Secret sharing

Security in Computing (3rd Edition)

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CS 161 Computer Security Fall 2005 Joseph/Tygar/Vazirani/Wagner Notes 16 In a secret sharing scheme there is a trusted authority TA and n users U 1 ,..., U n . The TA has a secret value K called the secret or key. The TA uses a share generation algorithm to split K into n shares s 1 ,..., s n . Each share s i is then transmitted to user U i by a secure channel. The secret sharing protocol guarantees that two properties hold: • A reconstruction algorithm can be used to efficiently reconstruct the secret K from any t of the n shares. • Any t - 1 of the n shares reveal no information about the secret K . Such a scheme is called an ( n , t ) threshold scheme. For example, if the secret K is an integer between 0 and M - 1, then an ( n , n ) threshold scheme can be obtained by selecting s 1 ,..., s n - 1 uniformly at randomly between 0 and M - 1, and setting s n = K - n - 1 i = i s i mod M . Now, K = n i = 1 s i mod M . • Given all shares except s j , K can take on any value modulo M .

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