Team, Visitors, External Collaborators
Overall Objectives
Research Program
Application Domains
Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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Section: New Results

Lattices: algorithms and cryptology

Approx-SVP in ideal lattices with pre-processing

In [28], 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|Δ| with 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(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(O˜((log|Δ|)max(2/3,1-2α))) in the classical setting, and exp(O˜((log|Δ|)1-2α)) in the quantum setting. The parameter 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 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.

An LLL algorithm for module lattices

The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors. In [23], we provide a generalization to R-modules contained in Kn for arbitrary number fields K and dimension n, with R denoting the ring of integers of K. Concretely, we introduce an algorithm that efficiently finds short vectors in rank-n modules when given access to an oracle that finds short vectors in rank-2 modules, and an algorithm that efficiently finds short vectors in rank-2 modules given access to a Closest Vector Problem oracle for a lattice that depends only on K. The second algorithm relies on quantum computations and its analysis is heuristic.

The general sieve kernel and new records in lattice reduction

In [14], we propose the General Sieve Kernel (G6K), 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.

Statistical zeroizing attack: cryptanalysis of candidates of BP obfuscation over GGH15 multilinear map

In [19], we present a new cryptanalytic algorithm on obfuscations based on GGH15 multilinear map. Our algorithm, statistical zeroizing attack, directly distinguishes two distributions from obfuscation while it follows the zeroizing attack paradigm, that is, it uses evaluations of zeros of obfuscated programs.

Our attack breaks the recent indistinguishability obfuscation candidate suggested by Chen et al. (CRYPTO'18) for the optimal parameter settings. More precisely, we show that there are two functionally equivalent branching programs whose CVW obfuscations can be efficiently distinguished by computing the sample variance of evaluations.

This statistical attack gives a new perspective on the security of the indistinguishability obfuscations: we should consider the shape of the distributions of evaluation of obfuscation to ensure security.

In other words, while most of the previous (weak) security proofs have been studied with respect to algebraic attack model or ideal model, our attack shows that this algebraic security is not enough to achieve indistinguishability obfuscation. In particular, we show that the obfuscation scheme suggested by Bartusek et al. (TCC'18) does not achieve the desired security in a certain parameter regime, in which their algebraic security proof still holds.

The correctness of statistical zeroizing attacks holds under a mild assumption on the preimage sampling algorithm with a lattice trapdoor. We experimentally verify this assumption for implemented obfuscation by Halevi et al. (ACM CCS'17).

Cryptanalysis of the CLT13 multilinear map

The reference [6] is the journal version of the Eurocrypt'15 article with the same title and authors.

Multi-Client Functional Encryption for Linear Functions in the Standard Model from LWE

Multi-client functional encryption (MCFE) allows clients to encrypt ciphertexts 𝐂t,1,𝐂t,2,...,𝐂t, under some label. Each client can encrypt his own data Xi for a label t using a private encryption key 𝖾𝗄i issued by a trusted authority in such a way that, as long as all 𝐂t,i share the same label t, an evaluator endowed with a functional key 𝖽𝗄f can evaluate f(X1,X2,...,X) without learning anything else on the underlying plaintexts Xi. Functional decryption keys can be derived by the central authority using the master secret key. Under the Decision Diffie-Hellman assumption, Chotard et al. (Asiacrypt 2018) recently described an adaptively secure MCFE scheme for the evaluation of linear functions over the integers. They also gave a decentralized variant (DMCFE) of their scheme which does not rely on a centralized authority, but rather allows encryptors to issue functional secret keys in a distributed manner. While efficient, their constructions both rely on random oracles in their security analysis. In [27], we build a standard-model MCFE scheme for the same functionality and prove it fully secure under adaptive corruptions. Our proof relies on the Learning-With-Errors (LWE) assumption and does not require the random oracle model. We also provide a decentralized variant of our scheme, which we prove secure in the static corruption setting (but for adaptively chosen messages) under the LWE assumption.

Zero-Knowledge Elementary Databases with More Expressive Queries

Zero-knowledge elementary databases (ZK-EDBs) are cryptographic schemes that allow a prover to commit to a set D of key-value pairs so as to be able to prove statements such as “x belongs to the support of D and D(x) = y” or “x is not in the support of D”. Importantly , proofs should leak no information beyond the proven statement and even the size of D should remain private. Chase et al. (Eurocrypt'05) showed that ZK-EDBs are implied by a special flavor of non-interactive commitment, called mercurial commitment, which enables efficient instantiations based on standard number theoretic assumptions. On the other hand, the resulting ZK-EDBs are only known to support proofs for simple statements like (non-)membership and value assignments. In [25], we show that mercurial commitments actually enable significantly richer queries. We show that, modulo an additional security property met by all known efficient constructions, they actually enable range queries over keys and values-even for ranges of super-polynomial size-as well as membership/non-membership queries over the space of values. Beyond that, we exploit the range queries to realize richer queries such as k-nearest neighbors and revealing the k smallest or largest records within a given range. In addition, we provide a new realization of trapdoor mercurial commitment from standard lattice asssumptions, thus obtaining the most expressive quantum-safe ZK-EDB construction so far.

Lossy Algebraic Filters With Short Tags

Lossy algebraic filters (LAFs) are function families where each function is parametrized by a tag, which determines if the function is injective or lossy. While initially introduced by Hofheinz (Eurocrypt 2013) as a technical tool to build encryption schemes with key-dependent message chosen-ciphertext (KDM-CCA) security, they also find applications in the design of robustly reusable fuzzy extractors. So far, the only known LAF family requires tags comprised of Θ(n2) group elements for functions with input space p, where p is the group order. In [26], we describe a new LAF family where the tag size is only linear in n and prove it secure under simple assumptions in asymmetric bilinear groups. Our construction can be used as a drop-in replacement in all applications of the initial LAF system. In particular, it can shorten the ciphertexts of Hofheinz's KDM-CCA-secure public-key encryption scheme by 19 group elements. It also allows substantial space improvements in a recent fuzzy extractor proposed by Wen and Liu (Asiacrypt 2018). As a second contribution , we show how to modify our scheme so as to prove it (almost) tightly secure, meaning that security reductions are not affected by a concrete security loss proportional to the number of adversarial queries.

Shorter Quadratic QA-NIZK Proofs

Despite recent advances in the area of pairing-friendly Non-Interactive Zero-Knowledge proofs, there have not been many efficiency improvements in constructing arguments of satisfiability of quadratic (and larger degree) equations since the publication of the Groth-Sahai proof system (J. of Cryptology 2012). In [20], we address the problem of aggregating such proofs using techniques derived from the interactive setting and recent constructions of SNARKs. For certain types of quadratic equations, this problem was investigated before by González et al. (Asiacrypt'15). Compared to their result, we reduce the proof size by approximately 50

Shorter Pairing-based Arguments under Standard Assumptions

The paper [22] constructs efficient non-interactive arguments for correct evaluation of arithmetic and Boolean circuits with proof size O(d) group elements, where d is the multiplicative depth of the circuit, under falsifiable assumptions. This is achieved by combining techniques from SNARKs and QA-NIZK arguments of membership in linear spaces. The first construction is very efficient (the proof size is 4d group elements and the verification cost is 4d pairings and O(n+n+d) exponentiations, where n is the size of the input and n of the output) but one type of attack can only be ruled out assuming the knowledge soundness of QA-NIZK arguments of membership in linear spaces. We give an alternative construction which replaces this assumption with a decisional assumption in bilinear groups at the cost of approximately doubling the proof size. The construction for Boolean circuits can be made zero-knowledge with Groth-Sahai proofs, resulting in a NIZK argument for circuit satisfiability based on falsifiable assumptions in bilinear groups of proof size O(n+d). Our main technical tool is what we call an “argument of knowledge transfer”. Given a commitment C1 and an opening x, such an argument allows to prove that some other commitment C2 opens to f(x), for some function f, even if C2 is not extractable. We construct very short, constant-size, pairing-based arguments of knowledge transfer with constant-time verification for any linear function and also for Hadamard products. These allow to transfer the knowledge of the input to lower levels of the circuit.

Shorter Ring Signatures from Standard Assumptions

Ring signatures, introduced by Rivest, Shamir and Tauman (ASIACRYPT 2001), allow to sign a message on behalf of a set of users while guaranteeing authenticity and anonymity. Groth and Kohlweiss (EUROCRYPT 2015) and Libert et al. (EUROCRYPT 2016) constructed schemes with signatures of size logarithmic in the number of users. An even shorter ring signature, of size independent from the number of users, was recently proposed by Malavolta and Schroeder (ASIACRYPT 2017). However, all these short signatures are obtained relying on strong and controversial assumptions. Namely, the former schemes are both proven secure in the random oracle model while the later requires non-falsifiable assumptions.

The most efficient construction under mild assumptions remains the construction of Chandran et al. (ICALP 2007) with a signature of size Θ(n), where n is the number of users, and security is based on the Diffie-Hellman assumption in bilinear groups (the SXDH assumption in asymmetric bilinear groups).

In [21], we construct an asymptotically shorter ring signature from the hardness of the Diffie-Hellman assumption in bilinear groups. Each signature comprises Θ(n1/3) group elements, signing a message requires computing Θ(n1/3) exponentiations, and verifying a signature requires Θ(n2/3) pairing operations.

Two-Party ECDSA from Hash Proof Systems and Efficient Instantiations

ECDSA is a widely adopted digital signature standard. Unfortunately, efficient distributed variants of this primitive are notoriously hard to achieve and known solutions often require expensive zero knowledge proofs to deal with malicious adversaries. For the two party case, Lindell (CRYPTO 2017) recently managed to get an efficient solution which, to achieve simulation-based security, relies on an interactive, non standard, assumption on Paillier's cryptosystem.

In this paper [18] we generalize Lindell's solution using hash proof systems. The main advantage of our generic method is that it results in a simulation-based security proof without resorting to non-standard interactive assumptions.

Moving to concrete constructions, we show how to instantiate our framework using class groups of imaginary quadratic fields. Our implementations show that the practical impact of dropping such interactive assumptions is minimal. Indeed, while for 128-bit security our scheme is marginally slower than Lindell's, for 256-bit security it turns out to be better both in key generation and signing time. Moreover, in terms of communication cost, our implementation significantly reduces both the number of rounds and the transmitted bits without exception.

Algebraic XOR-RKA-Secure Pseudorandom Functions from Post-Zeroizing Multilinear Maps

In [13], we construct the first pseudorandom functions that resist a strong class of attacks where an adversary is able to run the cryptosystem not only with the fixed secret key, but with related keys where bits of its choice of the original keys are flipped. This problem is motivated by practical attacks that have been performed against physical devices. Our construction guarantees that every output of our construction, for the original key or for tampered keys, are pseudorandom, i.e. are computationally hard to distinguish from truly random values. To achieve this, we rely on a recent tool introduced in cryptography and termed multilinear maps. While multilinear maps have been recently attacked by several techniques, we prove that our construction remains secure despite the numerous vulnerabilities of current constructions of multilinear maps.

Unifying Leakage Models on a Rényi Day

Most theoretical models in cryptography suppose that an attacker can only observe the input/output behavior of a cryptosystem and nothing more. Yet, in the real world, cryptosystems run on physical devices and auxiliary information leaks from these devices. This leakage can sometimes be used to attack the system, even though it is proven secure in theory. To circumvent these issues, cryptographers have introduces several new security models in an attempt to encompass the different forms of leakage. Some models are simple, such as the probing model, and simple compilers allow to transform a system into one secure in the probing model, while some more realistic problems such as the noisy-leakage model are very involved. In [29], we show that these models are actually equivalent, proving in particular that the simple compilers are sufficient to guarantee security in realistic environments.