Léo Ducas

Lattice Signatures and Bimodal Gaussians

Our main result is a construction of a lattice-based digital signature scheme that represents an improvement, both in theory and in practice, over today’s most efficient lattice schemes. The novel scheme is obtained as a result of a modification of the rejection sampling algorithm that is at the heart of Lyubashevsky’s signature scheme (Eurocrypt, 2012) and several other lattice primitives. Our new rejection sampling algorithm which samples from a bimodal Gaussian distribution, combined with a modified scheme instantiation, ends up reducing the standard deviation of the resulting signatures by a factor that is asymptotically square root in the security parameter. The implementations of our signature scheme for security levels of 128, 160, and 192 bits compare very favorably to existing schemes such as RSA and ECDSA in terms of efficiency. In addition, the new scheme has shorter signature and public key sizes than all previously proposed lattice signature schemes.

FHEW: Bootstrapping Homomorphic Encryption in Less Than a Second

The main bottleneck affecting the efficiency of all known fully homomorphic encryption (FHE) schemes is Gentry’s bootstrapping procedure, which is required to refresh noisy ciphertexts and keep computing on encrypted data. Bootstrapping in the latest implementation of FHE, the HElib library of Halevi and Shoup (Crypto 2014), requires about six minutes. We present a new method to homomorphically compute simple bit operations, and refresh (bootstrap) the resulting output, which runs on a personal computer in just about half a second. We present a detailed technical analysis of the scheme (based on the worst-case hardness of standard lattice problems) and report on the performance of our prototype implementation.

Anonymity from asymmetry: new constructions for anonymous HIBE

A Hierarchical Identity Based Encryption (HIBE) system is anonymous if the ciphertext reveals no information about the recipients identity. create it. While there are multiple constructions for secure HIBE, far fewer constructions exist for anonymous HIBE. In this paper we show how to use asymmetric pairings to convert a large family of IBE and HIBE constructions into anonymous IBE and HIBE systems. We also obtain a delegatable-HVE which is a generalization of anonymous HIBE.

Frodo: Take off the Ring! Practical, Quantum-Secure Key Exchange from LWE

Lattice-based cryptography offers some of the most attractive primitives believed to be resistant to quantum computers. Following increasing interest from both companies and government agencies in building quantum computers, a number of works have proposed instantiations of practical post-quantum key exchange protocols based on hard problems in ideal lattices, mainly based on the Ring Learning With Errors (R-LWE) problem. While ideal lattices facilitate major efficiency and storage benefits over their non-ideal counterparts, the additional ring structure that enables these advantages also raises concerns about the assumed difficulty of the underlying problems. Thus, a question of significant interest to cryptographers, and especially to those currently placing bets on primitives that will withstand quantum adversaries, is how much of an advantage the additional ring structure actually gives in practice. Despite conventional wisdom that generic lattices might be too slow and unwieldy, we demonstrate that LWE-based key exchange is quite practical: our constant time implementation requires around 1.3ms computation time for each party; compared to the recent NewHope R-LWE scheme, communication sizes increase by a factor of 4.7x, but remain under 12 KiB in each direction. Our protocol is competitive when used for serving web pages over TLS; when partnered with ECDSA signatures, latencies increase by less than a factor of 1.6x, and (even under heavy load) server throughput only decreases by factors of 1.5x and 1.2x when serving typical 1 KiB and 100 KiB pages, respectively. To achieve these practical results, our protocol takes advantage of several innovations. These include techniques to optimize communication bandwidth, dynamic generation of public parameters (which also offers additional security against backdoors), carefully chosen error distributions, and tight security parameters.

Enhanced Lattice-Based Signatures on Reconfigurable Hardware

The recent Bimodal Lattice Signature Scheme Bliss showed that lattice-based constructions have evolved to practical alternatives to RSA or ECC. Besides reasonably small signatures with 5600 bits for a 128-bit level of security, Bliss enables extremely fast signing and signature verification in software. However, due to the complex sampling of Gaussian noise with high precision, it is not clear whether this scheme can be mapped efficiently to embedded devices. Even though the authors of Bliss also proposed a new sampling algorithm using Bernoulli variables this approach is more complex than previous methods using large precomputed tables. The clear disadvantage of using large tables for high performance is that they cannot be used on constrained computing environments, such as FPGAs, with limited memory. In this work we thus present techniques for an efficient Cumulative Distribution Table CDT based Gaussian sampler on reconfigurable hardware involving Peikerts convolution lemma and the Kullback-Leibler divergence. Based on our enhanced sampler design, we provide a first Bliss architecture for Xilinx Spartan-6 FPGAs that integrates fast FFT/NTT-based polynomial multiplication, sparse multiplication, and a Keccak hash function. Additionally, we compare the CDT with the Bernoulli approach and show that for the particular Bliss-I parameter set the improved CDT approach is faster with lower area consumption. Our core uses 2,431 slices, 7.5 BRAMs, and 6 DSPs and performs a signing operation in 126 μs on average. Verification takes even less with 70 μs.

Learning a zonotope and more: cryptanalysis of NTRUSign countermeasures

NTRUSign is the most practical lattice signature scheme. Its basic version was broken by Nguyen and Regev in 2006: one can efficiently recover the secret key from about 400 signatures. However, countermeasures have been proposed to repair the scheme, such as the perturbation used in NTRUSign standardization proposals, and the deformation proposed by Hu et al. at IEEE Trans. Inform. Theory in 2008. These two countermeasures were claimed to prevent the NR attack. Surprisingly, we show that these two claims are incorrect by revisiting the NR gradient-descent attack: the attack is more powerful than previously expected, and actually breaks both countermeasures in practice, e.g. 8,000 signatures suffice to break NTRUSign-251 with one perturbation as submitted to IEEE P1363 in 2003. More precisely, we explain why the Nguyen-Regev algorithm for learning a parallelepiped is heuristically able to learn more complex objects, such as zonotopes and deformed parallelepipeds.

Faster gaussian lattice sampling using lazy floating-point arithmetic

Many lattice cryptographic primitives require an efficient algorithm to sample lattice points according to some Gaussian distribution. All algorithms known for this task require long-integer arithmetic at some point, which may be problematic in practice. We study how much lattice sampling can be sped up using floating-point arithmetic. First, we show that a direct floating-point implementation of these algorithms does not give any asymptotic speedup: the floating-point precision needs to be greater than the security parameter, leading to an overall complexity O(n3) where n is the lattice dimension. However, we introduce a laziness technique that can significantly speed up these algorithms. Namely, in certain cases such as NTRUSign lattices, laziness can decrease the complexity to O(n2) or even O(n). Furthermore, our analysis is practical: for typical parameters, most of the floating-point operations only require the double-precision IEEE standard.

Short Stickelberger Class Relations and Application to Ideal-SVP

The worst-case hardness of finding short vectors in ideals of cyclotomic number fields (Ideal-SVP) is a central matter in lattice based cryptography. Assuming the worst-case hardness of Ideal-SVP allows to prove the Ring-LWE and Ring-SIS assumptions, and therefore to prove the security of numerous cryptographic schemes and protocols — including key-exchange, digital signatures, public-key encryption and fully-homomorphic encryption.

Sanitization of FHE Ciphertexts

By definition, fully homomorphic encryption FHE schemes support homomorphic decryption, and all known FHE constructions are bootstrapped from a Somewhat Homomorphic Encryption SHE scheme via this technique. Additionally, when a public key is provided, ciphertexts are also re-randomizable, e.g., by adding to them fresh encryptions ofi¾?0. From those two operations we devise an algorithm to sanitize a ciphertext, by making its distribution canonical. In particular, the distribution of the ciphertext does not depend on the circuit that led to it via homomorphic evaluation, thus providing circuit privacy in the honest-but-curious model. Unlike the previous approach based on noise flooding, our approach does not degrade much the security/efficiency trade-off of the underlying FHE. The technique can be applied to all lattice-based FHE proposed so far, without substantially affecting their concrete parameters.

CRYSTALS-Dilithium: A Lattice-Based Digital Signature Scheme

In this paper, we present the lattice-based signature scheme Dilithium, which is a component of the CRYSTALS (Cryptographic Suite for Algebraic Lattices) suite that was submitted to NIST’s call for post-quantum cryptographic standards. The design of the scheme avoids all uses of discrete Gaussian sampling and is easily implementable in constant-time. For the same security levels, our scheme has a public key that is 2.5X smaller than the previously most efficient lattice-based schemes that did not use Gaussians, while having essentially the same signature size. In addition to the new design, we significantly improve the running time of the main component of many lattice-based constructions – the number theoretic transform. Our AVX2-based implementation results in a speed-up of roughly a factor of 2 over the previously best algorithms that appear in the literature. The techniques for obtaining this speed-up also have applications to other lattice-based schemes.