Cryptography - Sage

CHAPTERTENSecret SharingA secret sharing scheme is a means for n parties to carry shares or parts s i of a message s,called the secret, such that the complete set s 1 , . . . s n of the parts determines the message.The secret sharing scheme is said to be perfect if no proper subset of shares leaks anyinformation regarding the secret.Two party secret sharing. Let s be a secret, encoding as an integer in Z/mZ. Lets 1 ∈ Z/mZ be generated at random by a trusted party. Then the two shares are definedto be s 1 and s − s 1 . The secret is recovered as s = s 1 + s 2 .Multiple party secret sharing. Let s ∈ Z/mZ be a secret to be shared among nparties. Generate the first n − 1 shares s 1 , . . . , s n−1 at random and setThe secret is recovered as s = ∑ ni=1 s i.∑n−1s n = s − .A (t, n) threshold secret sharing scheme is a method for n parties to carry shares s i of amessage s such that any t of the them to reconstruct the message, but so that no t − 1 ofthem can easy do so. The threshold scheme is perfect if knowledge of t − 1 or fewer sharesprovides no information regarding s.Shamir’s (t, n)-threshold scheme. A scheme of Shamir provide an elegant constructionof a perfect (t, n)-threshold scheme using a classical algorithm called Lagrange interpolation.First we introduce Lagrange interpolation as a theorem.i=1Theorem 10.1 (Lagrange interpolation) Given t distinct points (x i , y i ) of the form(x i , f(x i )), where f(x) is a polynomial of degree less that t, then f(x) is determined byf(x) =t∑i=1y i∏1≤j≤ti≠jx − x jx i − x j. (10.1)83