We propose a framework for cryptanalysis of lattice-based schemes, when side information—in the form of ``hints’’— about the secret and/or error is available. Our framework generalizes the so-called primal lattice reduction attack, and allows the progressive integration of hints before running a final lattice reduction step. Our techniques for integrating hints include sparsifying the lattice, projecting onto and intersecting with hyperplanes, and/or altering the distribution of the secret vector. Our main contribution is to propose a toolbox and a methodology to integrate such hints into lattice reduction attacks and to predict the performance of those lattice attacks with side information.
While initially designed for side-channel information, our framework can also be used in other cases: exploiting decryption failures, or simply exploiting constraints imposed by certain schemes (LAC, Round5, NTRU).
We implement a Sage 9.0 toolkit to actually mount such attacks with hints when computationally feasible, and to predict their performances on larger instances. We provide several end-to-end application examples, such as an improvement of a single trace attack on Frodo by Bos et al (SAC 2018). In particular, our work can estimates security loss even given very little side information, leading to a smooth measurement/computation trade-off for side-channel attacks.
We construct a practical lattice-based zero-knowledge argument for proving multiplicative relations between committed values. The underlying commitment scheme that we use is the currently most efficient one of Baum et al. (SCN 2018), and the size of our multiplicative proof (9KB) is only slightly larger than the 7KB required for just proving knowledge of the committed values. We additionally expand on the work of Lyubashevsky and Seiler (Eurocrypt 2018) by showing that the above-mentioned result can also apply when working over rings Zq[X]/(Xd+1) where Xd+1 splits into low-degree factors, which is a desirable property for many applications (e.g. range proofs, multiplications over Zq) that take advantage of packing multiple integers into the NTT coefficients of the committed polynomial.
Trusted Platform Module (TPM) serves as a hardware-based root of trust that protects cryptographic keys from privileged system and physical adversaries. In this work, we per-form a black-box timing analysis of TPM 2.0 devices deployed on commodity computers. Our analysis reveals thatsome of these devices feature secret-dependent execution times during signature generation based on elliptic curves. In particular, we discovered timing leakage on an Intel firmware-based TPM as well as a hardware TPM. We show how this information allows an attacker to apply lattice techniques torecover 256-bit private keys for ECDSA and EC Schnorr signatures. On Intel fTPM, our key recovery succeeds after about1,300 observations and in less than two minutes. Similarly, weextract the private ECDSA key from a hardware TPM manufactured by STMicroelectronics, which is certified at Common Criteria (CC) EAL 4+, after fewer than 40,000 observations.We further highlight the impact of these vulnerabilities by demonstrating a remote attack against a StrongSwan IPsecVPN that uses a TPM to generate the digital signatures for authentication. In this attack, the remote client recovers the server’s private authentication key by timing only 45,000authentication handshakes via a network connection.The vulnerabilities we have uncovered emphasize the difficulty of correctly implementing known constant-time techniques, and show the importance of evolutionary testing and transparent evaluation of cryptographic implementations.Even certified devices that claim resistance against attacks require additional scrutiny by the community and industry, as we learn more about these attacks.
We devise new techniques for design and analysis of efficient lattice-based zero-knowledge proofs (ZKP). First, we introduce one-shot proof techniques for non-linear polynomial relations of degree k≥2, where the protocol achieves a negligible soundness error in a single execution, and thus performs significantly better in both computation and communication compared to prior protocols requiring multiple repetitions. Such proofs with degree k≥2 have been crucial ingredients for important privacy-preserving protocols in the discrete logarithm setting, such as Bulletproofs (IEEE S&P ’18) and arithmetic circuit arguments (EUROCRYPT ’16). In contrast, one-shot proofs in lattice-based cryptography have previously only been shown for the linear case (k=1) and a very specific quadratic case (k=2), which are obtained as a special case of our technique.
Moreover, we introduce two speedup techniques for lattice-based ZKPs: a CRT-packing technique supporting “inter-slot’’ operations, and “NTT-friendly’’ tools that permit the use of fully-splitting rings. The former technique comes at almost no cost to the proof length, and the latter one barely increases it, which can be compensated for by tweaking the rejection sampling parameters while still having faster computation overall.
To illustrate the utility of our techniques, we show how to use them to build efficient relaxed proofs for important relations, namely proof of commitment to bits, one-out-of-many proof, range proof and set membership proof. Despite their relaxed nature, we further show how our proof systems can be used as building blocks for advanced cryptographic tools such as ring signatures.
Our ring signature achieves a dramatic improvement in length over all the existing proposals from lattices at the same security level. The computational evaluation also shows that our construction is highly likely to outperform all the relevant works in running times. Being efficient in both aspects, our ring signature is particularly suitable for both small-scale and large-scale applications such as cryptocurrencies and e-voting systems. No trusted setup is required for any of our proposals.
We provide new zero-knowledge argument of knowledge systems that work directly for a wide class of language, namely, ones involving the satisfiability of matrix-vector relations and integer relations commonly found in constructions of lattice-based cryptography. Prior to this work, practical arguments for lattice-based relations either have a constant soundness error ( 2/3 ), or consider a weaker form of soundness, namely, extraction only guarantees that the prover is in possession of a witness that “approximates” the actual witness. Our systems do not suffer from these limitations.
The core of our new argument systems is an efficient zero-knowledge argument of knowledge of a solution to a system of linear equations, where variables of this solution satisfy a set of quadratic constraints. This argument enjoys standard soundness, a small soundness error ( 1/poly ), and a complexity linear in the size of the solution. Using our core argument system, we construct highly efficient argument systems for a variety of statements relevant to lattices, including linear equations with short solutions and matrix-vector relations with hidden matrices.
Based on our argument systems, we present several new constructions of common privacy-preserving primitives in the standard lattice setting, including a group signature, a ring signature, an electronic cash system, and a range proof protocol. Our new constructions are one to three orders of magnitude more efficient than the state of the art (in standard lattice). This illustrates the efficiency and expressiveness of our argument system.
We implemented (a simplified version of) the branching-program obfuscator due to Gentry et al. (GGH15), which is itself a variation of the first obfuscation candidate by Garg et al. (GGHRSW13). To keep within the realm of feasibility, we had to give up on some aspects of the construction, specifically the “multiplicative bundling” factors that protect against mixed-input attacks. Hence our implementation can only support read-once branching programs.
To be able to handle anything more than just toy problems, we developed a host of algorithmic and code-level optimizations. These include new variants of discrete Gaussian sampler and lattice trapdoor sampler, efficient matrix-manipulation routines, and many tradeoffs. We expect that these optimizations will find other uses in lattice-based cryptography beyond just obfuscation.
Our implementation is the first obfuscation attempt using the GGH15 graded encoding scheme, offering performance advantages over other graded encoding methods when obfuscating finite-state machines with many states. In out most demanding setting, we were able to obfuscate programs with input length of 20 nibbles (80 bits) and over 100 states, which seems out of reach for prior implementations. Although further optimizations are surely possible, we do not expect any implementation of current schemes to be able to handle much larger parameters.
We introduce MatRiCT, an efficient RingCT protocol for blockchain confidential transactions, whose security is based on ``post-quantum’’ (module) lattice assumptions. The proof length of the protocol is around two orders of magnitude shorter than the existing post-quantum proposal, and scales efficiently to large anonymity sets, unlike the existing proposal. Further, we provide the first full implementation of a post-quantum RingCT, demonstrating the practicality of our scheme. In particular, a typical transaction can be generated in a fraction of a second and verified in about 23 ms on a standard PC. Moreover, we show how our scheme can be extended to provide auditability, where a user can select a particular authority from a set of authorities to reveal her identity. The user also has the ability to select no auditing and all these auditing options may co-exist in the same environment.
The key ingredients, introduced in this work, of MatRiCT are 1) the shortest to date scalable ring signature from standard lattice assumptions with no Gaussian sampling required, 2) a novel balance zero-knowledge proof and 3) a novel extractable commitment scheme from (module) lattices. We believe these ingredients to be of independent interest for other privacy-preserving applications such as secure e-voting. Despite allowing 64-bit precision for transaction amounts, our new balance proof, and thus our protocol, does not require a range proof on a wide range (such as 32- or 64-bit ranges), which has been a major obstacle against efficient lattice-based solutions.
Further, we provide new formal definitions for RingCT-like protocols, where the real-world blockchain setting is captured more closely. The definitions are applicable in a generic setting, and thus are believed to contribute to the development of future confidential transaction protocols in general (not only in the lattice setting).
In this paper, we propose a constant-time implementation of the BLISS lattice-based signature scheme. BLISS is possibly the most efficient lattice-based signature scheme proposed so far, with a level of performance on par with widely used pre-quantum primitives like ECDSA. It is only one of the few postquantum signatures to have seen real-world deployment, as part of the strongSwan VPN software suite.
The outstanding performance of the BLISS signature scheme stems in large part from its reliance on discrete Gaussian distributions, which allow for better parameters and security reductions. However, that advantage has also proved to be its Achilles’ heel, as discrete Gaussians pose serious challenges in terms of secure implementations. Implementations of BLISS so far have included secret-dependent branches and memory accesses, both as part of the discrete Gaussian sampling and of the essential rejection sampling step in signature generation. These defects have led to multiple devastating timing attacks, and were a key reason why BLISS was not submitted to the NIST postquantum standardization effort. In fact, almost all of the actual candidates chose to stay away from Gaussians despite their efficiency advantage, due to the serious concerns surrounding implementation security.
Moreover, naive countermeasures will often not cut it: we show that a reasonable-looking countermeasure suggested in previous work to protect the BLISS rejection sampling can again be defeated using novel timing attacks, in which the timing information is fed to phase retrieval machine learning algorithm in order to achieve a full key recovery.
Fortunately, we also present careful implementation techniques that allow us to describe an implementation of BLISS with complete timing attack protection, achieving the same level of efficiency as the original unprotected code, without resorting on floating point arithmetic or platform-specific optimizations like AVX intrinsics. These techniques, including a new approach to the polynomial approximation of transcendental function, can also be applied to the masking of the BLISS signature scheme, and will hopefully make more efficient and secure implementations of lattice-based cryptography possible going forward.
In this age of massive data gathering for purposes of personalization, targeted ads, etc. there is an increased need for technology that allows for data analysis in a privacy-preserving manner. Private Stream Aggregation as introduced by Shi et al. (NDSS 2011) allows for the execution of aggregation operations over privacy-critical data from multiple data sources without placing trust in the aggregator and while maintaining differential privacy guarantees. We propose a generic PSA scheme, LaPS, based on the Learning With Error problem, which allows for a flexible choice of the utilized privacy-preserving mechanism while maintaining post-quantum security. We overcome the limitations of earlier schemes by relaxing previous assumptions in the security model and provide an efficient and compact scheme with high scalability. Our scheme is practical, for a plaintext space of 216 and 1000 participants we achieve a performance gain in decryption of roughly 150 times compared to previous results in Shi et al. (NDSS 2011).