International Association
for Cryptologic Research

Threshold signatures improve both availability and security of digital signatures by splitting the signing key into $N$ shares handed out to different parties. Later on, any subset of at least $T$ parties can cooperate to produce a signature on a given message. While threshold signatures have been extensively studied in the pre-quantum setting, they remain sparse from quantum-resilient assumptions. We present the first efficient lattice-based threshold signatures with signature size 13~KiB and communication cost 40~KiB per user, supporting a threshold size as large as 1024~signers. We provide an accompanying high performance implementation. The security of the scheme is based on the same assumptions as Dilithium, a signature recently selected by NIST for standardisation which, as far as we know, cannot easily be made threshold efficiently. All operations used during signing are due to symmetric primitives and simple lattice operations; in particular our scheme does not need heavy tools such as threshold fully homomorphic encryption or homomorphic trapdoor commitments as in prior constructions. The key technical idea is to use _one-time additive masks_ to mitigate the leakage of the partial signing keys through partial signatures.

We introduce a toolkit for transforming lattice-based hash-and-sign signature schemes into masking-friendly signatures secure in the t-probing model. Until now, efficiently masking lattice-based hash-and-sign schemes was an open problem unsuccessful attempts such as Mitaka. Our toolkit includes noise flooding to mitigate statistical leaks and an extended Strong Non-Interfering probing security property (SNIu) for masking gadgets to handle unshared inputs. Our main conceptual contribution lies in finding a systematic way to use noise flooding within the hash-and-sign paradigm. Our main technical contribution is to formalize, prove, instantiate and implement a hash-and-sign scheme based on these techniques. We showcase the efficiency of our techniques in a signature scheme, Plover-RLWE, based on (hint) Ring-LWE. It is the first lattice-based masked hash-and-sign scheme with quasi-linear complexity O(d log d) in the number of shares d. Our performances are competitive with the state-of-the-art masking-friendly signature, the Fiat-Shamir scheme Raccoon.

This paper present Raccoon, a lattice-based signature scheme submitted to the NIST 2022 call for additional post-quantum signatures. Raccoon has the specificity of always being masked. Concretely, all sensitive intermediate values are shared into $d$ parts. The main design rationale of Raccoon is to be easy to mask at high orders, and this dictated most of its design choices, such as the introduction of new algorithmic techniques for sampling small errors. As a result, Raccoon achieves a masking overhead $O(d \log d)$ that compares favorably with the overheads $O(d^2 \log q)$ observed when masking standard lattice signatures. In addition, we formally prove the security of Raccoon in the $t$-probing model: an attacker is able to probe $t <d-1$ shares during each execution of the main algorithms (key generation, signing, verification). While for most cryptographic schemes, the black-box security $t$-probing security can be studied in isolation, in Raccoon this analysis is performed jointly. To that end, a bridge must be made between the black-box game-based EUF-CMA proof and the usual simulation proofs of the ISW model (CRYPTO 2003). In this paper, we formalize an end-to-end masking proof by deploying the probing EUF-CMA introduced by Barthe et al. (Eurocrypt 2018) and exhibiting the simulators of the non-interference properties (Barthe et al. CCS 2016). More precisely, the proof is divided into three novel parts: - a simulation proof in the ISW model that allows to propagate the dependancy to a restricted number of inputs and random coins, - a game-based proof showing that the security of Raccoon with probes can be reduced to an instance of Raccoon with smaller parameters, - a parameter study to ensure that the smaller instance is secure, using a robust generalization of the Rényi divergence. While we apply our techniques to Raccoon, we expect that the algorithmic and proof techniques we introduce will be helpful for the design and analysis of future masking-friendly schemes.

We propose a new framework based on random submersions — that is projection over a random subspace blinded by a small Gaussian noise — for constructing verifiable short secret sharing and showcase it to construct efficient threshold lattice-based signatures in the hash-and-sign paradigm, when based on noise flooding. This is, to our knowledge, the first hash-and-sign lattice-based threshold signature. Our threshold signature enjoys the very desirable property of robustness, including at key generation. In practice, we are able to construct a robust hash-and-sign threshold signature for threshold and provide a typical parameter set for threshold T = 16 and signature size 13kB. Our constructions are provably secure under standard MLWE assumption in the ROM and only require basic primitives as building blocks. In particular, we do not rely on FHE-type schemes.

Mitaka is a lattice-based signature proposed at Eurocrypt 2022. A key advertised feature of Mitaka is that it can be masked at high orders efficiently, making it attractive in scenarios where side-channel attacks are a concern. Mitaka comes with a claimed security proof in the t-probing model. We uncover a flaw in the security proof of Mitaka, and subsequently show that it is not secure in the t-probing model. For any number of shares d ≥ 4, probing t < d variables per execution allows an attacker to recover the private key efficiently with approximately 2^21 executions. Our analysis shows that even a constant number of probes suffices (t = 3), as long as the attacker has access to a number of executions that is linear in d/t.

The Signal protocol is a secure instant messaging protocol that underlies the security of numerous applications such as WhatsApp, Skype, Facebook Messenger among many others. The Signal protocol consists of two sub-protocols known as the X3DH protocol and the double ratchet protocol, where the latter has recently gained much attention. For instance, Alwen, Coretti, and Dodis (Eurocrypt’19) provided a concrete security model along with a generic construction based on simple building blocks that are instantiable from versatile assumptions, including post-quantum ones. In contrast, as far as we are aware, works focusing on the X3DH protocol seem limited. In this work, we cast the X3DH protocol as a specific type of authenticated key exchange (AKE) protocol, which we call a Signal-conforming AKE protocol, and formally define its security model based on the vast prior works on AKE protocols. We then provide the first efficient generic construction of a Signal-conforming AKE protocol based on standard cryptographic primitives such as key encapsulation mechanisms (KEM) and signature schemes. Specifically, this results in the first post-quantum secure replacement of the X3DH protocol based on well-established assumptions. Similar to the X3DH protocol, our Signal-conforming AKE protocol offers a strong (or stronger) flavor of security, where the exchanged key remains secure even when all the non-trivial combinations of the long-term secrets and session-specific secrets are compromised. Moreover, our protocol has a weak flavor of deniability and we further show how to progressively strengthen it using ring signatures and/or non-interactive zero-knowledge proof systems. Finally, we provide a full-fledged, generic C implementation of our (weakly deniable) protocol. We instantiate it with several Round 3 candidates (finalists and alternates) to the NIST post-quantum standardization process and compare the resulting bandwidth and computation performances. Our implementation is publicly available.

We propose a new framework for (trapdoor) sampling over lattices. Our framework can be instantiated in a number of ways. It allows for example to sample from uniform, affine and “product affine” distributions. Another salient point of our framework is that the output distributions of our samplers are perfectly indistinguishable from ideal ones, in contrast with classical samplers that are statistically indistinguishable. One caveat of our framework is that all our current instantiations entail a rather large standard deviation.

The Signal protocol is a secure instant messaging protocol that underlies the security of numerous applications such as WhatsApp, Skype, Facebook Messenger among many others. The Signal protocol consists of two sub-protocols known as the X3DH protocol and the double ratchet protocol, where the latter has recently gained much attention. For instance, Alwen, Coretti, and Dodis (Eurocrypt'19) provided a concrete security model along with a generic construction based on simple building blocks that are instantiable from versatile assumptions, including post-quantum ones. In contrast, as far as we are aware, works focusing on the X3DH protocol seem limited. In this work, we cast the X3DH protocol as a specific type of authenticated key exchange (AKE) protocol, which we call a Signal-conforming AKE protocol, and formally define its security model based on the vast prior work on AKE protocols. We then provide the first efficient generic construction of a Signal-conforming AKE protocol based on standard cryptographic primitives such as key encapsulation mechanisms (KEM) and signature schemes. Specifically, this results in the first post-quantum secure replacement of the X3DH protocol on well-established assumptions. Similar to the X3DH protocol, our Signal-conforming AKE protocol offers a strong (or stronger) flavor of security, where the exchanged key remains secure even when all the non-trivial combinations of the long-term secrets and session-specific secrets are compromised. Moreover, our protocol has a weak flavor of deniability and we further show how to strengthen it using ring signatures. Finally, we provide a full-fledged, generic C implementation of our (weakly deniable) protocol. We instantiate it with several Round 3 candidates (finalists and alternates) to the NIST post-quantum standardization process and compare the resulting bandwidth and computation performances. Our implementation is publicly available.

The probing security model is widely used to formally prove the security of masking schemes. Whenever a masked implementation can be proven secure in this model with a reasonable leakage rate, it is also provably secure in a realistic leakage model known as the noisy leakage model. This paper introduces a new framework for the composition of probing-secure circuits. We introduce the security notion of input-output separation (IOS) for a refresh gadget. From this notion, one can easily compose gadgets satisfying the classical probing security notion –which does not ensure composability on its own– to obtain a region probing secure circuit. Such a circuit is secure against an adversary placing up to t probes in each gadget composing the circuit, which ensures a tight reduction to the more realistic noisy leakage model. After introducing the notion and proving our composition theorem, we compare our approach to the composition approaches obtained with the (Strong) Non-Interference (S/NI) notions as well as the Probe-Isolating Non-Interference (PINI) notion. We further show that any uniform SNI gadget achieves the IOS security notion, while the converse is not true. We further describe a refresh gadget achieving the IOS property for any linear sharing with a quasilinear complexity Θ(n log n) and a O(1/ log n) leakage rate (for an n-size sharing). This refresh gadget is a simplified version of the quasilinear SNI refresh gadget proposed by Battistello, Coron, Prouff, and Zeitoun (ePrint 2016). As an application of our composition framework, we revisit the quasilinear-complexity masking scheme of Goudarzi, Joux and Rivain (Asiacrypt 2018). We improve this scheme by generalizing it to any base field (whereas the original proposal only applies to field with nth powers of unity) and by taking advantage of our composition approach. We further patch a flaw in the original security proof and extend it from the random probing model to the stronger region probing model. Finally, we present some application of this extended quasilinear masking scheme to AES and MiMC and compare the obtained performances.

Many advanced lattice based cryptosystems require to sample lattice points from Gaussian distributions. One challenge for this task is that all current algorithms resort to floating-point arithmetic (FPA) at some point, which has numerous drawbacks in practice: it requires numerical stability analysis, extra storage for high-precision, lazy/backtracking techniques for efficiency, and may suffer from weak determinism which can completely break certain schemes. In this paper, we give techniques to implement Gaussian sampling over general lattices without using FPA. To this end, we revisit the approach of Peikert, using perturbation sampling. Peikert's approach uses continuous Gaussian sampling and some decomposition $\BSigma = \matA \matA^t$ of the target covariance matrix $\BSigma$. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix $\matA$ with real (not integer) entries. Our idea, in a nutshell, is to replace this decomposition by an integral one. While there is in general no integer solution if we restrict $\matA$ to being a square matrix, we show that such a decomposition can be efficiently found by allowing $\matA$ to be wider (say $n \times 9n$). This can be viewed as an extension of Lagrange's four-square theorem to matrices. In addition, we adapt our integral decomposition algorithm to the ring setting: for power-of-2 cyclotomics, we can exploit the tower of rings structure for improved complexity and compactness.

A multi-recipient key encapsulation mechanism, or mKEM, provides a scalable solution to securely communicating to a large group, and offers savings in both bandwidth and computational cost compared to the trivial solution of communicating with each member individually. All prior works on mKEM are only limited to classical assumptions and, although some generic constructions are known, they all require specific properties that are not shared by most post-quantum schemes. In this work, we first provide a simple and efficient generic construction of mKEM that can be instantiated from versatile assumptions, including post-quantum ones. We then study these mKEM instantiations at a practical level using 8 post-quantum KEMs (which are lattice and isogeny-based NIST candidates), and CSIDH, and show that compared to the trivial solution, our mKEM offers savings of at least one order of magnitude in the bandwidth, and make encryption time shorter by a factor ranging from 1.92 to 35. Additionally, we show that by combining mKEM with the TreeKEM protocol used by MLS – an IETF draft for secure group messaging – we obtain significant bandwidth savings.

NTRU lattices [13] are a class of polynomial rings which allow for compact and efficient representations of the lattice basis, thereby offering very good performance characteristics for the asymmetric algorithms that use them. Signature algorithms based on NTRU lattices have fast signature generation and verification, and relatively small signatures, public keys and private keys.A few lattice-based cryptographic schemes entail, generally during the key generation, solving the NTRU equation: $$\begin{aligned} f G - g F = q \mod x^n + 1 \end{aligned}$$Here f and g are fixed, the goal is to compute solutions F and G to the equation, and all the polynomials are in $${\mathbb {Z}}[x]/(x^n + 1)$$. The existing methods for solving this equation are quite cumbersome: their time and space complexities are at least cubic and quadratic in the dimension n, and for typical parameters they therefore require several megabytes of RAM and take more than a second on a typical laptop, precluding onboard key generation in embedded systems such as smart cards.In this work, we present two new algorithms for solving the NTRU equation. Both algorithms make a repeated use of the field norm in tower of fields; it allows them to be faster and more compact than existing algorithms by factors $${\tilde{O}}(n)$$. For lattice-based schemes considered in practice, this reduces both the computation time and RAM usage by factors at least 100, making key pair generation within range of smart card abilities.

In the last decade, several works have focused on finding the best way to model the leakage in order to obtain provably secure implementations. One of the most realistic models is the noisy leakage model, introduced in [PR13, DDF14] together with secure constructions. These works suffer from various limitations, in particular the use of ideal leak-free gates in [PR13] and an important loss (in the size of the field) in the reduction in [DDF14].In this work, we provide new strategies to prove the security of masked implementations and start by unifying the different noisiness metrics used in prior works by relating all of them to a standard notion in information theory: the pointwise mutual information. Based on this new interpretation, we define two new natural metrics and analyze the security of known compilers with respect to these metrics. In particular, we prove (1) a tighter bound for reducing the noisy leakage models to the probing model using our first new metric, (2) better bounds for amplification-based security proofs using the second metric.To support that the improvements we obtain are not only a consequence of the use of alternative metrics, we show that for concrete representation of leakage (e.g., “Hamming weight + Gaussian noise”), our approach significantly improves the parameters compared to prior works. Finally, using the Rényi divergence, we quantify concretely the advantage of an adversary in attacking a block cipher depending on the number of leakage acquisitions available to it.