## CryptoDB

### Recently updated IACR publications

CryptoDB is periodically updated by manual and automatic processes. Whenever a paper is added or modified it will appear in this list, e.g., when a video appears.

A separate history of changes tracks schema and process changes. There is further information about CryptoDB in the documentation.

Year
Venue
Title
2019
TOSC
The Feistel construction is one of the most studied ways of building block ciphers. Several generalizations were then proposed in the literature, leading to the Generalized Feistel Network, where the round function first applies a classical Feistel operation in parallel on an even number of blocks, and then a permutation is applied to this set of blocks. In 2010 at FSE, Suzaki and Minematsu studied the diffusion of such construction, raising the question of how many rounds are required so that each block of the ciphertext depends on all blocks of the plaintext. They thus gave some optimal permutations, with respect to this diffusion criteria, for a Generalized Feistel Network consisting of 2 to 16 blocks, as well as giving a good candidate for 32 blocks. Later at FSE’19, Cauchois et al. went further and were able to propose optimal even-odd permutations for up to 26 blocks.In this paper, we complete the literature by building optimal even-odd permutations for 28, 30, 32, 36 blocks which to the best of our knowledge were unknown until now. The main idea behind our constructions and impossibility proof is a new characterization of the total diffusion of a permutation after a given number of rounds. In fact, we propose an efficient algorithm based on this new characterization which constructs all optimal even-odd permutations for the 28, 30, 32, 36 blocks cases and proves a better lower bound for the 34, 38, 40 and 42 blocks cases. In particular, we improve the 32 blocks case by exhibiting optimal even-odd permutations with diffusion round of 9. The existence of such a permutation was an open problem for almost 10 years and the best known permutation in the literature had a diffusion round of 10. Moreover, our characterization can be implemented very efficiently and allows us to easily re-find all optimal even-odd permutations for up to 26 blocks with a basic exhaustive search
2019
TOSC
In this paper we study the problem of recovering a secret S-box from its difference distribution table (DDT). While being an interesting theoretical problem on its own, the ability to recover the S-box from the DDT of a secret S-box can be used in cryptanalytic attacks where the attacker can obtain the DDT (e.g., in Bar-On et al.’s attack on GOST), in supporting theoretical analysis of the properties of difference distribution tables (e.g., in Boura et al.’s work), or in some analysis of S-boxes with unknown design criteria (e.g., in Biryukov and Perrin’s analysis).We show that using the well established relation between the DDT and the linear approximation table (LAT), one can devise an algorithm different from the straightforward guess-and-determine (GD) algorithm proposed by Boura et al. Moreover, we show how to exploit this relation, and embed the knowledge obtained from it in the GD algorithm. We tested our new algorithm on random S-boxes of different sizes, and for random 14-bit bijective S-boxes, our results outperform the GD attack by several orders of magnitude.
2016
ASIACRYPT
Best Paper
2019
EUROCRYPT
In this work, we study privacy-preserving storage primitives that are suitable for use in data analysis on outsourced databases within the differential privacy framework. The goal in differentially private data analysis is to disclose global properties of a group without compromising any individual’s privacy. Typically, differentially private adversaries only ever learn global properties. For the case of outsourced databases, the adversary also views the patterns of access to data. Oblivious RAM (ORAM) can be used to hide access patterns but ORAM might be excessive as in some settings it could be sufficient to be compatible with differential privacy and only protect the privacy of individual accesses.We consider $(\epsilon ,\delta )$(ϵ,δ)-Differentially Private RAM, a weakening of ORAM that only protects individual operations and seems better suited for use in data analysis on outsourced databases. As differentially private RAM has weaker security than ORAM, there is hope that we can bypass the $\varOmega (\log (nb/c))$Ω(log(nb/c)) bandwidth lower bounds for ORAM by Larsen and Nielsen [CRYPTO ’18] for storing an array of nb-bit entries and a client with c bits of memory. We answer in the negative and present an $\varOmega (\log (nb/c))$Ω(log(nb/c)) bandwidth lower bound for privacy budgets of $\epsilon = O(1)$ϵ=O(1) and $\delta \le 1/3$δ≤1/3.The information transfer technique used for ORAM lower bounds does not seem adaptable for use with the weaker security guarantees of differential privacy. Instead, we prove our lower bounds by adapting the chronogram technique to our setting. To our knowledge, this is the first work that uses the chronogram technique for lower bounds on privacy-preserving storage primitives.
2019
EUROCRYPT
Encrypt-then-MAC (EtM) is a popular mode for authenticated encryption (AE). Unfortunately, almost all designs following the EtM paradigm, including the AE suites for TLS, are vulnerable against nonce misuse. A single repetition of the nonce value reveals the hash key, leading to a universal forgery attack. There are only two authenticated encryption schemes following the EtM paradigm which can resist nonce misuse attacks, the GCM-RUP (CRYPTO-17) and the $\mathsf {GCM/2}^{+}$ (INSCRYPT-12). However, they are secure only up to the birthday bound in the nonce respecting setting, resulting in a restriction on the data limit for a single key. In this paper we show that nEHtM, a nonce-based variant of EHtM (FSE-10) constructed using a block cipher, has a beyond birthday bound (BBB) unforgeable security that gracefully degrades under nonce misuse. We combine nEHtM with the CENC (FSE-06) mode of encryption using the EtM paradigm to realize a nonce-based AE, CWC+. CWC+ is very close (requiring only a few more xor operations) to the CWC AE scheme (FSE-04) and it not only provides BBB security but also gracefully degrading security on nonce misuse.
2019
EUROCRYPT
Concrete security proofs give upper bounds on the attacker’s advantage as a function of its time/query complexity. Cryptanalysis suggests however that other resource limitations – most notably, the attacker’s memory – could make the achievable advantage smaller, and thus these proven bounds too pessimistic. Yet, handling memory limitations has eluded existing security proofs.This paper initiates the study of time-memory trade-offs for basic symmetric cryptography. We show that schemes like counter-mode encryption, which are affected by the Birthday Bound, become more secure (in terms of time complexity) as the attacker’s memory is reduced.One key step of this work is a generalization of the Switching Lemma: For adversaries with S bits of memory issuing q distinct queries, we prove an n-to-n bit random function indistinguishable from a permutation as long as $S \times q \ll 2^n$S×q≪2n. This result assumes a combinatorial conjecture, which we discuss, and implies right away trade-offs for deterministic, stateful versions of CTR and OFB encryption.We also show an unconditional time-memory trade-off for the security of randomized CTR based on a secure PRF. Via the aforementioned conjecture, we extend the result to assuming a PRP instead, assuming only one-block messages are encrypted.Our results solely rely on standard PRF/PRP security of an underlying block cipher. We frame the core of our proofs within a general framework of indistinguishability for streaming algorithms which may be of independent interest.
2019
EUROCRYPT
We construct efficient non-malleable codes (NMC) that are (computationally) secure against tampering by functions computable in any fixed polynomial time. Our construction is in the plain (no-CRS) model and requires the assumptions that (1) $\mathbf {E}$E is hard for $\mathbf {NP}$NP circuits of some exponential $2^{\beta n}$2βn ($\beta >0$β>0) size (widely used in the derandomization literature), (2) sub-exponential trapdoor permutations exist, and (3) $\mathbf {P}$P-certificates with sub-exponential soundness exist.While it is impossible to construct NMC secure against arbitrary polynomial-time tampering (Dziembowski, Pietrzak, Wichs, ICS ’10), the existence of NMC secure against $O(n^c)$O(nc)-time tampering functions (for any fixedc), was shown (Cheraghchi and Guruswami, ITCS ’14) via a probabilistic construction. An explicit construction was given (Faust, Mukherjee, Venturi, Wichs, Eurocrypt ’14) assuming an untamperable CRS with length longer than the runtime of the tampering function. In this work, we show that under computational assumptions, we can bypass these limitations. Specifically, under the assumptions listed above, we obtain non-malleable codes in the plain model against $O(n^c)$O(nc)-time tampering functions (for any fixed c), with codeword length independent of the tampering time bound.Our new construction of NMC draws a connection with non-interactive non-malleable commitments. In fact, we show that in the NMC setting, it suffices to have a much weaker notion called quasi non-malleable commitments—these are non-interactive, non-malleable commitments in the plain model, in which the adversary runs in $O(n^c)$O(nc)-time, whereas the honest parties may run in longer (polynomial) time. We then construct a 4-tag quasi non-malleable commitment from any sub-exponential OWF and the assumption that $\mathbf {E}$E is hard for some exponential size $\mathbf {NP}$NP-circuits, and use tag amplification techniques to support an exponential number of tags.
2019
EUROCRYPT
Non-malleable codes (NMCs), introduced by Dziembowski, Pietrzak and Wichs [20], provide a useful message integrity guarantee in situations where traditional error-correction (and even error-detection) is impossible; for example, when the attacker can completely overwrite the encoded message. NMCs have emerged as a fundamental object at the intersection of coding theory and cryptography. In particular, progress in the study of non-malleable codes and the related notion of non-malleable extractors has led to new insights and progress on even more fundamental problems like the construction of multi-source randomness extractors. A large body of the recent work has focused on various constructions of non-malleable codes in the split-state model. Many variants of NMCs have been introduced in the literature, e.g., strong NMCs, super strong NMCs and continuous NMCs. The most general, and hence also the most useful notion among these is that of continuous non-malleable codes, that allows for continuous tampering by the adversary. We present the first efficient information-theoretically secure continuously non-malleable code in the constant split-state model. We believe that our main technical result could be of independent interest and some of the ideas could in future be used to make progress on other related questions.
2019
EUROCRYPT
In this paper, we consider the setting where a party uses correlated random tapes across multiple executions of a cryptographic algorithm. We ask if the security properties could still be preserved in such a setting. As examples, we introduce the notion of correlated-tape zero knowledge, and, correlated-tape multi-party computation, where, the zero-knowledge property, and, the ideal/real model security must still be preserved even if a party uses correlated random tapes in multiple executions.Our constructions are based on a new type of randomness extractor which we call correlated-source extractors. Correlated-source extractors can be seen as a dual of non-malleable extractors, and, allow an adversary to choose several tampering functions which are applied to the randomness source. Correlated-source extractors guarantee that even given the output of the extractor on the tampered sources, the output on the original source is still uniformly random. Given (seeded) correlated-source extractors, and, resettably-secure computation protocols, we show how to directly get a positive result for both correlated-tape zero-knowledge and correlated-tape multi-party computation in the CRS model. This is tight considering the known impossibility results on cryptography with imperfect randomness.Our main technical contribution is an explicit construction of a correlated-source extractor where the length of the seed is independent of the number of tamperings. Additionally, we also provide a (non-explicit) existential result for correlated source extractors with almost optimal parameters.
2019
EUROCRYPT
A threshold secret sharing scheme (with threshold t) allows a dealer to share a secret among a set of parties such that any group of t or more parties can recover the secret and no group of at most $t-1$ t-1 parties learn any information about the secret. A non-malleable threshold secret sharing scheme, introduced in the recent work of Goyal and Kumar (STOC’18), additionally protects a threshold secret sharing scheme when its shares are subject to tampering attacks. Specifically, it guarantees that the reconstructed secret from the tampered shares is either the original secret or something that is unrelated to the original secret.In this work, we continue the study of threshold non-malleable secret sharing against the class of tampering functions that tamper each share independently. We focus on achieving greater efficiency and guaranteeing a stronger security property. We obtain the following results:Rate Improvement. We give the first construction of a threshold non-malleable secret sharing scheme that has rate $> 0$ >0. Specifically, for every $n,t \ge 4$ n,t≥4, we give a construction of a t-out-of-n non-malleable secret sharing scheme with rate $\varTheta (\frac{1}{t\log ^2 n})$ Θ(1tlog2n). In the prior constructions, the rate was $\varTheta (\frac{1}{n\log m})$ Θ(1nlogm) where m is the length of the secret and thus, the rate tends to 0 as $m \rightarrow \infty$ m→∞. Furthermore, we also optimize the parameters of our construction and give a concretely efficient scheme.Multiple Tampering. We give the first construction of a threshold non-malleable secret sharing scheme secure in the stronger setting of bounded tampering wherein the shares are tampered by multiple (but bounded in number) possibly different tampering functions. The rate of such a scheme is $\varTheta (\frac{1}{k^3t\log ^2 n})$ Θ(1k3tlog2n) where k is an apriori bound on the number of tamperings. We complement this positive result by proving that it is impossible to have a threshold non-malleable secret sharing scheme that is secure in the presence of an apriori unbounded number of tamperings.General Access Structures. We extend our results beyond threshold secret sharing and give constructions of rate-efficient, non-malleable secret sharing schemes for more general monotone access structures that are secure against multiple (bounded) tampering attacks.
2019
EUROCRYPT
Smart contracts are self-executing agreements written in program code and are envisioned to be one of the main applications of blockchain technology. While they are supported by prominent cryptocurrencies such as Ethereum, their further adoption is hindered by fundamental scalability challenges. For instance, in Ethereum contract execution suffers from a latency of more than 15 s, and the total number of contracts that can be executed per second is very limited. State channel networks are one of the core primitives aiming to address these challenges. They form a second layer over the slow and expensive blockchain, thereby enabling instantaneous contract processing at negligible costs.In this work we present the first complete description of a state channel network that exhibits the following key features. First, it supports virtual multi-party state channels, i.e. state channels that can be created and closed without blockchain interaction and that allow contracts with any number of parties. Second, the worst case time complexity of our protocol is constant for arbitrary complex channels. This is in contrast to the existing virtual state channel construction that has worst case time complexity linear in the number of involved parties. In addition to our new construction, we provide a comprehensive model for the modular design and security analysis of our construction.
2019
EUROCRYPT
Mimblewimble is an electronic cash system proposed by an anonymous author in 2016. It combines several privacy-enhancing techniques initially envisioned for Bitcoin, such as Confidential Transactions (Maxwell, 2015), non-interactive merging of transactions (Saxena, Misra, Dhar, 2014), and cut-through of transaction inputs and outputs (Maxwell, 2013). As a remarkable consequence, coins can be deleted once they have been spent while maintaining public verifiability of the ledger, which is not possible in Bitcoin. This results in tremendous space savings for the ledger and efficiency gains for new users, who must verify their view of the system.In this paper, we provide a provable-security analysis for Mimblewimble. We give a precise syntax and formal security definitions for an abstraction of Mimblewimble that we call an aggregate cash system. We then formally prove the security of Mimblewimble in this definitional framework. Our results imply in particular that two natural instantiations (with Pedersen commitments and Schnorr or BLS signatures) are provably secure against inflation and coin theft under standard assumptions.
2019
EUROCRYPT
State Machine Replication (SMR) is an important abstraction for a set of nodes to agree on an ever-growing, linearly-ordered log of transactions. In decentralized cryptocurrency applications, we would like to design SMR protocols that (1) resist adaptive corruptions; and (2) achieve small bandwidth and small confirmation time. All past approaches towards constructing SMR fail to achieve either small confirmation time or small bandwidth under adaptive corruptions (without resorting to strong assumptions such as the erasure model or proof-of-work).We propose a novel paradigm for reaching consensus that departs significantly from classical approaches. Our protocol is inspired by a social phenomenon called herding, where people tend to make choices considered as the social norm. In our consensus protocol, leader election and voting are coalesced into a single (randomized) process: in every round, every node tries to cast a vote for what it views as the most popular item so far: such a voting attempt is not always successful, but rather, successful with a certain probability. Importantly, the probability that the node is elected to vote for v is independent from the probability it is elected to vote for $v' \ne v$v′≠v. We will show how to realize such a distributed, randomized election process using appropriate, adaptively secure cryptographic building blocks.We show that amazingly, not only can this new paradigm achieve consensus (e.g., on a batch of unconfirmed transactions in a cryptocurrency system), but it also allows us to derive the first SMR protocol which, even under adaptive corruptions, requires only polylogarithmically many rounds and polylogarithmically many honest messages to be multicast to confirm each batch of transactions; and importantly, we attain these guarantees under standard cryptographic assumptions.
2019
EUROCRYPT
Homomorphic secret sharing (HSS) is an analog of somewhat- or fully homomorphic encryption (S/FHE) to the setting of secret sharing, with applications including succinct secure computation, private manipulation of remote databases, and more. While HSS can be viewed as a relaxation of S/FHE, the only constructions from lattice-based assumptions to date build atop specific forms of threshold or multi-key S/FHE. In this work, we present new techniques directly yielding efficient 2-party HSS for polynomial-size branching programs from a range of lattice-based encryption schemes, without S/FHE. More concretely, we avoid the costly key-switching and modulus-reduction steps used in S/FHE ciphertext multiplication, replacing them with a new distributed decryption procedure for performing “restricted” multiplications of an input with a partial computation value. Doing so requires new methods for handling the blowup of “noise” in ciphertexts in a distributed setting, and leverages several properties of lattice-based encryption schemes together with new tricks in share conversion.The resulting schemes support a superpolynomial-size plaintext space and negligible correctness error, with share sizes comparable to SHE ciphertexts, but cost of homomorphic multiplication roughly one order of magnitude faster. Over certain rings, our HSS can further support some level of packed SIMD homomorphic operations. We demonstrate the practical efficiency of our schemes within two application settings, where we compare favorably with current best approaches: 2-server private database pattern-match queries, and secure 2-party computation of low-degree polynomials.
2019
EUROCRYPT
Since Cheon et al. introduced a homomorphic encryption scheme for approximate arithmetic (Asiacrypt ’17), it has been recognized as suitable for important real-life usecases of homomorphic encryption, including training of machine learning models over encrypted data. A follow up work by Cheon et al. (Eurocrypt ’18) described an approximate bootstrapping procedure for the scheme. In this work, we improve upon the previous bootstrapping result. We improve the amortized bootstrapping time per plaintext slot by two orders of magnitude, from $\sim$∼1 s to $\sim$∼0.01 s. To achieve this result, we adopt a smart level-collapsing technique for evaluating DFT-like linear transforms on a ciphertext. Also, we replace the Taylor approximation of the sine function with a more accurate and numerically stable Chebyshev approximation, and design a modified version of the Paterson-Stockmeyer algorithm for fast evaluation of Chebyshev polynomials over encrypted data.
2019
EUROCRYPT
Algebraic structure lies at the heart of Cryptomania as we know it. An interesting question is the following: instead of building (Cryptomania) primitives from concrete assumptions, can we build them from simple Minicrypt primitives endowed with some additional algebraic structure? In this work, we affirmatively answer this question by adding algebraic structure to the following Minicrypt primitives:One-Way Function (OWF)Weak Unpredictable Function (wUF)Weak Pseudorandom Function (wPRF) The algebraic structure that we consider is group homomorphism over the input/output spaces of these primitives. We also consider a “bounded” notion of homomorphism where the primitive only supports an a priori bounded number of homomorphic operations in order to capture lattice-based and other “noisy” assumptions. We show that these structured primitives can be used to construct many cryptographic protocols. In particular, we prove that: (Bounded) Homomorphic OWFs (HOWFs) imply collision-resistant hash functions, Schnorr-style signatures and chameleon hash functions.(Bounded) Input-Homomorphic weak UFs (IHwUFs) imply CPA-secure PKE, non-interactive key exchange, trapdoor functions, blind batch encryption (which implies anonymous IBE, KDM-secure and leakage-resilient PKE), CCA2 deterministic PKE, and hinting PRGs (which in turn imply transformation of CPA to CCA security for ABE/1-sided PE).(Bounded) Input-Homomorphic weak PRFs (IHwPRFs) imply PIR, lossy trapdoor functions, OT and MPC (in the plain model). In addition, we show how to realize any CDH/DDH-based protocol with certain properties in a generic manner using IHwUFs/IHwPRFs, and how to instantiate such a protocol from many concrete assumptions.We also consider primitives with substantially richer structure, namely Ring IHwPRFs and L-composable IHwPRFs. In particular, we show the following: Ring IHwPRFs with certain properties imply FHE.2-composable IHwPRFs imply (black-box) IBE, and L-composable IHwPRFs imply non-interactive $(L+1)$ (L+1)-party key exchange. Our framework allows us to categorize many cryptographic protocols based on which structured Minicrypt primitive implies them. In addition, it potentially makes showing the existence of many cryptosystems from novel assumptions substantially easier in the future.
2019
EUROCRYPT
We improve the attack of Durak and Vaudenay (CRYPTO’17) on NIST Format-Preserving Encryption standard FF3, reducing the running time from $O(N^5)$O(N5) to $O(N^{17/6})$O(N17/6) for domain $\mathbb {Z}_N \times \mathbb {Z}_N$ZN×ZN. Concretely, DV’s attack needs about $2^{50}$250 operations to recover encrypted 6-digit PINs, whereas ours only spends about $2^{30}$230 operations. In realizing this goal, we provide a pedagogical example of how to use distinguishing attacks to speed up slide attacks. In addition, we improve the running time of DV’s known-plaintext attack on 4-round Feistel of domain $\mathbb {Z}_N \times \mathbb {Z}_N$ZN×ZN from $O(N^3)$O(N3) time to just $O(N^{5/3})$O(N5/3) time. We also generalize our attacks to a general domain $\mathbb {Z}_M \times \mathbb {Z}_N$ZM×ZN, allowing one to recover encrypted SSNs using about $2^{50}$250 operations. Finally, we provide some proof-of-concept implementations to empirically validate our results.
2019
EUROCRYPT
The TLS 1.3 0-RTT mode enables a client reconnecting to a server to send encrypted application-layer data in “0-RTT” (“zero round-trip time”), without the need for a prior interactive handshake. This fundamentally requires the server to reconstruct the previous session’s encryption secrets upon receipt of the client’s first message. The standard techniques to achieve this are Session Caches or, alternatively, Session Tickets. The former provides forward security and resistance against replay attacks, but requires a large amount of server-side storage. The latter requires negligible storage, but provides no forward security and is known to be vulnerable to replay attacks.In this paper, we first formally define session resumption protocols as an abstract perspective on mechanisms like Session Caches and Session Tickets. We give a new generic construction that provably provides forward security and replay resilience, based on puncturable pseudorandom functions (PPRFs). This construction can immediately be used in TLS 1.3 0-RTT and deployed unilaterally by servers, without requiring any changes to clients or the protocol.We then describe two new constructions of PPRFs, which are particularly suitable for use for forward-secure and replay-resilient session resumption in TLS 1.3. The first construction is based on the strong RSA assumption. Compared to standard Session Caches, for “128-bit security” it reduces the required server storage by a factor of almost 20, when instantiated in a way such that key derivation and puncturing together are cheaper on average than one full exponentiation in an RSA group. Hence, a 1 GB Session Cache can be replaced with only about 51 MBs of storage, which significantly reduces the amount of secure memory required. For larger security parameters or in exchange for more expensive computations, even larger storage reductions are achieved. The second construction combines a standard binary tree PPRF with a new “domain extension” technique. For a reasonable choice of parameters, this reduces the required storage by a factor of up to 5 compared to a standard Session Cache. It employs only symmetric cryptography, is suitable for high-traffic scenarios, and can serve thousands of tickets per second.
2019
EUROCRYPT
We investigate the security properties of the three deterministic random bit generator (DRBG) mechanisms in NIST SP 800-90A [2]. The standard received considerable negative attention due to the controversy surrounding the now retracted $\mathsf{{DualEC\text {-}DRBG}}$DualEC-DRBG, which appeared in earlier versions. Perhaps because of the attention paid to the DualEC, the other algorithms in the standard have received surprisingly patchy analysis to date, despite widespread deployment. This paper addresses a number of these gaps in analysis, with a particular focus on $\mathsf{{HASH\text {-}DRBG}}$HASH-DRBG and $\mathsf{{HMAC\text {-}DRBG}}$HMAC-DRBG. We uncover a mix of positive and less positive results. On the positive side, we prove (with a caveat) the robustness [13] of $\mathsf{{HASH\text {-}DRBG}}$HASH-DRBG and $\mathsf{{HMAC\text {-}DRBG}}$HMAC-DRBG in the random oracle model (ROM). Regarding the caveat, we show that if an optional input is omitted, then – contrary to claims in the standard—$\mathsf{{HMAC\text {-}DRBG}}$HMAC-DRBG does not even achieve the (weaker) property of forward security. We then conduct a more informal and practice-oriented exploration of flexibility in the standard. Specifically, we argue that these DRBGs have the property that partial state leakage may lead security to break down in unexpected ways. We highlight implementation choices allowed by the overly flexible standard that exacerbate both the likelihood, and impact, of such attacks. While our attacks are theoretical, an analysis of two open source implementations of $\mathsf{{CTR\text {-}DRBG}}$CTR-DRBG shows that these potentially problematic implementation choices are made in the real world.
2019
EUROCRYPT
We initiate the study of structured encryption schemes with computationally-secure leakage. Specifically, we focus on the design of volume-hiding encrypted multi-maps; that is, of encrypted multi-maps that hide the response length to computationally-bounded adversaries. We describe the first volume-hiding STE schemes that do not rely on naïve padding; that is, padding all tuples to the same length. Our first construction has efficient query complexity and storage but can be lossy. We show, however, that the information loss can be bounded with overwhelming probability for a large class of multi-maps (i.e., with lengths distributed according to a Zipf distribution). Our second construction is not lossy and can achieve storage overhead that is asymptotically better than naïve padding for Zipf-distributed multi-maps. We also show how to further improve the storage when the multi-map is highly concentrated in the sense that it has a large number of tuples with a large intersection. We achieve these results by leveraging computational assumptions; not just for encryption but, more interestingly, to hide the volumes themselves. Our first construction achieves this using a pseudo-random function whereas our second construction achieves this by relying on the conjectured hardness of the planted densest subgraph problem which is a planted variant of the well-studied densest subgraph problem. This assumption was previously used to design public-key encryptions schemes (Applebaum et al., STOC ’10) and to study the computational complexity of financial products (Arora et al., ICS ’10).
2019
EUROCRYPT
Oblivious RAMs, introduced by Goldreich and Ostrovsky [JACM’96], compile any RAM program into one that is “memory oblivious”, i.e., the access pattern to the memory is independent of the input. All previous ORAM schemes, however, completely break the locality of data accesses (for instance, by shuffling the data to pseudorandom positions in memory).In this work, we initiate the study of locality-preserving ORAMs—ORAMs that preserve locality of the accessed memory regions, while leaking only the lengths of contiguous memory regions accessed. Our main results demonstrate the existence of a locality-preserving ORAM with poly-logarithmic overhead both in terms of bandwidth and locality. We also study the tradeoff between locality, bandwidth and leakage, and show that any scheme that preserves locality and does not leak the lengths of the contiguous memory regions accessed, suffers from prohibitive bandwidth.To the best of our knowledge, before our work, the only works combining locality and obliviousness were for symmetric searchable encryption [e.g., Cash and Tessaro (EUROCRYPT’14), Asharov et al. (STOC’16)]. Symmetric search encryption ensures obliviousness if each keyword is searched only once, whereas ORAM provides obliviousness to any input program. Thus, our work generalizes that line of work to the much more challenging task of preserving locality in ORAMs.
2019
EUROCRYPT
We consider a scenario where a server holds a huge database that it wants to make accessible to a large group of clients. After an initial setup phase, clients should be able to read arbitrary locations in the database while maintaining privacy (the server does not learn which locations are being read) and anonymity (the server does not learn which client is performing each read). This should hold even if the server colludes with a subset of the clients. Moreover, the run-time of both the server and the client during each read operation should be low, ideally only poly-logarithmic in the size of the database and the number of clients. We call this notion Private Anonymous Data Access (PANDA). PANDA simultaneously combines aspects of Private Information Retrieval (PIR) and Oblivious RAM (ORAM). PIR has no initial setup, and allows anybody to privately and anonymously access a public database, but the server’s run-time is linear in the data size. On the other hand, ORAM achieves poly-logarithmic server run-time, but requires an initial setup after which only a single client with a secret key can access the database. The goal of PANDA is to get the best of both worlds: allow many clients to privately and anonymously access the database as in PIR, while having an efficient server as in ORAM.In this work, we construct bounded-collusion PANDA schemes, where the efficiency scales linearly with a bound on the number of corrupted clients that can collude with the server, but is otherwise poly-logarithmic in the data size and the total number of clients. Our solution relies on standard assumptions, namely the existence of fully homomorphic encryption, and combines techniques from both PIR and ORAM. We also extend PANDA to settings where clients can write to the database.
2019
EUROCRYPT
Proofs of sequential work (PoSW) are proof systems where a prover, upon receiving a statement $\chi$ and a time parameter T computes a proof $\phi (\chi ,T)$ which is efficiently and publicly verifiable. The proof can be computed in T sequential steps, but not much less, even by a malicious party having large parallelism. A PoSW thus serves as a proof that T units of time have passed since $\chi$ was received.PoSW were introduced by Mahmoody, Moran and Vadhan [MMV11], a simple and practical construction was only recently proposed by Cohen and Pietrzak [CP18].In this work we construct a new simple PoSW in the random permutation model which is almost as simple and efficient as [CP18] but conceptually very different. Whereas the structure underlying [CP18] is a hash tree, our construction is based on skip lists and has the interesting property that computing the PoSW is a reversible computation.The fact that the construction is reversible can potentially be used for new applications like constructing proofs of replication. We also show how to “embed” the sloth function of Lenstra and Weselowski [LW17] into our PoSW to get a PoSW where one additionally can verify correctness of the output much more efficiently than recomputing it (though recent constructions of “verifiable delay functions” subsume most of the applications this construction was aiming at).
2019
EUROCRYPT
A proof of sequential work allows a prover to convince a verifier that a certain amount of sequential steps have been computed. In this work we introduce the notion of incremental proofs of sequential work where a prover can carry on the computation done by the previous prover incrementally, without affecting the resources of the individual provers or the size of the proofs.To date, the most efficient instance of proofs of sequential work [Cohen and Pietrzak, Eurocrypt 2018] for N steps require the prover to have $\sqrt{N}$N memory and to run for $N + \sqrt{N}$N+N steps. Using incremental proofs of sequential work we can bring down the prover’s storage complexity to $\log N$logN and its running time to N.We propose two different constructions of incremental proofs of sequential work: Our first scheme requires a single processor and introduces a poly-logarithmic factor in the proof size when compared with the proposals of Cohen and Pietrzak. Our second scheme assumes $\log N$logN parallel processors but brings down the overhead of the proof size to a factor of 9. Both schemes are simple to implement and only rely on hash functions (modelled as random oracles).
2019
EUROCRYPT
We construct a concretely practical proof-of-space (PoS) with arbitrarily tight security based on stacked depth robust graphs and constant-degree expander graphs. A proof-of-space (PoS) is an interactive proof system where a prover demonstrates that it is persistently using space to store information. A PoS is arbitrarily tight if the honest prover uses exactly N space and for any $\epsilon > 0$ϵ>0 the construction can be tuned such that no adversary can pass verification using less than $(1-\epsilon ) N$(1-ϵ)N space. Most notably, the degree of the graphs in our construction are independent of $\epsilon$ϵ, and the number of layers is only $O(\log (1/\epsilon ))$O(log(1/ϵ)). The proof size is $O(d/\epsilon )$O(d/ϵ). The degree d depends on the depth robust graphs, which are only required to maintain $\varOmega (N)$Ω(N) depth in subgraphs on 80% of the nodes. Our tight PoS is also secure against parallel attacks.Tight proofs of space are necessary for proof-of-replication (PoRep), which is a publicly verifiable proof that the prover is dedicating unique resources to storing one or more retrievable replicas of a specified file. Our main PoS construction can be used as a PoRep, but data extraction is as inefficient as replica generation. We present a second variant of our construction called ZigZag PoRep that has fast/parallelizable data extraction compared to replica generation and maintains the same space tightness while only increasing the number of levels by roughly a factor two.
2019
EUROCRYPT
We study the foundations of secure computation in the blockchain-hybrid model, where a blockchain – modeled as a global functionality – is available as an Oracle to all the participants of a cryptographic protocol. We demonstrate both destructive and constructive applications of blockchains:We show that classical rewinding-based simulation techniques used in many security proofs fail against blockchain-active adversaries that have read and post access to a global blockchain. In particular, we show that zero-knowledge (ZK) proofs with black-box simulation are impossible against blockchain-active adversaries.Nevertheless, we show that achieving security against blockchain-active adversaries is possible if the honest parties are also blockchain active. We construct an $\omega (1)$-round ZK protocol with black-box simulation. We show that this result is tight by proving the impossibility of constant-round ZK with black-box simulation.Finally, we demonstrate a novel application of blockchains to overcome the known impossibility results for concurrent secure computation in the plain model. We construct a concurrent self-composable secure computation protocol for general functionalities in the blockchain-hybrid model based on standard cryptographic assumptions. We develop a suite of techniques for constructing secure protocols in the blockchain-hybrid model that we hope will find applications to future research in this area.
2019
EUROCRYPT
A fundamental problem in the theory of secure multi-party computation (MPC) is to characterize functions with more than 2 parties which admit MPC protocols with information-theoretic security against passive corruption. This question has seen little progress since the work of Chor and Ishai (1996), which demonstrated difficulties in resolving it. In this work, we make significant progress towards resolving this question in the important case of aggregating functionalities, in which m parties $P_1,\dots ,P_m$ P1,⋯,Pm hold inputs $x_1,\dots ,x_m$ x1,⋯,xm and an aggregating party $P_0$ P0 must learn $f(x_1,\dots ,x_m)$ f(x1,⋯,xm).We uncover a rich class of algebraic structures that are closely related to secure computability, namely, “Commuting Permutations Systems” (CPS) and its variants. We present an extensive set of results relating these algebraic structures among themselves and to MPC, including new protocols, impossibility results and separations. Our results include a necessary algebraic condition and slightly stronger sufficient algebraic condition for a function to admit information-theoretically secure MPC protocols.We also introduce and study new models of minimally interactive MPC (called UNIMPC and ), which not only help in understanding our positive and negative results better, but also open up new avenues for studying the cryptographic complexity landscape of multi-party functionalities. Our positive results include novel protocols in these models, which may be of independent practical interest.Finally, we extend our results to a definition that requires UC security as well as semi-honest security (which we term strong security). In this model we are able to carry out the characterization of all computable functions, except for a gap in the case of aggregating functionalities.
2019
EUROCRYPT
Choosing safe post-quantum parameters for the new CSIDH isogeny-based key-exchange system requires concrete analysis of the cost of quantum attacks. The two main contributions to attack cost are the number of queries in hidden-shift algorithms and the cost of each query. This paper analyzes algorithms for each query, introducing several new speedups while showing that some previous claims were too optimistic for the attacker. This paper includes a full computer-verified simulation of its main algorithm down to the bit-operation level.
2019
EUROCRYPT
In privacy amplification, two mutually trusted parties aim to amplify the secrecy of an initial shared secret X in order to establish a shared private key K by exchanging messages over an insecure communication channel. If the channel is authenticated the task can be solved in a single round of communication using a strong randomness extractor; choosing a quantum-proof extractor allows one to establish security against quantum adversaries.In the case that the channel is not authenticated, this simple solution is no longer secure. Nevertheless, Dodis and Wichs (STOC’09) showed that the problem can be solved in two rounds of communication using a non-malleable extractor, a stronger pseudo-random construction than a strong extractor.We give the first construction of a non-malleable extractor that is secure against quantum adversaries. The extractor is based on a construction by Li (FOCS’12), and is able to extract from source of min-entropy rates larger than 1 / 2. Combining this construction with a quantum-proof variant of the reduction of Dodis and Wichs, due to Cohen and Vidick (unpublished) we obtain the first privacy amplification protocol secure against active quantum adversaries.
2019
EUROCRYPT
Secure multiparty computation ( $\mathsf {MPC}$ MPC) addresses the challenge of evaluating functions on secret inputs without compromising their privacy. A central question in multiparty computation is to understand the amount of communication needed to securely evaluate a circuit of size s. In this work, we revisit this fundamental question in the setting of information-theoretically secure $\mathsf {MPC}$ MPC in the correlated randomness model, where a trusted dealer distributes correlated random coins, independent of the inputs, to all parties before the start of the protocol. This setting is of strong theoretical interest, and has led to the most practically efficient $\mathsf {MPC}$ MPC protocols known to date.While it is known that protocols with optimal communication (proportional to input plus output size) can be obtained from the LWE assumption, and that protocols with sublinear communication o(s) can be obtained from the DDH assumption, the question of constructing protocols with o(s) communication remains wide open for the important case of information-theoretic $\mathsf {MPC}$ MPC in the correlated randomness model; all known protocols in this model require O(s) communication in the online phase.In this work, we exhibit the first generic multiparty computation protocol in the correlated randomness model with communication sublinear in the circuit size, for a large class of circuits. More precisely, we show the following: any size-slayered circuit (whose nodes can be partitioned into layers so that any edge connects adjacent layers) can be evaluated with $O(s/\log \log s)$ O(s/loglogs) communication. Our results holds for both boolean and arithmetic circuits, in the honest-but-curious setting, and do not assume honest majority. For boolean circuits, we extend our results to handle malicious corruption.
2019
EUROCRYPT
We show, via a non-interactive reduction, that the existence of a secure multi-party computation (MPC) protocol for degree-2 functions implies the existence of a protocol with the same round complexity for general functions. Thus showing that when considering the round complexity of MPC, it is sufficient to consider very simple functions.Our completeness theorem applies in various settings: information theoretic and computational, fully malicious and malicious with various types of aborts. In fact, we give a master theorem from which all individual settings follow as direct corollaries. Our basic transformation does not require any additional assumptions and incurs communication and computation blow-up which is polynomial in the number of players and in $S,2^D$S,2D, where S, D are the circuit size and depth of the function to be computed. Using one-way functions as an additional assumption, the exponential dependence on the depth can be removed.As a consequence, we are able to push the envelope on the state of the art in various settings of MPC, including the following cases. 3-round perfectly-secure protocol (with guaranteed output delivery) against an active adversary that corrupts less than 1/4 of the parties.2-round statistically-secure protocol that achieves security with “selective abort” against an active adversary that corrupts less than half of the parties.Assuming one-way functions, 2-round computationally-secure protocol that achieves security with (standard) abort against an active adversary that corrupts less than half of the parties. This gives a new and conceptually simpler proof to the recent result of Ananth et al. (Crypto 2018). Technically, our non-interactive reduction draws from the encoding method of Applebaum, Brakerski and Tsabary (TCC 2018). We extend these methods to ones that can be meaningfully analyzed even in the presence of malicious adversaries.
2019
EUROCRYPT
We provide the first constructions of two round information-theoretic (IT) secure multiparty computation (MPC) protocols in the plain model that tolerate any $t<n/2$t<n/2 malicious corruptions. Our protocols satisfy the strongest achievable standard notions of security in two rounds in different communication models.Previously, IT-MPC protocols in the plain model either required a larger number of rounds, or a smaller minority of corruptions.
2019
EUROCRYPT
We provide a generic construction of non-interactive zero-knowledge (NIZK) schemes. Our construction is a refinement of Dwork and Naor’s (FOCS 2000) implementation of the hidden bits model using verifiable pseudorandom generators (VPRGs). Our refinement simplifies their construction and relaxes the necessary assumptions considerably.As a result of this conceptual improvement, we obtain interesting new instantiations:A designated-verifier NIZK (with unbounded soundness) based on the computational Diffie-Hellman (CDH) problem. If a pairing is available, this NIZK becomes publicly verifiable. This constitutes the first fully secure CDH-based designated-verifier NIZKs (and more generally, the first fully secure designated-verifier NIZK from a non-generic assumption which does not already imply publicly-verifiable NIZKs), and it answers an open problem recently raised by Kim and Wu (CRYPTO 2018).A NIZK based on the learning with errors (LWE) assumption, and assuming a non-interactive witness-indistinguishable (NIWI) proof system for bounded distance decoding (BDD). This simplifies and improves upon a recent NIZK from LWE that assumes a NIZK for BDD (Rothblum et al., PKC 2019).
2019
EUROCRYPT
Non-interactive zero-knowledge proofs (NIZKs) are a fundamental cryptographic primitive. Despite a long history of research, we only know how to construct NIZKs under a few select assumptions, such as the hardness of factoring or using bilinear maps. Notably, there are no known constructions based on either the computational or decisional Diffie-Hellman (CDH/DDH) assumption without relying on a bilinear map.In this paper, we study a relaxation of NIZKs in the designated verifier setting (DV-NIZK), in which the public common-reference string is generated together with a secret key that is given to the verifier in order to verify proofs. In this setting, we distinguish between one-time and reusable schemes, depending on whether they can be used to prove only a single statement or arbitrarily many statements. For reusable schemes, the main difficulty is to ensure that soundness continues to hold even when the malicious prover learns whether various proofs are accepted or rejected by the verifier. One-time DV-NIZKs are known to exist for general NP statements assuming only public-key encryption. However, prior to this work, we did not have any construction of reusable DV-NIZKs for general NP statements from any assumption under which we didn’t already also have standard NIZKs.In this work, we construct reusable DV-NIZKs for general NP statements under the CDH assumption, without requiring a bilinear map. Our construction is based on the hidden-bits paradigm, which was previously used to construct standard NIZKs. We define a cryptographic primitive called a hidden-bits generator (HBG), along with a designated-verifier variant (DV-HBG), which modularly abstract out how to use this paradigm to get both standard NIZKs and reusable DV-NIZKs. We construct a DV-HBG scheme under the CDH assumption by relying on techniques from the Cramer-Shoup hash-proof system, and this yields our reusable DV-NIZK for general NP statements under CDH.We also consider a strengthening of DV-NIZKs to the malicious designated-verifier setting (MDV-NIZK) where the setup consists of an honestly generated common random string and the verifier then gets to choose his own (potentially malicious) public/secret key pair to generate/verify proofs. We construct MDV-NIZKs under the “one-more CDH” assumption without relying on bilinear maps.
2019
EUROCRYPT
In a non-interactive zero-knowledge (NIZK) proof, a prover can non-interactively convince a verifier of a statement without revealing any additional information. Thus far, numerous constructions of NIZKs have been provided in the common reference string (CRS) model (CRS-NIZK) from various assumptions, however, it still remains a long standing open problem to construct them from tools such as pairing-free groups or lattices. Recently, Kim and Wu (CRYPTO’18) made great progress regarding this problem and constructed the first lattice-based NIZK in a relaxed model called NIZKs in the preprocessing model (PP-NIZKs). In this model, there is a trusted statement-independent preprocessing phase where secret information are generated for the prover and verifier. Depending on whether those secret information can be made public, PP-NIZK captures CRS-NIZK, designated-verifier NIZK (DV-NIZK), and designated-prover NIZK (DP-NIZK) as special cases. It was left as an open problem by Kim and Wu whether we can construct such NIZKs from weak paring-free group assumptions such as DDH. As a further matter, all constructions of NIZKs from Diffie-Hellman (DH) type assumptions (regardless of whether it is over a paring-free or paring group) require the proof size to have a multiplicative-overhead $|C| \cdot \mathsf {poly}(\kappa )$|C|·poly(κ), where |C| is the size of the circuit that computes the $\mathbf {NP}$NP relation.In this work, we make progress of constructing (DV, DP, PP)-NIZKs with varying flavors from DH-type assumptions. Our results are summarized as follows:DV-NIZKs for $\mathbf {NP}$NP from the CDH assumption over pairing-free groups. This is the first construction of such NIZKs on pairing-free groups and resolves the open problem posed by Kim and Wu (CRYPTO’18).DP-NIZKs for $\mathbf {NP}$NP with short proof size from a DH-type assumption over pairing groups. Here, the proof size has an additive-overhead $|C|+\mathsf {poly}(\kappa )$|C|+poly(κ) rather then an multiplicative-overhead $|C| \cdot \mathsf {poly}(\kappa )$|C|·poly(κ). This is the first construction of such NIZKs (including CRS-NIZKs) that does not rely on the LWE assumption, fully-homomorphic encryption, indistinguishability obfuscation, or non-falsifiable assumptions.PP-NIZK for $\mathbf {NP}$NP with short proof size from the DDH assumption over pairing-free groups. This is the first PP-NIZK that achieves a short proof size from a weak and static DH-type assumption such as DDH. Similarly to the above DP-NIZK, the proof size is $|C|+\mathsf {poly}(\kappa )$|C|+poly(κ). This too serves as a solution to the open problem posed by Kim and Wu (CRYPTO’18). Along the way, we construct two new homomorphic authentication (HomAuth) schemes which may be of independent interest.
2019
EUROCRYPT
Many advanced lattice cryptography applications require efficient algorithms for inverting the so-called “gadget” matrices, which are used to formally describe a digit decomposition problem that produces an output with specific (statistical) properties. The common gadget inversion problems are the classical (often binary) digit decomposition, subgaussian decomposition, Learning with Errors (LWE) decoding, and discrete Gaussian sampling. In this work, we build and implement an efficient lattice gadget toolkit that provides a general treatment of gadget matrices and algorithms for their inversion/sampling. The main contribution of our work is a set of new gadget matrices and algorithms for efficient subgaussian sampling that have a number of major theoretical and practical advantages over previously known algorithms. Another contribution deals with efficient algorithms for LWE decoding and discrete Gaussian sampling in the Residue Number System (RNS) representation.We implement the gadget toolkit in PALISADE and evaluate the performance of our algorithms both in terms of runtime and noise growth. We illustrate the improvements due to our algorithms by implementing a concrete complex application, key-policy attribute-based encryption (KP-ABE), which was previously considered impractical for CPU systems (except for a very small number of attributes). Our runtime improvements for the main bottleneck operation based on subgaussian sampling range from 18x (for 2 attributes) to 289x (for 16 attributes; the maximum number supported by a previous implementation). Our results are applicable to a wide range of other advanced applications in lattice cryptography, such as GSW-based homomorphic encryption schemes, leveled fully homomorphic signatures, other forms of ABE, some program obfuscation constructions, and more.
2019
EUROCRYPT
We describe an algorithm to solve the approximate Shortest Vector Problem for lattices corresponding to ideals of the ring of integers of an arbitrary number field K. This algorithm has a pre-processing phase, whose run-time is exponential in  $\log |\varDelta |$ log|Δ| with  $\varDelta$ Δ the discriminant of K. Importantly, this pre-processing phase depends only on K. The pre-processing phase outputs an “advice”, whose bit-size is no more than the run-time of the query phase. Given this advice, the query phase of the algorithm takes as input any ideal I of the ring of integers, and outputs an element of I which is at most $\exp (\widetilde{O}((\log |\varDelta |)^{\alpha +1}/n))$ exp(O~((log|Δ|)α+1/n)) times longer than a shortest non-zero element of I (with respect to the Euclidean norm of its canonical embedding). This query phase runs in time and space $\exp (\widetilde{O}( (\log |\varDelta |)^{\max (2/3, 1-2\alpha )}))$ exp(O~((log|Δ|)max(2/3,1-2α))) in the classical setting, and $\exp (\widetilde{O}((\log |\varDelta |)^{1-2\alpha }))$ exp(O~((log|Δ|)1-2α)) in the quantum setting. The parameter $\alpha$ α can be chosen arbitrarily in [0, 1 / 2]. Both correctness and cost analyses rely on heuristic assumptions, whose validity is consistent with experiments.The algorithm builds upon the algorithms from Cramer et al. [EUROCRYPT 2016] and Cramer et al. [EUROCRYPT 2017]. It relies on the framework from Buchmann [Séminaire de théorie des nombres 1990], which allows to merge them and to extend their applicability from prime-power cyclotomic fields to all number fields. The cost improvements are obtained by allowing precomputations that depend on the field only.
2019
EUROCRYPT
We propose the General Sieve Kernel (G6K, pronounced / e.si.ka/), an abstract stateful machine supporting a wide variety of lattice reduction strategies based on sieving algorithms. Using the basic instruction set of this abstract stateful machine, we first give concise formulations of previous sieving strategies from the literature and then propose new ones. We then also give a light variant of BKZ exploiting the features of our abstract stateful machine. This encapsulates several recent suggestions (Ducas at Eurocrypt 2018; Laarhoven and Mariano at PQCrypto 2018) to move beyond treating sieving as a blackbox SVP oracle and to utilise strong lattice reduction as preprocessing for sieving. Furthermore, we propose new tricks to minimise the sieving computation required for a given reduction quality with mechanisms such as recycling vectors between sieves, on-the-fly lifting and flexible insertions akin to Deep LLL and recent variants of Random Sampling Reduction.Moreover, we provide a highly optimised, multi-threaded and tweakable implementation of this machine which we make open-source. We then illustrate the performance of this implementation of our sieving strategies by applying G6K to various lattice challenges. In particular, our approach allows us to solve previously unsolved instances of the Darmstadt SVP (151, 153, 155) and LWE (e.g. (75, 0.005)) challenges. Our solution for the SVP-151 challenge was found 400 times faster than the time reported for the SVP-150 challenge, the previous record. For exact-SVP, we observe a performance crossover between G6K and FPLLL’s state of the art implementation of enumeration at dimension 70.
2019
EUROCRYPT
Many post-quantum cryptosystems which have been proposed in the National Institute of Standards and Technology (NIST) standardization process follow the same meta-algorithm, but in different algebras or different encoding methods. They usually propose two constructions, one being weaker and the other requiring a random oracle. We focus on the weak version of nine submissions to NIST. Submitters claim no security when the secret key is used several times. In this paper, we analyze how easy it is to run a key recovery under multiple key reuse. We mount a classical key recovery under plaintext checking attacks (i.e., with a plaintext checking oracle saying if a given ciphertext decrypts well to a given plaintext) and a quantum key recovery under chosen ciphertext attacks. In the latter case, we assume quantum access to the decryption oracle.
2019
EUROCRYPT
Private set intersection (PSI) is an important area of research and has been the focus of many works over the past decades. It describes the problem of finding an intersection between the input sets of at least two parties without revealing anything about the input sets apart from their intersection.In this paper, we present a new approach to compute the intersection between sets based on a primitive called Oblivious Linear Function Evaluation (OLE). On an abstract level, we use this primitive to efficiently add two polynomials in a randomized way while preserving the roots of the added polynomials. Setting the roots of the input polynomials to be the elements of the input sets, this directly yields an intersection protocol with optimal asymptotic communication complexity $O(m\kappa )$. We highlight that the protocol is information-theoretically secure against a malicious adversary assuming OLE.We also present a natural generalization of the 2-party protocol for the fully malicious multi-party case. Our protocol does away with expensive (homomorphic) threshold encryption and zero-knowledge proofs. Instead, we use simple combinatorial techniques to ensure the security. As a result we get a UC-secure protocol with asymptotically optimal communication complexity $O((n^2+nm)\kappa )$, where n is the number of parties, m is the set size and $\kappa$ is the security parameter. Apart from yielding an asymptotic improvement over previous works, our protocols are also conceptually simple and require only simple field arithmetic. Along the way we develop techniques that might be of independent interest.
2019
EUROCRYPT
In (single-server) Private Information Retrieval (PIR), a server holds a large database ${\mathtt {DB}}$ of size n, and a client holds an index $i \in [n]$ and wishes to retrieve ${\mathtt {DB}}[i]$ without revealing i to the server. It is well known that information theoretic privacy even against an “honest but curious” server requires $\varOmega (n)$ communication complexity. This is true even if quantum communication is allowed and is due to the ability of such an adversarial server to execute the protocol on a superposition of databases instead of on a specific database (“input purification attack”).Nevertheless, there have been some proposals of protocols that achieve sub-linear communication and appear to provide some notion of privacy. Most notably, a protocol due to Le Gall (ToC 2012) with communication complexity $O(\sqrt{n})$ , and a protocol by Kerenidis et al. (QIC 2016) with communication complexity $O(\log (n))$ , and O(n) shared entanglement.We show that, in a sense, input purification is the only potent adversarial strategy, and protocols such as the two protocols above are secure in a restricted variant of the quantum honest but curious (a.k.a specious) model. More explicitly, we propose a restricted privacy notion called anchored privacy, where the adversary is forced to execute on a classical database (i.e. the execution is anchored to a classical database). We show that for measurement-free protocols, anchored security against honest adversarial servers implies anchored privacy even against specious adversaries.Finally, we prove that even with (unlimited) pre-shared entanglement it is impossible to achieve security in the standard specious model with sub-linear communication, thus further substantiating the necessity of our relaxation. This lower bound may be of independent interest (in particular recalling that PIR is a special case of Fully Homomorphic Encryption).
2019
EUROCRYPT
A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size $2^{n-o(n)}$ and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to $O(2^{0.994n})$. Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is $O(k^2)$ times the size of the secret.A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $\tilde{O}(2^{h(k/n)n/2})$ (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $2^{\tilde{O}(\sqrt{k \log n})}$. Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.
2019
EUROCRYPT
Robust secret sharing enables the reconstruction of a secret-shared message in the presence of up to t (out of n) incorrect shares. The most challenging case is when $n = 2t+1$, which is the largest t for which the task is still possible, up to a small error probability $2^{-\kappa }$ and with some overhead in the share size.Recently, Bishop, Pastro, Rajaraman and Wichs [3] proposed a scheme with an (almost) optimal overhead of $\widetilde{O}(\kappa )$. This seems to answer the open question posed by Cevallos et al. [6] who proposed a scheme with overhead of $\widetilde{O}(n+\kappa )$ and asked whether the linear dependency on n was necessary or not. However, a subtle issue with Bishop et al.’s solution is that it (implicitly) assumes a non-rushing adversary, and thus it satisfies a weaker notion of security compared to the scheme by Cevallos et al. [6], or to the classical scheme by Rabin and BenOr [13].In this work, we almost close this gap. We propose a new robust secret sharing scheme that offers full security against a rushing adversary, and that has an overhead of $O(\kappa n^\varepsilon )$, where $\varepsilon > 0$ is arbitrary but fixed. This $n^\varepsilon$-factor is obviously worse than the $\mathrm {polylog}(n)$-factor hidden in the $\widetilde{O}$ notation of the scheme of Bishop et al. [3], but it greatly improves on the linear dependency on n of the best known scheme that features security against a rushing adversary (when $\kappa$ is substantially smaller than n).A small variation of our scheme has the same $\widetilde{O}(\kappa )$ overhead as the scheme of Bishop et al. and achieves security against a rushing adversary, but suffers from a (slightly) superpolynomial reconstruction complexity.
2019
EUROCRYPT
The bounded storage model promises unconditional security proofs against computationally unbounded adversaries, so long as the adversary’s space is bounded. In this work, we develop simple new constructions of two-party key agreement, bit commitment, and oblivious transfer in this model. In addition to simplicity, our constructions have several advantages over prior work, including an improved number of rounds and enhanced correctness. Our schemes are based on Raz’s lower bound for learning parities.
2019
EUROCRYPT
A chosen-prefix collision attack is a stronger variant of a collision attack, where an arbitrary pair of challenge prefixes are turned into a collision. Chosen-prefix collisions are usually significantly harder to produce than (identical-prefix) collisions, but the practical impact of such an attack is much larger. While many cryptographic constructions rely on collision-resistance for their security proofs, collision attacks are hard to turn into break of concrete protocols, because the adversary has a limited control over the colliding messages. On the other hand, chosen-prefix collisions have been shown to break certificates (by creating a rogue CA) and many internet protocols (TLS, SSH, IPsec).In this article, we propose new techniques to turn collision attacks into chosen-prefix collision attacks. Our strategy is composed of two phases: first a birthday search that aims at taking the random chaining variable difference (due to the chosen-prefix model) to a set of pre-defined target differences. Then, using a multi-block approach, carefully analysing the clustering effect, we map this new chaining variable difference to a colliding pair of states using techniques developed for collision attacks.We apply those techniques to MD5 and SHA-1, and obtain improved attacks. In particular, we have a chosen-prefix collision attack against SHA-1 with complexity between $2^{66.9}$ and $2^{69.4}$ (depending on assumptions about the cost of finding near-collision blocks), while the best-known attack has complexity $2^{77.1}$. This is within a small factor of the complexity of the classical collision attack on SHA-1 (estimated as $2^{64.7}$). This represents yet another warning that industries and users have to move away from using SHA-1 as soon as possible.
2019
EUROCRYPT
We present new preimage attacks on standard Keccak-224 and Keccak-256 that are reduced to 3 and 4 rounds. An allocating approach is used in the attacks, and the whole complexity is allocated to two stages, such that fewer constraints are considered and the complexity is lowered in each stage. Specifically, we are trying to find a 2-block preimage, instead of a 1-block one, for a given hash value, and the first and second message blocks are found in two stages, respectively. Both the message blocks are constrained by a set of newly proposed conditions on the middle state, which are weaker than those brought by the initial values and the hash values. Thus, the complexities in the two stages are both lower than that of finding a 1-block preimage directly. Together with the basic allocating approach, an improved method is given to balance the complexities of two stages, and hence, obtains the optimal attacks. As a result, we present the best theoretical preimage attacks on Keccak-224 and Keccak-256 that are reduced to 3 and 4 rounds. Moreover, we practically found a (second) preimage for 3-round Keccak-224 with a complexity of $2^{39.39}$.
2019
EUROCRYPT
We give the first practical instance – bison – of the Whitened Swap-Or-Not construction. After clarifying inherent limitations of the construction, we point out that this way of building block ciphers allows easy and very strong arguments against differential attacks.
2019
EUROCRYPT
We present a worst case decoding problem whose hardness reduces to that of solving the Learning Parity with Noise (LPN) problem, in some parameter regime. Prior to this work, no worst case hardness result was known for LPN (as opposed to syntactically similar problems such as Learning with Errors). The caveat is that this worst case problem is only mildly hard and in particular admits a quasi-polynomial time algorithm, whereas the LPN variant used in the reduction requires extremely high noise rate of $1/2-1/\mathrm{poly}(n)$ . Thus we can only show that “very hard” LPN is harder than some “very mildly hard” worst case problem. We note that LPN with noise $1/2-1/\mathrm{poly}(n)$ already implies symmetric cryptography.Specifically, we consider the (n, m, w)-nearest codeword problem ((n, m, w)-NCP) which takes as input a generating matrix for a binary linear code in m dimensions and rank n, and a target vector which is very close to the code (Hamming distance at most w), and asks to find the codeword nearest to the target vector. We show that for balanced (unbiased) codes and for relative error $w/m \approx {\log ^2 n}/{n}$ , (n, m, w)-NCP can be solved given oracle access to an LPN distinguisher with noise ratio $1/2-1/\mathrm{poly}(n)$ .Our proof relies on a smoothing lemma for codes which we show to have further implications: We show that (n, m, w)-NCP with the aforementioned parameters lies in the complexity class $\mathrm {{Search}\hbox {-}\mathcal {BPP}}^\mathcal {SZK}$ (i.e. reducible to a problem that has a statistical zero knowledge protocol) implying that it is unlikely to be $\mathcal {NP}$ -hard. We then show that the hardness of LPN with very low noise rate $\log ^2(n)/n$ implies the existence of collision resistant hash functions (our aforementioned result implies that in this parameter regime LPN is also in $\mathcal {BPP}^\mathcal {SZK}$ ).
2019
EUROCRYPT
A conjunction is a function $f(x_1,\dots ,x_n) = \bigwedge _{i \in S} l_i$ where $S \subseteq [n]$ and each $l_i$ is $x_i$ or $\lnot x_i$. Bishop et al. (CRYPTO 2018) recently proposed obfuscating conjunctions by embedding them in the error positions of a noisy Reed-Solomon codeword and placing the codeword in a group exponent. They prove distributional virtual black box (VBB) security in the generic group model for random conjunctions where $|S| \ge 0.226n$. While conjunction obfuscation is known from LWE [31, 47], these constructions rely on substantial technical machinery.In this work, we conduct an extensive study of simple conjunction obfuscation techniques. We abstract the Bishop et al. scheme to obtain an equivalent yet more efficient “dual” scheme that can handle conjunctions over exponential size alphabets. This scheme admits a straightforward proof of generic group security, which we combine with a novel combinatorial argument to obtain distributional VBB security for |S| of any size.If we replace the Reed-Solomon code with a random binary linear code, we can prove security from standard LPN and avoid encoding in a group. This addresses an open problem posed by Bishop et al. to prove security of this simple approach in the standard model.We give a new construction that achieves information theoretic distributional VBB security and weak functionality preservation for $|S| \ge n - n^\delta$ and $\delta < 1$. Assuming discrete log and $\delta < 1/2$, we satisfy a stronger notion of functionality preservation for computationally bounded adversaries while still achieving information theoretic security.
2019
EUROCRYPT
Distributional collision resistance is a relaxation of collision resistance that only requires that it is hard to sample a collision (x, y) where x is uniformly random and y is uniformly random conditioned on colliding with x. The notion lies between one-wayness and collision resistance, but its exact power is still not well-understood. On one hand, distributional collision resistant hash functions cannot be built from one-way functions in a black-box way, which may suggest that they are stronger. On the other hand, so far, they have not yielded any applications beyond one-way functions.Assuming distributional collision resistant hash functions, we construct constant-round statistically hiding commitment scheme. Such commitments are not known based on one-way functions, and are impossible to obtain from one-way functions in a black-box way. Our construction relies on the reduction from inaccessible entropy generators to statistically hiding commitments by Haitner et al. (STOC ’09). In the converse direction, we show that two-message statistically hiding commitments imply distributional collision resistance, thereby establishing a loose equivalence between the two notions.A corollary of the first result is that constant-round statistically hiding commitments are implied by average-case hardness in the class ${\textsf {SZK}}$ (which is known to imply distributional collision resistance). This implication seems to be folklore, but to the best of our knowledge has not been proven explicitly. We provide yet another proof of this implication, which is arguably more direct than the one going through distributional collision resistance.
2019
TOSC
In this paper we analyze for the first time the post-quantum security of AES. AES is the most popular and widely used block cipher, established as the encryption standard by the NIST in 2001. We consider the secret key setting and, in particular, AES-256, the recommended primitive and one of the few existing ones that aims at providing a post-quantum security of 128 bits. In order to determine the new security margin, i.e., the lowest number of non-attacked rounds in time less than 2128 encryptions, we first provide generalized and quantized versions of the best known cryptanalysis on reduced-round AES, as well as a discussion on attacks that don’t seem to benefit from a significant quantum speed-up. We propose a new framework for structured search that encompasses both the classical and quantum attacks we present, and allows to efficiently compute their complexity. We believe this framework will be useful for future analysis.Our best attack is a quantum Demirci-Selçuk meet-in-the-middle attack. Unexpectedly, using the ideas underlying its design principle also enables us to obtain new, counter-intuitive classical TMD trade-offs. In particular, we can reduce the memory in some attacks against AES-256 and AES-128.One of the building blocks of our attacks is solving efficiently the AES S-Box differential equation, with respect to the quantum cost of a reversible S-Box. We believe that this generic quantum tool will be useful for future quantum differential attacks. Judging by the results obtained so far, AES seems a resistant primitive in the post-quantum world as well as in the classical one, with a bigger security margin with respect to quantum generic attacks.
2019
TOSC
The conditional cube attack on round-reduced Keccak keyed modes was proposed by Huang et al. at EUROCRYPT 2017. In their attack, a conditional cube variable was introduced, whose diffusion was significantly reduced by certain key bit conditions. The attack requires a set of cube variables which are not multiplied in the first round while the conditional cube variable is not multiplied with other cube variables (called ordinary cube variables) in the first two rounds. This has an impact on the degree of the output of Keccak and hence gives a distinguisher. Later, the MILP method was applied to find ordinary cube variables. However, for some Keccak based versions with few degrees of freedom, one could not find enough ordinary cube variables, which weakens or even invalidates the conditional cube attack.In this paper, a new conditional cube attack on Keccak is proposed. We remove the limitation that no cube variables multiply with each other in the first round. As a result, some quadratic terms may appear in the first round. We make use of some new bit conditions to prevent the quadratic terms from multiplying with other cube variables in the second round, so that there will be no cubic terms in the first two rounds. Furthermore, we introduce the kernel quadratic term and construct a 6-2-2 pattern to reduce the diffusion of quadratic terms significantly, where the Θ operation even in the second round becomes an identity transformation (CP-kernel property) for the kernel quadratic term. Previous conditional cube attacks on Keccak only explored the CP-kernel property of Θ operation in the first round. Therefore, more degrees of freedom are available for ordinary cube variables and fewer bit conditions are used to remove the cubic terms in the second round, which plays a key role in the conditional cube attack on versions with very few degrees of freedom. We also use the MILP method in the search of cube variables and give key-recovery attacks on round-reduced Keccak keyed modes.As a result, we reduce the time complexity of key-recovery attacks on 7-round Keccak-MAC-512 and 7-round Ketje Sr v2 from 2111, 299 to 272, 277, respectively. Additionally, we have reduced the time complexity of attacks on 9-round KMAC256 and 7-round Ketje Sr v1. Besides, practical attacks on 6-round Ketje Sr v1 and v2 are also given in this paper for the first time.
2019
TOSC
PMAC is a rate-1, parallelizable, block-cipher-based message authentication code (MAC), proposed by Black and Rogaway (EUROCRYPT 2002). Improving the security bound is a main research topic for PMAC. In particular, showing a tight bound is the primary goal of the research, since Luykx et al.’s paper (EUROCRYPT 2016). Regarding the pseudo-random-function (PRF) security of PMAC, a collision of the hash function, or the difference between a random permutation and a random function offers the lower bound Ω(q2/2n) for q queries and the block cipher size n. Regarding the MAC security (unforgeability), a hash collision for MAC queries, or guessing a tag offers the lower bound Ω(q2m /2n + qv/2n) for qm MAC queries and qv verification queries (forgery attempts). The tight upper bound of the PRF-security O(q2/2n) of PMAC was given by Gaži et el. (ToSC 2017, Issue 1), but their proof requires a 4-wise independent masking scheme that uses 4 n-bit random values. Open problems from their work are: (1) find a masking scheme with three or less random values with which PMAC has the tight upper bound for PRF-security; (2) find a masking scheme with which PMAC has the tight upper bound for MAC-security.In this paper, we consider PMAC with two powering-up masks that uses two random values for the masking scheme. Using the structure of the powering-up masking scheme, we show that the PMAC has the tight upper bound O(q2/2n) for PRF-security, which answers the open problem (1), and the tight upper bound O(q2m /2n + qv/2n) for MAC-security, which answers the open problem (2). Note that these results deal with two-key PMACs, thus showing tight upper bounds of PMACs with single-key and/or with one powering-up mask are open problems.
2019
TOSC
ISO/IEC 9797-1 is an international standard for block-cipher-based Message Authentication Code (MAC). The current version ISO/IEC 9797-1:2011 specifies six single-pass CBC-like MAC structures that are capped at the birthday bound security. For a higher security that is beyond-birthday bound, it recommends to use the concatenation combiner of two single-pass MACs. In this paper, we reveal the invalidity of the suggestion, by presenting a birthday bound forgery attack on the concatenation combiner, which is essentially based on Joux’s multi-collision. Notably, our new forgery attack for the concatenation of two MAC Algorithm 1 with padding scheme 2 only requires 3 queries. Moreover, we look for patches by revisiting the development of ISO/IEC 9797-1 with respect to the beyond-birthday bound security. More specifically, we evaluate the XOR combiner of single-pass CBC-like MACs, which was used in previous version of ISO/IEC 9797-1.
2019
TOSC
S-boxes, typically the only nonlinear part of a block cipher, are the heart of symmetric cryptographic primitives. They significantly impact the cryptographic strength and the implementation characteristics of an algorithm. Due to their simplicity, quadratic vectorial Boolean functions are preferred when efficient implementations for a variety of applications are of concern. Many characteristics of a function stay invariant under affine equivalence. So far, all 6-bit Boolean functions, 3- and 4-bit permutations have been classified up to affine equivalence. At FSE 2017, Bozoliv et al. presented the first classification of 5-bit quadratic permutations. In this work, we propose an adaptation of their work resulting in a highly efficient algorithm to classify n x m functions for n ≥ m. Our algorithm enables for the first time a complete classification of 6-bit quadratic permutations as well as all balanced quadratic functions for n ≤ 6. These functions can be valuable for new cryptographic algorithm designs with efficient multi-party computation or side-channel analysis resistance as goal. In addition, we provide a second tool for finding decompositions of length two. We demonstrate its use by decomposing existing higher degree S-boxes and constructing new S-boxes with good cryptographic and implementation properties.
2019
TOSC
We define ZOCB and ZOTR for nonce-based authenticated encryption with associated data, and analyze their provable security. These schemes use a tweakable blockcipher (TBC) as the underlying primitive, and fully utilize its input to process a plaintext and associated data (AD). This property is commonly referred to as full absorption, and this has been explored for schemes based on a permutation or a pseudorandom function (PRF). Our schemes improve the efficiency of TBC-based counterparts of OCB and OTR called OCB3 (Krovetz and Rogaway, FSE 2011) and OTR (Minematsu, EUROCRYPT 2014). Specifically, ΘCB3 and OTR have an independent part to process AD, and our schemes integrate this process into the encryption part of a plaintext by using the tweak input of the TBC. Up to a certain length of AD, ZOCB and ZOTR completely eliminate the independent process for it. Even for longer AD, our schemes process it efficiently by fully using the tweak input of the TBC. For this purpose, based on previous tweak extension schemes for TBCs, we introduce a scheme called XTX*. To our knowledge, ZOCB and ZOTR are the first efficiency improvement of ΘCB3 and OTR in terms of the number of TBC calls. Compared to Sponge-based and PRF-based schemes, ZOCB and ZOTR allow fully parallel computation of the underlying primitive, and have a unique design feature that an authentication tag is independent of a part of AD. We present experimental results illustrating the practical efficiency gain and clarifying the efficiency cost for it with a concrete instantiation. The results show that for long input data, our schemes have gains, while we have efficiency loss for short input data.
2004
EUROCRYPT
2019 IACR Test of Time Award
1999
PKC
PKC Test of Time Award
2006
TCC
TCC Test of Time Award
2005
TCC
TCC Test of Time Award
2006
TCC
TCC Test of Time Award
2004
ASIACRYPT
2019 IACR Test of Time Award
2004
TCC
TCC Test of Time Award
2004
TCC
TCC Test of Time Award
2000
PKC
PKC Test of Time Award
2001
JOFC
PKC Test of Time Award
2001
PKC
PKC Test of Time Award
2004
CRYPTO
2019 IACR Test of Time Award
2009
EUROCRYPT
Best paper
2011
FSE
2011
CHES
2012
CRYPTO
Best Paper
2011
CRYPTO
Best Paper
2010
ASIACRYPT
Best Paper
2016
CRYPTO
Early Career Award
2017
CHES
Best Paper
Synchrotron-based X-ray nanobeams are investigated as a tool to perturb microcontroller circuits. An intense hard X-ray focused beam of a few tens of nanometers is used to target the flash, EEPROM and RAM memory of a circuit. The obtained results show that it is possible to corrupt a single transistor in a semi-permanent state. A simple heat treatment can remove the induced effect, thus making the corruption reversible. An attack on a code stored in flash demonstrates unambiguously that this new technique can be a threat to the security of integrated circuits.
2008
TCC
2013
CRYPTO
Best Paper
2015
FSE
Best Paper
2005
CRYPTO
Best paper
2006
PKC
Best Paper
2007
CHES
Best Paper
2009
PKC
2011
ASIACRYPT
Best Paper
2011
ASIACRYPT
Best Paper
2006
EUROCRYPT
Best Paper
2005
EUROCRYPT
Best Paper
2016
CRYPTO
Best Paper
2005
EUROCRYPT
Best Paper
2015
CHES
Best Paper
2017
CRYPTO
Best paper
2008
PKC
2011
EUROCRYPT
Best paper
2010
TCC
Best Student Paper
2013
CRYPTO
Best Young-Author Paper
2005
CHES
Best Paper
2017
CRYPTO
Best paper
2006
ASIACRYPT
2018
TCC
Best Student Paper
The possibility of basing cryptography on the minimal assumption $\mathbf{NP }\nsubseteq \mathbf{BPP }$ NP⊈BPP is at the very heart of complexity-theoretic cryptography. The closest we have gotten so far is lattice-based cryptography whose average-case security is based on the worst-case hardness of approximate shortest vector problems on integer lattices. The state-of-the-art is the construction of a one-way function (and collision-resistant hash function) based on the hardness of the $\tilde{O}(n)$ O~(n)-approximate shortest independent vector problem ${\textsf {SIVP}}_{\tilde{O}(n)}$ SIVPO~(n).Although ${\textsf {SIVP}}$ SIVP is NP-hard in its exact version, Guruswami et al. (CCC 2004) showed that ${\textsf {gapSIVP}}_{\sqrt{n/\log n}}$ gapSIVPn/logn is in $\mathbf{NP } \cap \mathbf{coAM }$ NP∩coAM and thus unlikely to be $\mathbf{NP }$ NP-hard. Indeed, any language that can be reduced to ${\textsf {gapSIVP}}_{\tilde{O}(\sqrt{n})}$ gapSIVPO~(n) (under general probabilistic polynomial-time adaptive reductions) is in $\mathbf{AM } \cap \mathbf{coAM }$ AM∩coAM by the results of Peikert and Vaikuntanathan (CRYPTO 2008) and Mahmoody and Xiao (CCC 2010). However, none of these results apply to reductions to search problems, still leaving open a ray of hope: can $\mathbf{NP }$ NPbe reduced to solving search SIVP with approximation factor $\tilde{O}(n)$ O~(n)?We eliminate such possibility, by showing that any language that can be reduced to solving search ${\textsf {SIVP}}$ SIVP with any approximation factor $\lambda (n) = \omega (n\log n)$ λ(n)=ω(nlogn) lies in AM intersect coAM.
2008
CRYPTO
Best Paper