International Association
for Cryptologic Research

Zero-knowledge proof (ZKP) is a cryptographic primitive that enables a prover to convince a verifier that a statement is true, without revealing any other information beyond the correctness of the statement itself. Due to its powerful capabilities, its most practical type, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK), has been widely deployed in various privacypreserving applications such as cryptocurrencies and verifiable computation. Although state-of-the-art zkSNARKs are highly efficient for the verifier, the computational overhead for the prover is still orders of magnitude too high to warrant use in many applications. This overhead arises from several time-consuming operations, including large-scale matrix-vector multiplication (MUL), number-theoretic transform (NTT), and especially the multi-scalar multiplication (MSM) which constitutes the largest proportion. Therefore, further efficiency improvements are needed.In this paper, we focus on comprehensive optimization of running time and storage space required by the MSM algorithm on GPUs. Specifically, we propose a novel, modular and adaptive parameter configuration technique—elastic MSM to enable us to adjust the scale of MSM according to our own wishes by performing a corresponding amount of preprocessing. This technique enables us to fully unleash the potential of various efficient parallel MSM algorithms. We have implemented and tested elastic MSM over three prevailing parallel Pippenger algorithms on GPUs. Across various preprocessing space limitations (across various MSM scales), our constructions achieve up to about 1.90×, 1.08× and 1.36× (2.58×, 1.39× and 1.91×) speedup versus three state-of-the-art parallel Pippenger algorithms on GPUs, respectively.From another perspective, elastic MSM could also be regarded as a preprocessing technique over the well-known Pippenger algorithm, which is modular and could be used to accelerate almost all the most advanced parallel Pippenger algorithms on GPUs. Meanwhile, elastic MSM provides an adaptive trade-off between the running time and the extra storage space needed by parallel Pippenger algorithms on GPUs. This is the first preprocessing technique to retain the improved MSM computation brought by preprocessing under varying storage space limitations. Specifically, across various preprocessing space limitations (across various MSM scales), our constructions achieve up to about 192× and 223× (159× and 174×) speedup versus two state-ofthe- art preprocessing parallel Pippenger algorithms on GPUs, respectively.

Recent attacks on NTRU lattices given by Ducas and van Woerden (ASIACRYPT 2021) showed that for moduli $q$ larger than the so-called fatigue point $n^{2.484+o(1)}$, the security of NTRU is noticeably less than that of (ring)-LWE. Unlike NTRU-based PKE with $q$ typically lying in the secure regime of NTRU lattices (i.e., $q<n^{2.484+o(1)}$), the security of existing NTRU-based multi-key FHEs (MK-FHEs) requiring $q=O(n^k)$ for $k$ keys could be significantly affected by those attacks. In this paper, we first propose a (matrix) NTRU-based MK-FHE for super-constant number $k$ of keys without using overstretched NTRU parameters. Our scheme is essentially a combination of two components following the two-layer framework of TFHE/FHEW: - a simple first-layer matrix NTRU-based encryption which naturally supports multi-key NAND operations with moduli $q=O(k\cdot n^{1.5})$ only linear in the number $k$ of keys; - and a crucial second-layer NTRU-based encryption which supports efficient hybrid product between a single-key ciphertext and a multi-key ciphertext for gate bootstrapping. Then, by replacing the first-layer with a more efficient LWE-based multi-key encryption, we obtain an improved MK-FHE scheme with better performance. We also employ a light key-switching technique to reduce the key-switching key size from previous $O(n^2)$ bits to $O(n)$ bits. A proof-of-concept implementation shows that our two MK-FHE schemes outperform the state-of-the-art TFHE-like MK-FHE schemes in both computation efficiency and bootstrapping key size. Concretely, for $k=8$ at the same 100-bit security level, our improved MK-FHE scheme can bootstrap a ciphertext in {0.54s} on a laptop and only has a bootstrapping key of size {13.89}MB, which are respectively 2.2 times faster and 7.4 times smaller than the MK-FHE scheme (which relies on a second-layer encryption from the ring-LWE assumption) due to Chen, Chillotti and Song (ASIACRYPT 2019).

Blind rotation is one of the key techniques to construct fully homomorphic encryptions with the best known bootstrapping algorithms running in less than one second. Currently, the two main approaches, namely, AP and GINX, for realizing blind rotation are first introduced by Alperin-Sheriff and Peikert (CRYPTO 2014) and Gama, Izabachene, Nguyen and Xie (EUROCRYPT 2016), respectively. In this paper, we propose a new blind rotation algorithm based on a GSW-like encryption from the NTRU assumption. Our algorithm has performance asymptotically independent from the key distributions, and outperforms AP and GINX in both the evaluation key size and the computational efficiency (especially for large key distributions). By using our blind rotation algorithm as a building block, we present new bootstrapping algorithms for both LWE and RLWE ciphertexts. We implement our bootstrapping algorithm for LWE ciphertexts, and compare the actual performance with two bootstrapping algorithms, namely, FHEW/AP by Ducas and Micciancio (EUROCRYPT 2015) and TFHE/GINX by Chillotti, Gama, Georgieva and Izabach\`ene (Journal of Cryptology 2020), that were implemented in the OpenFHE library. For parameters with ternary key distribution at 128-bit security, our bootstrapping only needs to store evaluation key of size 18.65MB for blind rotation, which is about 89.8 times smaller than FHEW/AP and 2.9 times smaller than TFHE/GINX. Moreover, our bootstrapping can be done in 112ms on a laptop, which is about 3.2 times faster than FHEW/AP and 2.1 times faster than TFHE/GINX. More improvements are available for large key distributions such as Gaussian distributions.

Zero-knowledge elementary databases (ZK-EDBs) enable a prover to commit a database $D$ of key-value $(x,v)$ pairs and later provide a convincing answer to the query ``send me the value $D(x)$ associated with $x$'' without revealing any extra knowledge (including the size of $D$). After its introduction, several works extended it to allow more expressive queries, but the expressiveness achieved so far is still limited: only a relatively simple queries--range queries over the keys and values-- can be handled by known constructions. In this paper we introduce a new notion called zero knowledge functional elementary databases (ZK-FEDBs), which allows the most general functional queries. Roughly speaking, for any Boolean circuit $f$, ZK-FEDBs allows the ZK-EDB prover to provide convincing answers to the queries of the form ``send me all records ${(x,v)}$ in ${{D}}$ satisfying $f(x,v)=1$,'' without revealing any extra knowledge (including the size of ${D}$). We present a construction of ZK-FEDBs in the random oracle model and generic group model, whose proof size is only linear in the length of record and the size of query circuit, and is independent of the size of input database $D$. Our technical constribution is two-fold. Firstly, we introduce a new variant of zero-knowledge sets (ZKS) which supports combined operations on sets. We present a concrete construction that is based on groups with unknown order. Secondly, we develop a tranformation that tranforms the query of Boolean circuit into a query of combined operations on related sets, which may be of independent interest.

We introduce a new notion of public key encryption, knowledge encryption, for which its ciphertexts can be reduced to the public-key, i.e., any algorithm that can break the ciphertext indistinguishability can be used to extract the (partial) secret key. We show that knowledge encryption can be built solely on any two-round oblivious transfer with game-based security, which are known based on various standard (polynomial-hardness) assumptions, such as the DDH, the Quadratic($N^{th}$) Residuosity or the LWE assumption. We use knowledge encryption to construct the first three-round (weakly) simulatable oblivious transfer. This protocol satisfies (fully) simulatable security for the receiver, and weakly simulatable security ($(T,\epsilon)$-simulatability) for the sender in the following sense: for any polynomial $T$ and any inverse polynomial $\epsilon$, there exists an efficient simulator such that the distinguishing gap of any distinguisher of size less than $T$ is at most $\epsilon$. Equipped with these tools, we construct a variety of fundamental cryptographic protocols with low round-complexity, assuming only the existence of two-round oblivious transfer with game-based security. These protocols include three-round delayed-input weak zero knowledge argument, three-round weakly secure two-party computation, three-round concurrent weak zero knowledge in the BPK model, and a two-round commitment with weak security under selective opening attack. These results improve upon the assumptions required by the previous constructions. Furthermore, all our protocols enjoy the above $(T,\epsilon)$-simulatability (stronger than the distinguisher-dependent simulatability), and are quasi-polynomial time simulatable under the same (polynomial hardness) assumption.

We formally study “non-malleable functions” (NMFs), a general cryptographic primitive which simplifies and relaxes “non-malleable one-way/hash functions” (NMOWHFs) introduced by Boldyreva et al. (in: Advances in cryptology—ASIACRYPT 2009, pp 524–541, 2009) and refined by Baecher et al. (in: CT-RSA 2011, pp 268–283, 2011). NMFs focus on basic functions, rather than one-way/hash functions considered in the literature of NMOWHFs. We formalize a game-based definition for NMFs. Roughly, a function f is non-malleable if given an image $$y^* \leftarrow f(x^*)$$ y ∗ ← f ( x ∗ ) for a randomly chosen $$x^*$$ x ∗ , it is hard to output a value y together with a transformation $$\phi $$ ϕ from some prefixed transformation class such that y is an image of $$\phi (x^*)$$ ϕ ( x ∗ ) under f . Our non-malleable notion is strong in the sense that only trivial copy solution $$(\mathsf {id}, y^*)$$ ( id , y ∗ ) is forbidden, where $$\mathsf {id}$$ id is the identity transformation. We also consider the adaptive notion, which stipulates that non-malleability holds even when an inversion oracle is available. We investigate the relations between non-malleability and one-wayness in depth. In the non-adaptive setting, we show that non-malleability generally implies one-wayness for poly-to-one functions but not vice versa. In the adaptive setting, we show that for most algebra-induced transformation classes, adaptive non-malleability (ANM) is equivalent to adaptive one-wayness (AOW) for injective functions. These results establish theoretical connections between non-malleability and one-wayness for functions and extend to trapdoor functions as well, resolving the open problems left by Kiltz et al. (in: Advances in cryptology—EUROCRYPT 2010, pp 673–692, 2010). We also study the relations between standard OW/NM and hinting OW/NM, where the latter notions are typically more useful in practice. Toward efficient realizations of NMFs, we give a deterministic construction from adaptive trapdoor functions as well as a randomized construction from all-but-one lossy functions and one-time signature. This partially solves an open problem posed by Boldyreva et al. (in: Advances in cryptology—ASIACRYPT 2009, pp 524–541, 2009). Finally, we explore applications of NMFs in security against related-key attacks (RKA). We first show that, somewhat surprisingly, the implication AOW $$\Rightarrow $$ ⇒ ANM sheds light on addressing non-trivial copy attacks in RKA security. We then show that NMFs give rise to a generic construction of RKA-secure authenticated key derivation functions, which have proven to be very useful in achieving RKA security for numerous cryptographic primitives. Particularly, our construction simplifies and unifies the result due to Qin et al. (in: Public-key cryptography—PKC 2015, volume 9020 of LNCS. Springer, Berlin, pp 557–578, 2015).

Threshold Signatures allow $n$ parties to share the ability of issuing digital signatures so that any coalition of size at least $t+1$ can sign, whereas groups of $t$ or less players cannot. The currently known class-group-based threshold ECDSA constructions are either inefficient (requiring parallel-repetition of the underlying zero knowledge proof with small challenge space) or requiring rather non-standard assumptions. In this paper, under \emph{standard assumptions} we present efficient threshold ECDSA protocols from encryption schemes based on class groups \emph{without parallel repeating the underlying zero knowledge proof}, yielding a significant efficiency improvement in the key generation over previous constructions (even those based on non-standard assumptions). Along the way we introduce a new notion of \emph{promise} $\Sigma$-protocol that satisfies only a weaker soundness called \emph{promise extractability}. An accepting \emph{promise} $\Sigma$-proof for statements related to class-group-based encryptions does not establish the truth of the statement but provides security guarantees (promise extractability) that are sufficient for our applications. We also show how to simulate homomorphic operations on a (possibly invalid) class-group-based encryption whose correctness has been proven via our \emph{promise} $\Sigma$-protocol. We believe that these techniques are of independent interest and applicable to other scenarios where efficient zero knowledge proofs for statements related to class-group is required.

We develop an individual simulation technique that explicitly makes use of particular properties/structures of a given adversary's functionality. Using this simulation technique, we obtain the following results. 1. We construct the first protocols that break previous black-box barriers under the standard hardness of factoring, both of which are polynomial time simulatable against all a-priori bounded polynomial size distinguishers: a)Two-round selective opening secure commitment scheme. b)Three-round concurrent zero knowledge and concurrent witness hiding argument for NP in the bare public-key model. 2. We present a simpler two-round weak zero knowledge and witness hiding argument for NP in the plain model under the sub-exponential hardness of factoring. Our technique also yields a significantly simpler proof that existing distinguisher-dependent simulatable zero knowledge protocols are also polynomial time simulatable against all distinguishers of a-priori bounded polynomial size. The core conceptual idea underlying our individual simulation technique is an observation of the existence of nearly optimal extractors for all hard distributions: For any NP-instance(s) sampling algorithm, there exists a polynomial-size witness extractor (depending on the sampler's functionality) that almost outperforms any circuit of a-priori bounded polynomial size in terms of the success probability.

We revisit the problem of whether the known classic constant-round public-coin argument/proof systems are witness hiding for languages/distributions with unique witnesses. Though strong black-box impossibility results are known, we provide some less unexpected positive results on the witness hiding security of these classic protocols:We give sufficient conditions on a hard distribution over unique witness NP relation for which all witness indistinguishable protocols (including all public-coin ones, such as ZAPs, Blum protocol and GMW protocol) are indeed witness hiding. We also show a wide range of cryptographic problems with unique witnesses satisfy these conditions, and thus admit constant-round public-coin witness hiding proof system.For the classic Schnorr protocol (for which the distribution of statements being proven seems not to satisfy the above sufficient conditions), we develop an embedding technique and extend the result of Bellare and Palacio to base the witness hiding property of the Schnorr protocol in the standalone setting on a relaxed version of one-more like discrete logarithm (DL) assumption, which essentially assumes there does not exist instance compression scheme for the DL problem, and show that breaking this assumption would lead to some surprising consequences, such as zero knowledge protocols for the AND-DL language with extremely efficient communication and highly non-trivial hash combiner for hash functions based on the DL problem. Similar results hold for the Guillou-Quisquater protocol.