We introduce a new primitive called intrusion-resilient secret sharing (IRSS), whose security proof exploits the fact that there exist functions which can be efficiently computed interactively using low communication complexity in k, but not in k-1 rounds. IRSS is a means of sharing a secret message amongst a set of players which comes with a very strong security guarantee. The shares in an IRSS are made artificially large so that it is hard to retrieve them completely, and the reconstruction procedure is interactive requiring the players to exchange k short messages. The adversaries considered can attack the scheme in rounds, where in each round the adversary chooses some player to corrupt and some function, and retrieves the output of that function applied to the share of the corrupted player. This model captures for example computers connected to a network which can occasionally he infected by malicious software like viruses, which can compute any function on the infected machine, but cannot sent out a huge amount of data. Using methods from the bounded-retrieval model, we construct an IRSS scheme which is secure against any computationally unbounded adversary as long as the total amount of information retrieved by the adversary is somewhat less than the length of the shares, and the adversary makes at most k-1 corruption rounds (as described above, where k rounds are necessary for reconstruction). We extend our basic scheme in several ways in order to allow the shares sent by the dealer to be short (the players then blow them up locally) and to handle even stronger adversaries who can learn some of the shares completely. As mentioned, there is an obvious connection between IRSS schemes and the fact that there exist functions with an exponential gap in their communication complexity for k and k-1 rounds. Our scheme implies such a separation which is in several aspects stronger than the previously known ones.
We introduce a new primitive called intrusion-resilient secret sharing (IRSS), whose security proof exploits the fact that there exist functions which can be efficiently computed interactively using low communication complexity in k, but not in k-1 rounds. IRSS is a means of sharing a secret message amongst a set of players which comes with a very strong security guarantee. The shares in an IRSS are made artificially large so that it is hard to retrieve them completely, and the reconstruction procedure is interactive requiring the players to exchange k short messages. The adversaries considered can attack the scheme in rounds, where in each round the adversary chooses some player to corrupt and some function, and retrieves the output of that function applied to the share of the corrupted player. This model captures for example computers connected to a network which can occasionally he infected by malicious software like viruses, which can compute any function on the infected machine, but cannot sent out a huge amount of data. Using methods from the bounded-retrieval model, we construct an IRSS scheme which is secure against any computationally unbounded adversary as long as the total amount of information retrieved by the adversary is somewhat less than the length of the shares, and the adversary makes at most k-1 corruption rounds (as described above, where k rounds are necessary for reconstruction). We extend our basic scheme in several ways in order to allow the shares sent by the dealer to be short (the players then blow them up locally) and to handle even stronger adversaries who can learn some of the shares completely. As mentioned, there is an obvious connection between IRSS schemes and the fact that there exist functions with an exponential gap in their communication complexity for k and k-1 rounds. Our scheme implies such a separation which is in several aspects stronger than the previously known ones.
We introduce a new primitive called intrusion-resilient secret sharing (IRSS), whose security proof exploits the fact that there exist functions which can be efficiently computed interactively using low communication complexity in k, but not in k-1 rounds. IRSS is a means of sharing a secret message amongst a set of players which comes with a very strong security guarantee. The shares in an IRSS are made artificially large so that it is hard to retrieve them completely, and the reconstruction procedure is interactive requiring the players to exchange k short messages. The adversaries considered can attack the scheme in rounds, where in each round the adversary chooses some player to corrupt and some function, and retrieves the output of that function applied to the share of the corrupted player. This model captures for example computers connected to a network which can occasionally he infected by malicious software like viruses, which can compute any function on the infected machine, but cannot sent out a huge amount of data. Using methods from the bounded-retrieval model, we construct an IRSS scheme which is secure against any computationally unbounded adversary as long as the total amount of information retrieved by the adversary is somewhat less than the length of the shares, and the adversary makes at most k-1 corruption rounds (as described above, where k rounds are necessary for reconstruction). We extend our basic scheme in several ways in order to allow the shares sent by the dealer to be short (the players then blow them up locally) and to handle even stronger adversaries who can learn some of the shares completely. As mentioned, there is an obvious connection between IRSS schemes and the fact that there exist functions with an exponential gap in their communication complexity for k and k-1 rounds. Our scheme implies such a separation which is in several aspects stronger than the previously known ones.
Broadcast and verifiable secret sharing (VSS) are central building blocks for secure multi-party computation. These protocols are required to be resilient against a Byzantine adversary who controls at most t out of the n parties running the protocol. In this dissertation, we consider the design of fault-tolerant protocols for broadcast and verifiable secret sharing with stronger security guarantees and improved round complexity. Broadcast allows a party to send the same message to all parties, and all parties are assured they have received identical messages. Given a public-key infrastructure (PKI) and digital signatures, it is possible to construct broadcast protocols tolerating any number of corrupted parties. We address two important issues related to broadcast: (1) Almost all existing protocols do not distinguish between corrupted parties (who do not follow the protocol) and honest parties whose secret (signing) keys have been compromised (but who continue to behave honestly); (2) all existing protocols for broadcast are insecure against an adaptive adversary who can choose which parties to corrupt as the protocol progresses. We propose new security models that capture these issues, and present tight feasibility and impossibility results. In the problem of verifiable secret sharing, there is a designated player who shares a secret during an initial sharing phase such that the secret is hidden from an adversary that corrupts at most t parties. In a subsequent reconstruction phase of the protocol, a unique secret, well-defined by the view of honest players in the sharing phase, is reconstructed. The round complexity of VSS protocols is a very important metric of their efficiency. We show two improvements regarding the round complexity of information-theoretic VSS. First, we construct an efficient perfectly secure VSS protocol tolerating t < n/3 corrupted parties that is simultaneously optimal in both the number of rounds and the number of invocations of broadcast. Second, we construct a statistically secure VSS protocol tolerating t < n/2 corrupted parties that has optimal round complexity, and an efficient statistical VSS protocol tolerating t < n/2 corrupted parties that requires one additional round.
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.
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.
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.
Function secret sharing (FSS) is a cryptographic primitive that is introduced by Boyle et al. (Eurocrypt 2015), motivated by application scenarios involving private access to large distributed data while minimising the overhead of communication, for example, private information retrieval. Informally, an n-party FSS scheme splits a function f into n functions f1,...,fnsuch that f = f1+...+ fnand every strict subset of the function shares hide f . Most of the known FSS constructions only have computational hiding, namely, the hiding property holds only against a computationally bounded adversary. We consider information-theoretic hiding in this work while allowing f to be recovered from t function shares and correspondingly, any (t - 1) function shares unconditionally hide f . Call it (t, n)-threshold function secret sharing ((t, n)-TFSS for short). Using information-theoretic tools and through a series of optimizations, we show that our (t, n)-TFSS have better performance than FSS in terms of communication complexity, a criterion that measures the efficiency of such protocols. Specifically, a (t, n)-TFSS scheme with communication complexity O(l) is designed in this paper and it is better than the existing FSS schemes with lowest communication complexity O(λl), where λ is the length of pseudo-random generator's seeds. In addition, the (t, n)-TFSS have an extra robustness property in the sense that even if up to (n - t) function shares are not available, the protocol still computes the function value at a given point correctly.