International Association
for Cryptologic Research

Consider a coin tossing protocol in which n processors P_1,...,P_n agree on a random bit b in n rounds, where in round i P_i sends a single message w_i. Imagine a full-information adversary who prefers the output 1, and in every round i it knows all the finalized messages w_1,...,w_{i-1} so far as well as the prepared message w_i. A k-replacing attack will have a chance to replace the prepared w_i with its own choice w'_i \neq w_i in up to k rounds. Taking majority protocol over uniformly random bits w_i = b_i is robust in the following strong sense. Any k-replacing adversary can only increase the probability of outputting 1 by at most O(k/\sqrt{n}). In this work, we ask if the above simple protocol is tight. For the same setting, but restricted to uniformly random bit messages, Lichtenstein, Linial, and Saks [Combinatorica'89] showed how to achieve bias \Omega(k/\sqrt{n}) for any k \in [n]. Kalai, Komargodski, and Raz [DISC'18, Combinatorica'21] gave an alternative polynomial-time attack when k \geq \Theta(\sqrt{n}). Etesami, Mahloujifar, and Mahmoody [ALT'19, SODA'20] extended the result of KKR18 to arbitrary long messages. In this work, we resolve both of these problems. - For arbitrary length messages, we show that k-replacing polynomial-time attacks can indeed increase the probability of outputting 1 by \Omega(k/\sqrt{n}) for any k, which is optimal up to a constant factor. By plugging in our attack into the framework of Mahloujifar Mahmoody [TCC'17] we obtain similar data poisoning attacks against deterministic learners when adversary is limited to changing k=o(\sqrt{n}) of the n training examples. - For uniformly random bits b_1,...,b_n, we show that whenever Pr[b=1]=Pr[\sum b_i \geq t]=\beta[t]_n for t \in [n] is the probability of a Hamming ball, then online polynomial-time k-replacing attacks can increase Pr[b=1] from \beta[t]_n to \beta[t-k]_n , which is optimal due to the majority protocol. In comparison, the (information-theoretic) attack of LLS89 increased Pr[b=1] to \beta[t-k]_{n-k}, which is optimal for adaptive adversaries who cannot see the message before changing it. Thus, we obtain a computational variant of Harper's celebrated vertex isoperimetric inequality.

The notion of Registration-Based Encryption (RBE) was recently introduced by Garg, Hajiabadi, Mahmoody, and Rahimi [TCC’18] with the goal of removing the private-key generator (PKG) from IBE. Specifically, RBE allows encrypting to identities using a (compact) master public key, like how IBE is used, with the benefit that the PKG is substituted with a weaker entity called “key curator” who has no knowledge of any secret keys. Here individuals generate their secret keys on their own and then publicly register their identities and their corresponding public keys to the key curator. Finally, individuals obtain “rare” decryption-key updates from the key curator as the population grows. In their work, they gave a construction of RBE schemes based on the combination of indistinguishability obfuscation and somewhere statistically binding hash functions. However, they left open the problem of constructing RBE schemes based on standard assumptions.In this work, we resolve the above problem and construct RBE schemes based on standard assumptions (e.g., CDH or LWE). Furthermore, we show a new application of RBE in a novel context. In particular, we show that anonymous variants of RBE (which we also construct under standard assumptions) can be used for realizing abstracts forms of anonymous messaging tasks in simple scenarios in which the parties communicate by writing messages on a shared board in a synchronized way.

Understanding whether public-key encryption can be based on one-way functions is a fundamental open problem in cryptography. The seminal work of Impagliazzo and Rudich [STOC’89] shows that black-box constructions of public-key encryption from one-way functions are impossible. However, this impossibility result leaves open the possibility of using non-black-box techniques for achieving this goal.One of the most powerful classes of non-black-box techniques, which can be based on one-way functions (OWFs) alone, is Yao’s garbled circuit technique [FOCS’86]. As for the non-black-box power of this technique, the recent work of Döttling and Garg [CRYPTO’17] shows that the use of garbling allows us to circumvent known black-box barriers in the context of identity-based encryption.We prove that garbling of circuits that have OWF (or even random oracle) gates in them are insufficient for obtaining public-key encryption. Additionally, we show that this model also captures (non-interactive) zero-knowledge proofs for relations with OWF gates. This indicates that currently known OWF-based non-black-box techniques are perhaps insufficient for realizing public-key encryption.

We show that any OT extension protocol based on one-way functions (or more generally any symmetric-key primitive) either requires an additional round compared to the base OTs or must make a non-black-box use of one-way functions. This result also holds in the semi-honest setting or in the case of certain setup models such as the common random string model. This implies that OT extension in any secure computation protocol must come at the price of an additional round of communication or the non-black-box use of symmetric key primitives. Moreover, we observe that our result is tight in the sense that positive results can indeed be obtained using non-black-box techniques or at the cost of one additional round of communication.

In this work, we introduce the notion of registration-based encryption (RBE for short) with the goal of removing the trust parties need to place in the private-key generator in an IBE scheme. In an RBE scheme, users sample their own public and secret keys. There will also be a “key curator” whose job is only to aggregate the public keys of all the registered users and update the “short” public parameter whenever a new user joins the system. Encryption can still be performed to a particular recipient using the recipient’s identity and any public parameters released subsequent to the recipient’s registration. Decryption requires some auxiliary information connecting users’ public (and secret) keys to the public parameters. Because of this, as the public parameters get updated, a decryptor may need to obtain “a few” additional auxiliary information for decryption. More formally, if n is the total number of identities and $$\mathrm {\kappa }$$κ is the security parameter, we require the following.Efficiency requirements: (1) A decryptor only needs to obtain updated auxiliary information for decryption at most $$O(\log n)$$O(logn) times in its lifetime, (2) each of these updates are computed by the key curator in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn), and (3) the key curator updates the public parameter upon the registration of a new party in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). Properties (2) and (3) require the key curator to have random access to its data.Compactness requirements: (1) Public parameters are always at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bit, and (2) the total size of updates a user ever needs for decryption is also at most $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) bits.We present feasibility results for constructions of RBE based on indistinguishably obfuscation. We further provide constructions of weakly efficient RBE, in which the registration step is done in $${\text {poly}}(\mathrm {\kappa },n)$$poly(κ,n), based on CDH, Factoring or LWE assumptions. Note that registration is done only once per identity, and the more frequent operation of generating updates for a user, which can happen more times, still runs in time $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn). We leave open the problem of obtaining standard RBE (with $${\text {poly}}(\mathrm {\kappa },\log n)$$poly(κ,logn) registration time) from standard assumptions.