Switching Lemma for Bilinear Tests and Constant-size NIZK Proofs for Linear Subspaces

We state a switching lemma for tests on adversarial inputs involving bilinear pairings in hard groups, where the tester can effectively switch the randomness used in the test from being given to the adversary at the outset to being chosen after the adversary commits its input. The switching lemma can be based on any $k$-linear hardness assumptions on one of the groups. In particular, this enables convenient information theoretic arguments in the construction of sequence of games proving security of cryptographic schemes, mimicking proofs and constructions in the random oracle model. As an immediate application, we show that the quasi-adaptive NIZK proofs of Jutla and Roy [AsiaCrypt 2013] for linear subspaces can be further shortened to \emph{constant}-size proofs, independent of the number of witnesses and equations. In particular, under the XDH assumption, a length $n$ vector of group elements can be proven to belong to a subspace of rank $t$ with a quasi-adaptive NIZK proof consisting of just a single group element. Similar quasi-adaptive aggregation of proofs is also shown for Groth-Sahai NIZK proofs of linear multi-scalar multiplication equations, as well as linear pairing-product equations (equations without any quadratic terms)

Shorter Quasi-Adaptive NIZK Proofs for Linear Subspaces

We define a novel notion of quasi-adaptive non-interactive zero knowledge (NIZK) proofs for probability distributions on parametrized languages. It is quasi-adaptive in the sense that the common reference string (CRS) generator can generate the CRS depending on the language parameters. However, the simulation is required to be uniform, i.e., a single efficient simulator should work for the whole class of parametrized languages. For distributions on languages that are linear subspaces of vector spaces over bilinear groups, we give quasi-adaptive computationally sound NIZKs that are shorter and more efficient than Groth-Sahai NIZKs. For many cryptographic applications quasi-adaptive NIZKs suffice, and our constructions can lead to significant improvements in the standard model. Our construction can be based on any k-linear assumption, and in particular under the eXternal Diffie Hellman (XDH) assumption our proofs are even competitive with Random-Oracle based Sigma-protocol NIZK proofs. We also show that our system can be extended to include integer tags in the defining equations, where the tags are provided adaptively by the adversary. This leads to applicability of our system to many applications that use tags, e.g. applications using Cramer-Shoup projective hash proofs. Our techniques also lead to the shortest known (ciphertext) fully secure identity based encryption (IBE) scheme under standard static assumptions (SXDH). Further, we also get a short publicly-verifiable CCA2-secure IBE scheme

CRS-Updatable Asymmetric Quasi-Adaptive NIZK Arguments

A critical aspect for the practical use of non-interactive zero-knowledge (NIZK) arguments in the common reference string (CRS) model is the demand for a trusted setup, i.e., a trusted generation of the CRS. Recently, motivated by its increased use in real-world applications, there has been a growing interest in concepts that allow to reduce the trust in this setup. In particular one demands that the zero-knowledge and ideally also the soundness property hold even when the CRS generation is subverted. One important line of work in this direction is the so-called updatable CRS for NIZK by Groth et al. (CRYPTO’18). The basic idea is that everyone can update a CRS and there is a way to check the correctness of an update. This guarantees that if at least one operation (the generation or one update) have been performed honestly, the zero-knowledge and the soundness properties hold. Later, Lipmaa (SCN’20) adopted this notion of updatable CRS to quasi-adaptive NIZK (QA-NIZK) arguments. In this work, we continue the study of CRS-updatable QA-NIZK and analyse the most efficient asymmetric QA-NIZKs by González et al. (ASIACRYPT’15) in a setting where the CRS is fully subverted and propose an updatable version of it. In contrast to the updatable QA- NIZK by Lipmaa (SCN’20) which represents a symmetric QA-NIZK and requires a new non-standard knowledge assumption for the subversion zero-knowledge property, our technique to construct updatable asymmetric QA-NIZK is under a well-known standard knowledge assumption, i.e., the Bilinear Diffie-Hellman Knowledge of Exponents assumption. Furthermore, we show the knowledge soundness of the (updatable) asymmetric QA-NIZKs, an open problem posed by Lipmaa, which makes them compatible with modular zk-SNARK frameworks such as LegoS- NARK by Campanelli et al. (ACM CCS’19)

Short Group Signatures via Structure-Preserving Signatures: Standard Model Security from Simple Assumptions

International audienceGroup signatures are a central cryptographic primitive which allows users to sign messages while hiding their identity within a crowd of group members. In the standard model (without the random oracle idealization), the most efficient constructions rely on the Groth-Sahai proof systems (Euro-crypt'08). The structure-preserving signatures of Abe et al. (Asiacrypt'12) make it possible to design group signatures based on well-established, constant-size number theoretic assumptions (a.k.a. " simple assumptions ") like the Symmetric eXternal Diffie-Hellman or Decision Linear assumptions. While much more efficient than group signatures built on general assumptions, these constructions incur a significant overhead w.r.t. constructions secure in the idealized random oracle model. Indeed, the best known solution based on simple assumptions requires 2.8 kB per signature for currently recommended parameters. Reducing this size and presenting techniques for shorter signatures are thus natural questions. In this paper, our first contribution is to significantly reduce this overhead. Namely, we obtain the first fully anonymous group signatures based on simple assumptions with signatures shorter than 2 kB at the 128-bit security level. In dynamic (resp. static) groups, our signature length drops to 1.8 kB (resp. 1 kB). This improvement is enabled by two technical tools. As a result of independent interest, we first construct a new structure-preserving signature based on simple assumptions which shortens the best previous scheme by 25%. Our second tool is a new method for attaining anonymity in the strongest sense using a new CCA2-secure encryption scheme which is simultaneously a Groth-Sahai commitment

Subversion-Resistant Quasi-Adaptive NIZK and Applications to Modular zk-SNARKs

Quasi-adaptive non-interactive zero-knowledge (QA-NIZK) arguments are NIZK arguments where the common reference string (CRS) is allowed to depend on the language and they can be very efficient for specific languages. Thus, they are for instance used within the modular LegoSNARK toolbox by Campanelli et al. (ACM CCS\u2719) as succinct NIZKs (aka zkSNARKs) for linear subspace languages. Such modular frameworks are interesting, as they provide gadgets for a flexible design of privacy-preserving blockchain applications. Recently, there has been an increasing interest to reduce the trust required in the generator of the CRS. One important line of work in this direction is subversion zero-knowledge by Bellare et al. (ASIACRYPT\u2716), where the zero-knowledge property even holds when the CRS is generated maliciously. In this paper, we firstly analyze the security of the most efficient QA-NIZK constructions of Kiltz and Wee (EUROCRYPT\u2715) and the asymmetric QA-NIZKs by Gonzalez et al. (ASIACRYPT\u2715) when the CRS is subverted and propose subversion versions of them. Secondly, for the first time, we construct unbounded (strong) true-simulation extractable (tSE) variants of them. Thirdly, we show how to integrate our subversion QA-NIZKs into the LegoSNARK toolbox, which so far does not consider subversion resistance. Our results together with existing results on (SE) subversion zk-SNARKS represent an important step towards a subversion variant of the LegoSNARK toolbox

Improved (Almost) Tightly-Secure Structure-Preserving Signatures

Structure Preserving Signatures (SPS) allow the signatures and the messages signed to be further encrypted while retaining the ability to be proven valid under zero-knowledge. In particular, SPS are tailored to have structure suitable for Groth-Sahai NIZK proofs. More precisely, the messages, signatures, and verification keys are required to be elements of groups that support efficient bilinear-pairings (bilinear groups), and the signature verification consists of just evaluating one or more bilinear-pairing product equations. Since Groth-Sahai NIZK proofs can (with zero-knowledge) prove the validity of such pairing product equations, it leads to interesting applications such as blind signatures, group signatures, traceable signatures, group encryption, and delegatable credential systems. In this paper, we further improve on the SPS scheme of Abe, Hofheinz, Nishimaki, Ohkubo and Pan (CRYPTO 2017) while maintaining only an $O(\lambda)$-factor security reduction loss to the SXDH assumption. In particular, we compress the size of the signatures by almost 40%, and reduce the number of pairing-product equations in the verifier from fifteen to seven. Recall that structure preserving signatures are used in applications by encrypting the messages and/or the signatures, and hence these optimizations are further amplified as proving pairing-product equations in Groth-Sahai NIZK system is not frugal. While our scheme uses an important novel technique introduced by Hofheinz (EuroCrypt 2017), i.e., structure-preserving adaptive partitioning, our approach to building the signature scheme is different and this leads to the optimizations mentioned. Thus we make progress towards an open problem stated by Abe et al (CRYPTO 2017) to design more compact SPS-es with smaller number of group elements

On QA-NIZK in the BPK Model

Recently, Bellare et al. defined subversion-resistance (security in the case the CRS creator may be malicious) for NIZK. In particular, a Sub-ZK NIZK is zero-knowledge, even in the case of subverted CRS. We study Sub-ZK QA-NIZKs, where the CRS can depend on the language parameter. First, we observe that subversion zero-knowledge (Sub-ZK) in the CRS model corresponds to no-auxiliary-string non-black-box NIZK in the Bare Public Key model, and hence, the use of non-black-box techniques is needed to obtain Sub-ZK. Second, we give a precise definition of Sub-ZK QA-NIZKs that are (knowledge-)sound if the language parameter but not the CRS is subverted and zero-knowledge even if both are subverted. Third, we prove that the most efficient known QA-NIZK for linear subspaces by Kiltz and Wee is Sub-ZK under a new knowledge assumption that by itself is secure in (a weaker version of) the algebraic group model. Depending on the parameter setting, it is (knowledge-)sound under different non-falsifiable assumptions, some of which do not belong to the family of knowledge assumptions

Smooth NIZK Arguments with Applications to Asymmetric UC-PAKE and Threshold-IBE

We introduce a novel notion of smooth (-verifier) non-interactive zero-knowledge proofs (NIZK) which parallels the familiar notion of smooth projective hash functions (SPHF). We also show that the recent single group element quasi-adaptive NIZK (QA-NIZK) of Jutla and Roy (CRYPTO 2014) for linear subspaces can be easily extended to be computationally smooth. One important distinction of the new notion from SPHFs is that in a smooth NIZK the public evaluation of the hash on a language member using the projection key does not require the witness of the language member, but instead just requires its NIZK proof. This has the remarkable consequence that in the Gennaro-Lindell paradigm of designing universally-composable password-authenticated key-exchange (UC-PAKE) protocols, if one replaces the traditionally employed SPHFs with the novel smooth QA-NIZK, one gets highly efficient UC-PAKE protocols that are secure even under dynamic corruption. This simpler and modular design methodology allows us to give the first single-round asymmetric UC-PAKE protocol, which is also secure under dynamic corruption in the erasure model. We also define a related concept of smooth signatures, which we show is black-box equivalent to identity-based encryption (IBE). The novel abstraction allows us to give the first threshold (private-key generation) fully-secure IBE in the standard model

Mitte-interaktiivsed nullteadmusprotokollid nõrgemate usalduseeldustega

Väitekirja elektrooniline versioon ei sisalda publikatsiooneTäieliku koosluskindlusega (TK) kinnitusskeemid ja nullteadmustõestused on ühed põhilisemad krüptograafilised primitiivid, millel on hulgaliselt päriselulisi rakendusi. (TK) Kinnitusskeem võimaldab osapoolel arvutada salajasest sõnumist kinnituse ja hiljem see verifitseeritaval viisil avada. Täieliku koosluskindlusega protokolle saab vabalt kombineerida teiste täieliku koosluskindlusega protokollidega ilma, et see mõjutaks nende turvalisust. Nullteadmustõestus on protokoll tõestaja ja verifitseerija vahel, mis võimaldab tõestajal veenda verifitseerijat mingi väite paikapidavuses ilma rohkema informatsiooni lekitamiseta. Nullteadmustõestused pakuvad suurt huvi ka praktilistes rakendustes, siinkohal on olulisemateks näideteks krüptorahad ja hajusandmebaasid üldisemalt. Siin on eriti asjakohased just lühidad mitteinteraktiivsed nullteadmustõestused (SNARKid) ning kvaasiadaptiivsed mitteinteraktiivsed nullteadmustõestused (QA-NIZKid). Mitteinteraktiivsetel nullteadmustõestustel juures on kaks suuremat praktilist nõrkust. Esiteks on tarvis usaldatud seadistusfaasi osapoolte ühisstringi genereerimiseks ja teiseks on tarvis täielikku koosluskindlust. Käesolevas doktoritöös me uurime neid probleeme ja pakume välja konkreetseid konstruktsioone nende leevendamiseks. Esmalt uurime me õõnestuskindlaid SNARKe juhu jaoks, kus seadistusfaasi ühisstring on õõnestatud. Me konstrueerime õõnestuskindla versiooni seni kõige tõhusamast SNARKist. Samuti uurime me QA-NIZKide õõnestuskindlust ja konstrueerime kõige efektiivsemate QA-NIZKide õõnestuskindla versiooni. Mis puutub teise uurimissuunda, nimelt täielikku koosluskindlusesse, siis sel suunal kasutame me pidevaid projektiivseid räsifunktsioone. Me pakume välja uue primitiivi, kus eelmainitud räsifunktsioonid on avalikult verifitseeritavad. Nende abil me konstrueerime seni kõige tõhusama mitteinteraktiivse koosluskindla kinnitusskeemi. Lõpetuseks me töötame välja uue võtte koosluskindlate kinnitusskeemide jaoks, mis võimaldab ühisarvutuse abil luua nullteadmustõestuste ühisstringe.Quite central primitives in cryptographic protocols are (Universally composable (UC)) commitment schemes and zero-knowledge proofs that getting frequently employed in real-world applications. A (UC) commitment scheme enables a committer to compute a commitment to a secret message, and later open it in a verifiable manner (UC protocols can seamlessly be combined with other UC protocols and primitives while the entire protocol remains secure). A zero-knowledge proof is a protocol usually between a prover and a verifier that allows the prover to convince the verifier of the legality of a statement without disclosing any more information. Zero-knowledge proofs and in particular Succinct non-interactive zero-knowledge proofs (SNARKs) and quasi adaptive NIZK (QA-NIZK) are of particular interest in the real-world applications, with cryptocurrencies or more generally distributed ledger technologies being the prime examples. The two serious issues and the main drawbacks of the practical usage of NIZKs are (i) the demand for a trusted setup for generating the common reference string (CRS) and (ii) providing the UC security. In this thesis, we essentially investigate the aforementioned issues and propose concrete constructions for them. We first investigate subversion SNARKs (Sub zk-SNARKs) when the CRS is subverted. In particular, we build a subversion of the most efficient SNARKs. Then we initiate the study of subversion QA-NIZK (Sub-QA-NIZK) and construct subversion of the most efficient QA-NIZKs. For the second issue, providing UC-security, we first using hash proof systems or smooth projective hash functions (SPHFs), we introduce a new cryptographic primitive called publicly computable SPHFs (PC-SPHFs) and construct the currently most efficient non-interactive UC-secure commitment. Finally, we develop a new technique for constructing UC-secure commitments schemes that enables one to generate CRS of NIZKs by using MPC in a UC-secure mannerhttps://www.ester.ee/record=b535926

Updatable Trapdoor SPHFs: Modular Construction of Updatable Zero-Knowledge Arguments and More

Recently, motivated by its increased use in real-world applications, there has been a growing interest on the reduction of trust in the generation of the common reference string (CRS) for zero-knowledge (ZK) proofs. This line of research was initiated by the introduction of subversion non-interactive ZK (NIZK) proofs by Bellare et al. (ASIACRYPT\u2716). Here, the zero-knowledge property needs to hold even in case of a malicious generation of the CRS. Groth et al. (CRYPTO\u2718) then introduced the notion of updatable zk-SNARKS, later adopted by Lipmaa (SCN\u2720) to updatable quasi-adaptive NIZK (QA-NIZK) proofs. In contrast to the subversion setting, in the updatable setting one can achieve stronger soundness guarantees at the cost of reintroducing some trust, resulting in a model in between the fully trusted CRS generation and the subversion setting. It is a promising concept, but all previous updatable constructions are ad-hoc and tailored to particular instances of proof systems. Consequently, it is an interesting question whether it is possible to construct updatable ZK primitives in a more modular way from simpler building blocks. In this work we revisit the notion of trapdoor smooth projective hash functions (TSPHFs) in the light of an updatable CRS. TSPHFs have been introduced by Benhamouda et al. (CRYPTO\u2713) and can be seen as a special type of a 2-round ZK proof system. In doing so, we first present a framework called lighter TSPHFs (L-TSPHFs). Building upon it, we introduce updatable L-TSPHFs as well as instantiations in bilinear groups. We then show how one can generically construct updatable quasi-adaptive zero-knowledge arguments from updatable L-TSPHFs. Our instantiations are generic and more efficient than existing ones. Finally, we discuss applications of (updatable) L-TSPHFs to efficient (updatable) 2-round ZK arguments as well as updatable password-authenticated key-exchange (uPAKE)