Additive Secret-Sharing Schemes in Abelian Groups Quiz

The lecture discusses additive secret-sharing, a special case of threshold secret-sharing where t equals n-1.
In additive secret-sharing, only the entire set of n shareholders can reconstruct the secret, and any subset of n-1 or fewer parties cannot.
The access structure consists of the entire set of n parties, making unauthorized subsets those containing n-1 or fewer parties.
The intuition behind this secret-sharing scheme is that the sum of any n-1 shares should be independent of the secret.
The sharing algorithm divides the secret into n random shares, each a random element of the underlying group. The sum of these shares should equal the secret.
The nth share is computed by adding the first n-1 shares and finding their additive inverse, then adding it to the secret.
The sharing algorithm is randomized, meaning different inputs may result in different shares.
The reconstruction algorithm involves adding all n shares to obtain the secret.
The operations are performed over an abstract group with an abstract plus operation, and the secret and shares are also elements of the group.
The correctness condition is trivial to verify, as there is only one authorized subset.
The privacy condition requires formally arguing that any subset of n-1 or fewer shareholders' shares have an independent distribution from the secret.
The intuition behind privacy is that without all n pieces, the missing piece could have been any value in the group, and hence any value could have been shared.
Two specific groups, one of all bit strings of length l with the bitwise XOR operation as the plus operation, are given as examples to analyze the privacy property.- The text discusses a secret sharing scheme in an abelian group using XOR operation for summation and subtraction, ensuring that any subset of 3 parties among 4 cannot determine the underlying secret based on their shares.
The dealer shares a secret, a binary string, by randomly selecting 3 shares and calculating the fourth share as the XOR of the first three shares and the secret.
The shares are distributed to respective parties.
The text then analyzes the privacy property of the scheme, focusing on a subset of 3 parties trying to infer the secret based on their shares.
The parties cannot pinpoint the secret based on their shares as each share is randomly selected and independent of the secret.
The secret can take any value from the candidate set, and for each candidate, there is a corresponding share vector that can result in the observed shares.
The text also considers the case where the dealer runs the sharing algorithm again with the same secret, resulting in a different set of shares.
The parties again cannot infer the secret based on their shares as the candidate sets and corresponding missing shares are equally likely.
The text also discusses the case where an unauthorized subset includes the fourth party, who holds a share depending on the secret. However, since the missing share is randomly selected, it doesn't reveal any information about the secret.
The text then discusses an additive secret-sharing scheme using integers and modulo arithmetic, with the dealer randomly selecting the first 3 shares and computing the fourth share as the summation of the first 3 shares and the secret modulo the underlying group size.
The dealer then distributes the shares, and the text analyzes the privacy property, concluding that any subset of 3 parties cannot determine the secret based on their shares.
The text also mentions that if the fourth party is part of the unauthorized subset, they can learn a difference between the secret and one of the shares, but this doesn't reveal any information about the secret itself as any secret can result in the observed difference.
The text concludes by summarizing the n-1 out of n secret-sharing scheme as a simple additive secret-sharing method.
The share size for each party is not explicitly stated in the text.- The text discusses the concept of secret-sharing schemes where a secret is divided among multiple parties.
The size of each party's share is equal to the size of the underlying secret.
This is the best possible scenario in secret-sharing schemes, with the size of a share being no less than the size of the secret.
The text also mentions the concept of threshold secret-sharing, where a subset of shareholders must come together to learn the secret.
The additive secret-sharing scheme described in the text is only optimal when the threshold is n-1, and other threshold values may require different schemes.