Secret Sharing and Secure Computing from Monotone Formulae

We present a construction of log-depth formulae for various threshold functions based on atomic threshold gates of constant size. From this, we build a new family of linear secret sharing schemes that are multiplicative, scale well as the number of players increases and allows to raise a shared value to the characteristic of the underlying field without interaction. Some of these schemes are in addition strongly multiplicative. Our formulas can also be used to construct multiparty protocols from protocols for a constant number of parties. In particular we implement black-box multiparty computation over non-Abelian groups in a way that is much simpler than previously known and we also show how to get a protocol in this setting that is efficient and actively secure against a constant fraction of corrupted parties, a long standing open problem. Finally, we show a negative result on usage of our scheme for pseudorandom secret sharing as defined by Cramer, Damgård and Ishai

Proofs of partial knowledge and simplified design of witness hiding protocols

Suppose we are given a proof of knowledge P in which a prover demonstrates that he knows a solution to a given problem instance. Suppose also that we have a secret sharing scheme S on n participants. Then under certain assumptions on P and S , we show how to transform P into a witness indistinguishable protocol, in which the prover demonstrates knowledge of the solution to some subset of n problem instances out of a collection of subsets defined by S . For example, using a threshold scheme, the prover can show that he knows at least d out of n solutions without revealing which d instances are involved. If the instances are independently generated, we get a witness hiding protocol, even if P did not have this property. Our results can be used to efficiently implement general forms of group oriented identification and signatures. Our transformation produces a protocol with the same number of rounds as P and communication complexity n times that of P . Our results use no unproven complexity assumptions

Randomness Recoverable Secret Sharing Schemes

It is well-known that randomness is essential for secure cryptography. The randomness used in cryptographic primitives is not necessarily recoverable even by the party who can, e.g., decrypt or recover the underlying secret/message. Several cryptographic primitives that support randomness recovery have turned out useful in various applications. In this paper, we study randomness recoverable secret sharing schemes (RR-SSS), in both information-theoretic and computational settings and provide two results. First, we show that while every access structure admits a perfect RR-SSS, there are very simple access structures (e.g., in monotone AC?) that do not admit efficient perfect (or even statistical) RR-SSS. Second, we show that the existence of efficient computational RR-SSS for certain access structures in monotone AC? implies the existence of one-way functions. This stands in sharp contrast to (non-RR) SSS schemes for which no such results are known. RR-SSS plays a key role in making advanced attributed-based encryption schemes randomness recoverable, which in turn have applications in the context of designated-verifier non-interactive zero knowledge

Unbounded Dynamic Predicate Compositions in ABE from Standard Assumptions

At Eurocrypt\u2719, Attrapadung presented several transformations that dynamically compose a set of attribute-based encryption (ABE) schemes for simpler predicates into a new ABE scheme for more expressive predicates. Due to the powerful unbounded and modular nature of his compositions, many new ABE schemes can be obtained in a systematic manner. However, his approach heavily relies on $q$-type assumptions, which are not standard. Devising such powerful compositions from standard assumptions was left as an important open problem. In this paper, we present a new framework for constructing ABE schemes that allow unbounded and dynamic predicate compositions among them, and show that the adaptive security of these composed ABE will be preserved by relying only on the standard matrix Diffie-Hellman (MDDH) assumption. This thus resolves the open problem posed by Attrapadung. As for applications, we obtain various ABEs that are the first such instantiations of their kinds from standard assumptions.These include the following adaptively secure large-universe ABEs for Boolean formulae under MDDH: - The first completely unbounded monotone key-policy (KP)/ciphertext-policy (CP) ABE. Such ABE was recently proposed, but only for the KP and small-universe flavor (Kowalczyk and Wee, Eurocrypt\u2719). - The first completely unbounded non-monotone KP/CP-ABE. Especially, our ABEs support a new type of non-monotonicity that subsumes previous two types of non-monotonicity, namely, by Ostrovsky et al. (CCS\u2707) and by Okamoto and Takashima (CRYPTO\u2710). - The first (non-monotone) KP and CP-ABE with constant-size ciphertexts and secret keys, respectively. - The first KP and CP-ABE with constant-size secret keys and ciphertexts, respectively. At the core of our framework lies a new partially symmetric design of the core 1-key 1-ciphertext oracle component called Key Encoding Indistinguishability, which exploits the symmetry so as to obtain compositions

Fast, Compact, and Expressive Attribute-Based Encryption

Attribute-based encryption (ABE) is an advanced cryptographic tool and useful to build various types of access control systems. Toward the goal of making ABE more practical, we propose key-policy (KP) and ciphertext-policy (CP) ABE schemes, which first support unbounded sizes of attribute sets and policies with negation and multi-use of attributes, allow fast decryption, and are fully secure under a standard assumption, simultaneously. The proposed schemes are more expressive than previous schemes and efficient enough. We also implement our schemes in 128-bit security level and present their benchmarks for an ordinary personal computer and smartphones. They show that all algorithms run in one second with the personal computer when they handle any policy or attribute set with one hundred attributes

Efficient Multiparty Protocols via Log-Depth Threshold Formulae

We put forward a new approach for the design of efficient multiparty protocols: 1. Design a protocol for a small number of parties (say, 3 or 4) which achieves security against a single corrupted party. Such protocols are typically easy to construct as they may employ techniques that do not scale well with the number of corrupted parties. 2. Recursively compose with itself to obtain an efficient n-party protocol which achieves security against a constant fraction of corrupted parties. The second step of our approach combines the player emulation technique of Hirt and Maurer (J. Cryptology, 2000) with constructions of logarithmic-depth formulae which compute threshold functions using only constant fan-in threshold gates. Using this approach, we simplify and improve on previous results in cryptography and distributed computing. In particular: - We provide conceptually simple constructions of efficient protocols for Secure Multiparty Computation (MPC) in the presence of an honest majority, as well as broadcast protocols from point-to-point channels and a 2-cast primitive. - We obtain new results on MPC over blackbox groups and other algebraic structures. The above results rely on the following complexity-theoretic contributions, which may be of independent interest: - We show that for every integers j,k such that m = (k-1)/(j-1) is an integer, there is an explicit (poly(n)-time) construction of a logarithmic-depth formula which computes a good approximation of an (n/m)-out-of-n threshold function using only j-out-of-k threshold gates and no constants. - For the special case of n-bit majority from 3-bit majority gates, a non-explicit construction follows from the work of Valiant (J. Algorithms, 1984). For this special case, we provide an explicit construction with a better approximation than for the general threshold case, and also an exact explicit construction based on standard complexity-theoretic or cryptographic assumptions