We explain how to extend the Bitcoin backbone model of Garay et al. (Eurocrypt, 2015) to accommodate for redactable blockchains. Our extension captures fluid blockchain-based databases (with mutability requirements) and compliance with existing legislation, such as the GDPR right to be forgotten, or the need to erase offending data from nodes’ databases that would otherwise provoke legal shutdowns. Our redactable backbone protocol retains the essential properties of blockchains. Leveraging zero-knowledge proofs, old data can be erased without requiring trusted third parties or heuristics about past chain validation. Our solution can be implemented on Bitcoin immediately without hard-forks, and it is scalable. It allows the redaction of data from UTXOs or unconfirmed transactions that have not yet flooded the network, while guaranteeing invariance of the Bitcoin state. Thus, offending data does not need to persist in the system, not even temporarily.

We present a unified view of the two-party and multi-party computation protocols based on oblivious transfer first outlined in Nielsen et al. (CRYPTO 2012) and Larraia et al. (CRYPTO 2014). We present a number of modifications and improvements to these earlier presentations, as well as full proofs of the entire protocol. Improvements include a unified pre-processing and online MAC methodology, mechanisms to pass between different MAC variants and fixing a minor bug in the protocol of Larraia et al. in relation to a selective failure attack. It also fixes a minor bug in Nielsen et al. resulting from using Jensen’s inequality in the wrong direction in an analysis.

Abstract We provide constructions of multilinear groups equipped with natural hard problems from indistinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the $${\text {DDH}} $$ DDH assumption hold for them. Our first construction is symmetric and comes with a $$\kappa $$ κ -linear map $$\mathbf{e }: {{\mathbb {G}}}^\kappa \longrightarrow {\mathbb {G}}_T$$ e:Gκ⟶GT for prime-order groups $${\mathbb {G}}$$ G and $${\mathbb {G}}_T$$ GT . To establish the hardness of the $$\kappa $$ κ -linear $${\text {DDH}} $$ DDH problem, we rely on the existence of a base group for which the $$\kappa $$ κ -strong $${\text {DDH}} $$ DDH assumption holds. Our second construction is for the asymmetric setting, where $$\mathbf{e }: {\mathbb {G}}_1 \times \cdots \times {\mathbb {G}}_{\kappa } \longrightarrow {\mathbb {G}}_T$$ e:G1×⋯×Gκ⟶GT for a collection of $$\kappa +1$$ κ+1 prime-order groups $${\mathbb {G}}_i$$ Gi and $${\mathbb {G}}_T$$ GT , and relies only on the 1-strong $${\text {DDH}} $$ DDH assumption in its base group. In both constructions, the linearity $$\kappa $$ κ can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: probabilistic indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness-indistinguishability, and zero knowledge), and additively homomorphic encryption for the group $$\mathbb {Z}_N^{+}$$ ZN+ . At a high level, we enable “bootstrapping” multilinear assumptions from their simpler counterparts in standard cryptographic groups and show the equivalence of PIO and multilinear maps under the existence of the aforementioned primitives.

SPDZ (pronounced “Speedz”) is the nickname of the MPC protocol of Damgård et al. from Crypto 2012. In this paper we both resolve a number of open problems with SPDZ; and present several theoretical and practical improvements to the protocol. In detail, we start by designing and implementing a covertly secure key generation protocol for obtaining a BGV public key and a shared associated secret key. We then construct both a covertly and actively secure preprocessing phase, both of which compare favourably with previous work in terms of efficiency and provable security. We also build a new online phase, which solves a major problem of the SPDZ protocol: namely prior to this work preprocessed data could be used for only one function evaluation and then had to be recomputed from scratch for the next evaluation, while our online phase can support reactive functionalities. This improvement comes mainly from the fact that our construction does not require players to reveal the MAC keys to check correctness of MAC’d values. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.