We consider the problem of increasing the threshold parameter of a secret-sharing scheme after the setup (share distri- bution) phase, without further communication between the dealer and the shareholders. Previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose, or to have communication between shareholders. In contrast, we show how to increase the threshold parameter of the standard Shamir secret-sharing scheme without communica- tion between the shareholders. Our technique can thus be applied to existing Shamir schemes even if they were set up without con- sideration to future threshold increases. Our method is a new positive cryptographic application for lat- tice reduction algorithms, inspired by recent work on lattice-based list decoding of Reed-Solomon codes with noise bounded in the Lee norm. We use fundamental results from the theory of lattices (geometry of numbers) to prove quantitative statements about the information-theoretic security of our construction. These lattice- based security proof techniques may be of independent interest.
We consider the problem of increasing the threshold parameter of a secret-sharing scheme after the setup (share distri- bution) phase, without further communication between the dealer and the shareholders. Previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose, or to have communication between shareholders. In contrast, we show how to increase the threshold parameter of the standard Shamir secret-sharing scheme without communica- tion between the shareholders. Our technique can thus be applied to existing Shamir schemes even if they were set up without con- sideration to future threshold increases. Our method is a new positive cryptographic application for lat- tice reduction algorithms, inspired by recent work on lattice-based list decoding of Reed-Solomon codes with noise bounded in the Lee norm. We use fundamental results from the theory of lattices (geometry of numbers) to prove quantitative statements about the information-theoretic security of our construction. These lattice- based security proof techniques may be of independent interest.
In this paper, we introduce a method of threshold secret sharing scheme in which secret reconstruction is based on celebrated Babai lattice algorithm. In order to supply secure public channels for transmitting shares to parties, we need to ensure that there is no quantum threats to these channels. One solution for this problem can be utilization of lattice cryptosystems for these channels which requires designing lattice based secret sharing schemes. We indicate that our scheme is asymptotically correct. Moreover, we analyze the security of our scheme by giving a quantitative proof of security from the view point of information theory.
In this paper, we introduce a method of threshold secret sharing scheme (TSSS) in which secret reconstruction is based on Babai's nearest plane algorithm. In order to supply secure public channels for transmitting shares to parties, we need to ensure that there are no quantum threats to these channels. A solution to this problem can be utilization of lattice-based cryptosystems for these channels which requires designing lattice-based TSSSs. We investigate the effect of lattice dimension on the security and correctness of the proposed scheme. Moreover, we prove that for a fixed lattice dimension the proposed scheme is asymptotically correct. We also give a quantitative proof of security from information theoretic viewpoint.
In this paper, we propose a threshold increasing algorithm for a (t; n) latticebased Threshold Multi-Stage Secret Sharing (TMSSS) scheme. To realize the changeability feature, we use the zero addition protocol to construct a new (t0; n) TMSSS scheme. Therefore, the new scheme enjoys the signi cant feature of threshold changeability along with the inherited features of being multi-stage, multi-use, and veri able derived fromour previously proposed lattice-based TMSSS scheme. Furthermore, we use the improved TMSSS scheme to propose a threshold decryption algorithm for the Learning With Error (LWE) based public key encryption scheme based on the study of Lindner and Peikert. For threshold decryption, each authorized subset of participants decrypts the ciphertext partially and sends the result to the combiner. The combiner can decrypt the ciphertext using the partial decryptions. The security of both schemes is based on hardness of lattice problems, i.e. LWE and Inhomogeneous Small Integer Solution (ISIS) problems, which are believed to resist against the quantum algorithms. The proposed schemes are eficient, especially on the participants' side, making them suitable for the applications in which the participants have limited processing capacities.
We prove that a known approach to improve Shamir's celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ ∈ (0; 1/2). This result holds in the so-called “nonrushing” model in which the n shares are submitted simultaneously for reconstruction. We thus finally obtain a simple, fully explicit, and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1+o(1))+O(κ), where k is the secret length and κ is the security parameter. Like Shamir's scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δn(1 + ρ) honest parties, for arbitrarily small ρ > 0, can efficiently reconstruct the secret.
We prove that a known approach to improve Shamir’s celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ ∈ (0, 1/2). This result holds in the so-called “non-rushing ” model in which the n shares are submitted simultaneously for reconstruction. We thus finally obtain a fully explicit and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1 + o(1)) +O(κ), where k is the secret length and κ is the security parameter. Like Shamir’s scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow indepen-dently. In this case, when n is large enough, the scheme satisfies the “threshold ” requirement in an approximate sense; i.e., any set of δn(1 + ρ) honest parties, for arbitrarily small ρ> 0, can efficiently reconstruct the secret. 1
In a k-threshold secret sharing scheme, a dealer who holds a secret s distributes parts of this secret (the shares) to n players; If k or more of these players pool their shares, they are able to determine s, but if less than k of them pool their shares, they cannot infer any information on s. In 1979, Shamir introduced such a scheme, based on polynomial interpolation in a finite field. This scheme is widely used in other cryptographic protocols, because it is simple, elegant and above all information-theoretically secure. There are a few proofs of the scheme's security, but to our knowledge, none of them is entirely based on the information-theoretical entropy function introduced by Shannon in 1948. We propose such a demonstration.