4 Secret Sharing over the Reals

We would like to thank Shai Ben-David, Oded Goldreich and Hugo Krawczyk for helpful discussions on the topics of this paper. We would also like to thank Bob Blakley for acquainting us with his work 8, 9]. In this section we deal with secret-sharing schemes over the real numbers. Although it has no practical implications, it is interesting to ask the question whether secret-sharing schemes do not exist over every innnite set, or maybe some properties of countable sets are the cause of the results of section 2. We introduce a simple secret-sharing scheme using real numbers. Since there is a 1 ? 1 and onto transformation from the real numbers to the unit interval 0; 1), it is more convenient to use this interval as the set of secrets. We use the same interval as the set of shares, as it allows us to use the uniform probability distribution. We rst have to deene what we mean by a secret-sharing scheme over the reals. More speciically, we have to deene what we mean by saying that no \illegal" set (i.e., T = 2 F n) of shares reveals any information about the secret. All the schemes we present in this section are perfect (= 1), and therefore is omitted from the notations. The following natural deenition is used: For every two secrets a 1 ; a 2 2 A, for any set of indices T = 2 F n and for any jTj-tuple of measurable sets fC i g i2T 0; 1) the following holds: Pr(8i 2 T : s i 2 C i j a 1) = Pr(8i 2 T : s i 2 C i j a 2) We can now present a secret-sharing scheme for every good family of sets F n (n 2), using ideas that were used in the nite case 4, 5]. We rst introduce a (k; k) secret-sharing scheme which distributes a secret a taken from the interval 0; 1). We use the Lebesgue measure on 0; 1). interval 0; 1). 2. Choose s k 2 0; 1) which satisses s 1 + : : : + s k?1 + s k = a (mod 1). The proof that this is indeed a secret-sharing scheme is similar to the proof of its analogue in the nite case. For introducing a F n secret-sharing scheme for every good family of sets F n …