Non-malleable secret sharing (NMSS) schemes, introduced by Goyal and Kumar (STOC 2018), ensure that a secret m can be distributed into shares m₁,⋯,m_n (for some n), such that any t (a parameter ≤ n) shares can be reconstructed to recover the secret m, any t-1 shares doesn't leak information about m and even if the shares that are used for reconstruction are tampered, it is guaranteed that the reconstruction of these tampered shares will either result in the original m or something independent of m. Since their introduction, non-malleable secret sharing schemes sparked a very impressive line of research. In this work, we introduce a feature of local reconstructability in NMSS, which allows reconstruction of any portion of a secret by reading just a few locations of the shares. This is a useful feature, especially when the secret is long or when the shares are stored in a distributed manner on a communication network. In this work, we give a compiler that takes in any non-malleable secret sharing scheme and compiles it into a locally reconstructable non-malleable secret sharing scheme. To secret share a message consisting of k blocks of length ρ each, our scheme would only require reading ρ + log k bits (in addition to a few more bits, whose quantity is independent of ρ and k) from each party’s share (of a reconstruction set) to locally reconstruct a single block of the message. We show an application of our locally reconstructable non-malleable secret sharing scheme to a computational non-malleable secure message transmission scheme in the pre-processing model, with an improved communication complexity, when transmitting multiple messages.
Non-malleable secret sharing (NMSS) schemes, introduced by Goyal and Kumar (STOC 2018), ensure that a secret m can be distributed into shares m₁,⋯,m_n (for some n), such that any t (a parameter ≤ n) shares can be reconstructed to recover the secret m, any t-1 shares doesn't leak information about m and even if the shares that are used for reconstruction are tampered, it is guaranteed that the reconstruction of these tampered shares will either result in the original m or something independent of m. Since their introduction, non-malleable secret sharing schemes sparked a very impressive line of research. In this work, we introduce a feature of local reconstructability in NMSS, which allows reconstruction of any portion of a secret by reading just a few locations of the shares. This is a useful feature, especially when the secret is long or when the shares are stored in a distributed manner on a communication network. In this work, we give a compiler that takes in any non-malleable secret sharing scheme and compiles it into a locally reconstructable non-malleable secret sharing scheme. To secret share a message consisting of k blocks of length ρ each, our scheme would only require reading ρ + log k bits (in addition to a few more bits, whose quantity is independent of ρ and k) from each party’s share (of a reconstruction set) to locally reconstruct a single block of the message. We show an application of our locally reconstructable non-malleable secret sharing scheme to a computational non-malleable secure message transmission scheme in the pre-processing model, with an improved communication complexity, when transmitting multiple messages.
In this work, we consider the natural goal of designing secret sharing schemes that ensure security against an adversary who may learn some “leaked'' information about all the shares. We say that a secret sharing scheme is p-party leakage-resilient, if the secret remains statistically hidden even after a computationally unbounded adversary learns a bounded amount of leakage, where each bit of leakage adaptively and jointly depends on the shares of an adaptively chosen subset of p parties. Existing multi-party secret sharing schemes (Dziembowski and Pietrzak FOCS 07), (Goyal and Kumar STOC 18) and (Benhamouda, Degwekar, Ishai and Rabin CRYPTO 18) have focused on handling non-adaptive and individual leakage for (limited special cases of) threshold secret sharing schemes. (1) We give an unconditional compiler that transforms any secret sharing scheme on n parties into a p-party leakage-resilient one for p upto O(log n). This yields the first multi-party secret sharing schemes that are secure against adaptive or joint leakage. (2) As a natural extension, we initiate the study of leakage-resilient non-malleable secret sharing. We empower the adversary to adaptively leak from each of the shares and then use the leakage to tamper with all of them arbitrarily and independently. Leveraging our p-party leakage-resilient schemes, we compile any secret sharing scheme into a non-malleable one ensuring that any such tampering either preserves the secret or completely `destroys' it. This improves upon the non-malleable secret sharing scheme of (Goyal and Kumar CRYPTO 18) where no leakage was permitted. Leakage-resilient non-malleable codes can be seen as 2-out-of-2 schemes satisfying our guarantee and have already found many applications in cryptography. (3) Our constructions rely on a clean connection we draw to communication complexity in the well-studied number-on-forehead (NOF) model and rely on functions that have strong communication-complexity lower bounds in the NOF model (in a black-box way). We get efficient p-party leakage-resilient schemes for p upto O(log n) as our share sizes have exponential dependence on p. We observe that improving this exponential dependence, even for simultaneous, non-adaptive leakage, will lead to progress on longstanding open problems in complexity theory.
Non-malleable codes are fundamental objects at the intersection of cryptography and coding theory. These codes provide security guarantees even in settings where error correction and detection are impossible, and have found applications to several other cryptographic tasks. One of the strongest and most well-studied adversarial tampering models is 2-split-state tampering. Here, a codeword is split into two parts which are stored in physically distant servers, and the adversary can then independently tamper with each part using arbitrary functions. This model can be naturally extended to the secret sharing setting with several parties by having the adversary independently tamper with each share. Previous works on non-malleable coding and secret sharing in the split-state tampering model only considered the encoding of classical messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans. Inf. Theory 2024 & arXiv 2022), adversaries with quantum capabilities and shared entanglement had not been considered, and it is a priori not clear whether previous schemes remain secure in this model. In this work, we introduce the notions of split-state non-malleable codes and secret sharing schemes for quantum messages secure against quantum adversaries with shared entanglement. Then, we present explicit constructions of such schemes that achieve low-error non-malleability. More precisely, for some constantc>0, we construct efficiently encodable and decodable split-state non-malleable codes and secret sharing schemes for quantum messages preserving entanglement with external systems and achieving security against quantum adversaries having shared entanglement with codeword length n, any message length at mostnc, and errorε=2−nc. In the easier setting of average-case non-malleability, we achieve efficient non-malleable coding with rate close to1/11.
Leakage-resilient secret sharing has mostly been studied in the compartmentalized models, where a leakage oracle can arbitrarily leak bounded number of bits from all shares, provided that the oracle only has access to a bounded number of shares when the leakage is taking place. We start a systematic study of leakage-resilient secret sharing against global leakage, where the leakage oracle can access the full set of shares simultaneously, but the access is restricted to a special class of leakage functions. More concretely, the adversary can corrupt several players and obtain their shares, as well as applying a leakage function from a specific class to the full share vector. We explicitly construct such leakage-resilient secret sharing with respect to affine leakage functions and low-degree multi-variate polynomial leakage functions, respectively. For affine leakage functions, we obtain schemes with threshold access structure that are leakage-resilient as long as there is a substantial difference between the total amount of information obtained by the adversary, through corrupting individual players and leaking from the full share vector, and the amount that the reconstruction algorithm requires for reconstructing the secret. Furthermore, if we assume the adversary is non-adaptive, we can even make the secret length asymptotically equal to the difference, as the share length grows. Specifically, we have a threshold scheme with parameters similar to Shamir’s scheme and is leakage-resilient against affine leakage. For multi-variate polynomial leakage functions with degree bigger than one, our constructions here only yield ramp schemes that are leakage-resilient against such leakage. Finally, as a result of independent interest, we show that our approach to leakage-resilient secret sharing also yields a competitive scheme compared with the state-of-the-art construction in the compartmentalized models.
Non-malleable codes are fundamental objects at the intersection of cryptography and coding theory. These codes provide security guarantees even in settings where error correction and detection are impossible, and have found applications to several other cryptographic tasks. One of the strongest and most well-studied adversarial tampering models is 2-split-state tampering. Here, a codeword is split into two parts and the adversary can then independently tamper with each part using arbitrary functions. This model can be naturally extended to the secret sharing setting with several parties by having the adversary independently tamper with each share. Previous works on non-malleable coding and secret sharing in the split-state tampering model only considered the encoding of \emph{classical} messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans.\ Inf.\ Theory 2024), adversaries with quantum capabilities and \emph{shared entanglement} had not been considered, and it is a priori not clear whether previous schemes remain secure in this model. In this work, we introduce the notions of split-state non-malleable codes and secret sharing schemes for quantum messages secure against quantum adversaries with shared entanglement. Then, we present explicit constructions of such schemes that achieve low-error non-malleability. More precisely, we construct efficiently encodable and decodable split-state non-malleable codes and secret sharing schemes for quantum messages preserving entanglement with external systems and achieving security against quantum adversaries having shared entanglement with codeword length n, any message length at most nΩ(1), and error ε=2−nΩ(1). In the easier setting of \emph{average-case} non-malleability, we achieve efficient non-malleable coding with rate close to 1/11.
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
Leakage-resilient secret sharing has mostly been studied in the compartmentalized models, where a leakage oracle can arbitrarily leak bounded number of bits from all shares, provided that the oracle only has access to a bounded number of shares when the leakage is taking place. We start a systematic study of leakage-resilient secret sharing against global leakage, where the leakage oracle can access the full set of shares simultaneously, but the access is restricted to a special class of leakage functions. More concretely, the adversary can corrupt several players and obtain their shares, as well as applying a leakage function from a specific class to the full share vector. We explicitly construct such leakage-resilient secret sharing with respect to affine leakage functions and low-degree multi-variate polynomial leakage functions, respectively. For affine leakage functions, we obtain schemes with threshold access structure that are leakage-resilient as long as there is a substantial difference between the total amount of information obtained by the adversary, through corrupting individual players and leaking from the full share vector, and the amount that the reconstruction algorithm requires for reconstructing the secret. Furthermore, if we assume the adversary is non-adaptive, we can even make the secret length asymptotically equal to the difference, as the share length grows. Specifically, we have a threshold scheme with parameters similar to Shamir’s scheme and is leakage-resilient against affine leakage. For multi-variate polynomial leakage functions with degree bigger than one, our constructions here only yield ramp schemes that are leakage-resilient against such leakage. Finally, as a result of independent interest, we show that our approach to leakage-resilient secret sharing also yields a competitive scheme compared with the state-of-the-art construction in the compartmentalized models.