An aggregate signature scheme is a digital signature that supports aggregation: Given n signatures on n distinct messages from n distinct users, it is possible to aggregate all these signatures into a single short signature. This single signature (and the n original messages) will convince the verifier that the n users did indeed sign the n original messages (i.e., user i signed message M i for i = 1; : : : ; n). In this paper we introduce the concept of an aggregate signature scheme, present security models for such signatures, and give several applications for aggregate signatures. We construct an efficient aggregate signature from a recent short signature scheme based on bilinear maps due to Boneh, Lynn, and Shacham. Aggregate signatures are useful for reducing the size of certificate chains (by aggregating all signatures in the chain) and for reducing message size in secure routing protocols such as SBGP. We also show that aggregate signatures give rise to verifiably encrypted signatures. Such signatures enable the verifier to test that a given ciphertext C is the encryption of a signature on a given message M . Verifiably encrypted signatures are used in contract-signing protocols. Finally, we show that similar ideas can be used to extend the short signature scheme to give simple ring signatures.

Abstract. This paper presents efficient off-line anonymous e-cash schemes where a user can withdraw a wallet containing 2 ℓ coins each of which she can spend unlinkably. Our first result is a scheme, secure under the strong RSA and the y-DDHI assumptions, where the complexity of the withdrawal and spend operations is O(ℓ + k) andtheuser’s wallet can be stored using O(ℓ + k) bits,wherek is a security parameter. The best previously known schemes require at least one of these complexities to be O(2 ℓ · k). In fact, compared to previous e-cash schemes, our whole wallet of 2 ℓ coins has about the same size as one coin in these schemes. Our scheme also offers exculpability of users, that is, the bank can prove to third parties that a user has double-spent. We then extend our scheme to our second result, the first e-cash scheme that provides traceable coins without a trusted third party. That is, once a user has double spent one of the 2 ℓ coins in her wallet, all her spendings of these coins can be traced. However, the price for this is that the complexity of the spending and of the withdrawal protocols becomes O(ℓ · k) and O(ℓ · k + k 2) bits, respectively, and wallets take O(ℓ · k) bitsofstorage. All our schemes are secure in the random oracle model.

We create a credential system that lets a user anonymously authenticate at most n times in a single time period. A user withdraws a dispenser of n e-tokens. She shows an e-token to a verifier to authenticate herself; each e-token can be used only once, however, the dispenser automatically refreshes every time period. The only prior solution to this problem, due to Damg˚ard et al. [30], uses protocols that are a factor of k slower for the user and verifier, where k is the security parameter. Damg˚ard et al. also only support one authentication per time period, while we support n. Because our construction is based on e-cash, we can use existing techniques to identify a cheating user, trace all of her e-tokens, and revoke her dispensers. We also offer a new anonymity service: glitch protection for basically honest users who (occasionally) reuse etokens. The verifier can always recognize a reused e-token; however, we preserve the anonymity of users who do not reuse e-tokens too often. 1

A zap is a two-round, witness-indistinguishable protocol in which the first round, consisting of a message from the verifier to the prover, can be fixed "once-and-for-all" and applied to any instance, and where the verifier does not use any private coins. We present a zap for every language in NP, based on the existence of non-interactive zero-knowledge proofs in the shared random string model. The zap is in the standard model, and hence requires no common guaranteed random string.

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Abstract. A signature system is said to be strongly unforgeable if the signature is existentially unforgeable and, given signatures on some message m the adversary cannot produce a new signature on m. Strongly unforgeable signatures are used for constructing chosen ciphertext secure systems and group signatures. Current efficient constructions in the standard model (i.e. without random oracles) depend on relatively strong assumptions such as Strong-RSA or Strong-Diffie-Hellman. We construct an efficient strongly unforgeable signature system based on the standard Computational Diffie-Hellman problem in bilinear groups. 1

Abstract. This paper introduces simulatable verifiable random functions (sVRF). VRFs are similar to pseudorandom functions, except that they are also verifiable: corresponding to each seed SK, there is a public key PK, and for y = FPK(x), it is possible to prove that y is indeed the value of the function seeded by SK. A simulatable VRF is a VRF for which this proof can be simulated, so a simulator can pretend that the value of FPK(x) is any y. Our contributions are as follows. We introduce the notion of sVRF. We give two constructions: one from general assumptions (based on NIZK), but inefficient, just as a proof of concept; the other construction is practical and based on a special assumption about composite-order groups with bilinear maps. We then use an sVRF to get a direct transformation from a single-theorem non-interactive zero-knowledge proof system for a language L to a multi-theorem non-interactive proof system for the same language L. 1

Digital signatures provide authenticity and nonrepudiation. They are a standard cryptographic primitive with many applications in higher-level protocols. Groups featuring a computable bilinear map are particularly well suited for signature-related primitives. For some signature variants the only construction known uses bilinear maps. Where constructions based on, e.g., RSA are known, bilinear-map–based constructions are simpler, more efficient, and yield shorter signatures. We describe several constructions that support this claim. First, we present the Boneh-Lynn-Shacham (BLS) short signature scheme. BLS signatures with 1024-bit security are 160 bits long, the shortest of any scheme based on standard assumptions. Second, we present Boneh-Gentry-Lynn-Shacham (BGLS) aggregate signatures. In an aggregate signature scheme it is possible to combine n signatures on n distinct messages from n distinct users into a single aggregate that provides nonrepudiation for all of them. BGLS aggregates are 160 bits long, regardless of how many signatures are aggregated. No construction is known for aggregate signatures that does not employ bilinear maps. BGLS aggregates give rise to verifiably encrypted signatures, a signature variant with applications in contract signing.

In an electronic cash (e-cash) system, a user can withdraw coins from the bank, and then spend each coin anonymously and unlinkably. For some applications, it is desirable to set a limit on the dollar amounts of anonymous transactions. For example, governments require that large transactions be reported for tax purposes. In this work, we present the first e-cash system that makes this possible without a trusted party. In our system, a user’s anonymity is guaranteed so long as she does not: (1) double-spend a coin, or (2) exceed the publicly-known spending limit with any merchant. The spending limit may vary with the merchant. Violation of either condition can be detected, and can (optionally) lead to identification of the user and discovery of her other activities. While it is possible to balance accountability and privacy this way using e-cash, this is impossible to do using regular cash. Our scheme is based on our recent compact e-cash system. It is secure under the same complexity assumptions in the random-oracle model. We inherit its efficiency: 2 ℓ coins can be stored in O(ℓ + k) bitsandthe complexity of the withdrawal and spend protocols is O(ℓ + k), where k is the security parameter.

We propose a new methodology for rational secret sharing leading to various instantiations (in both the two-party and multi-party settings) that are simple and efficient in terms of computation, share size, and round complexity. Our protocols do not require physical assumptions or simultaneous channels, and can even be run over asynchronous, point-to-point networks. We also propose new equilibrium notions (namely, computational versions of strict Nash equilibrium and stability with respect to trembles) and prove that our protocols satisfy them. These notions guarantee, roughly speaking, that at each point in the protocol there is a unique legal message a party can send. This, in turn, ensures that protocol messages cannot be used as subliminal channels, something achieved in prior work only by making strong assumptions on the communication network. 1