schemes, including Shamir's t-out-of-n and generalized secret sharing. Important applications of secret sharing in secure multiparty computation, threshold cryptography, Byzantine agreement, access control, and more are discussed. The aim is to ensure that authorized parties can recover secrets while preventing unauthorized access, highlighting the correctness and privacy implications of these schemes.

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Understanding Secret Sharing: Foundations and Applications in Cryptography

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  1. Amos Beimel How to Share a Secret

  2. Secret Sharing[Shamir79,Blakley79,ItoSaitoNishizeki87] 1706 1706 2 bad ? 2538 3441 1329 6634

  3. Applications • Original motivation: Secure storage • Building box for cryptographic protocols • Secure multiparty computation • Threshold cryptography • Byzantine agreement • Access control • Private information retrieval • Attribute-based encryption • Generalized oblivious transfer BGU - Graduate Day

  4. 2-out-of-2 Secret Sharing Scheme • Input: secret • Choose at random a bit • Share of P1: • Share of P2: BGU - Graduate Day

  5. Shamir’s t-out-of-n Secret Sharing Scheme • Input: secrets • Choose at random apolynomial p(x)=s+r1x+r2x2+…+ rt-1xt-1 • Share of Pj: sj= p(j) s BGU - Graduate Day

  6. Open Question: Generalized Secret Sharing • Not all sets are equal. • There is a collection  of authorized sets • Correctness:Every authorized set can recover s. • Privacy:Every unauthorized set cannot learn s. • Are there efficient schemes for every ? • number of parties: n • Upper bound 2O(n) • Lower bound n2/log n • Open problem: BGU - Graduate Day

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