Sharper Ring-LWE Signatures

We present Tesla# (pronounced Tesla Sharp ), a digital signature scheme based on the RLWE assumption that continues a recent line of proposals of lattice-based digital signature schemes originating in work by Lyubashevsky as well as by Bai and Galbraith. It improves upon all of its predecessors in that it attains much faster key pair generation, signing, and verification, outperforming most (conventional or lattice-based) signature schemes on modern processors. We propose a selection of concrete parameter sets, including a high-security instance that aims at achieving post-quantum security. Based on these parameters, we present a full-fledged software implementation protected against timing and cache attacks that supports two scheme variants: one providing 128 bits of classical security and another providing 128 bits of post-quantum security

Integration of post-quantum cryptography in the TLS protocol (LWE Option)

Dissertação de mestrado em Computer ScienceWith the possibility of quantum computers making an appearance, possibly capable of breaking several well established and widespread crytposystems (especially those that implement public key cryptography), necessity has arisen to create new cryptographic algorithms which remain safe even against adversaries using quantum computers. Several algorithms based on different mathematical problems have been proposed which are considered to be hard to solve with quantum computers. In recent years, a new lattice-based mathematical problem called Learning With Errors (and its variant Ring - Learning With Errors) was introduced, and several cryptosystems based on this problem were introduced, some of which are becoming practical enough to compete with traditional schemes that have been used for decades. The primary focus in this work is the implementation of two Ring - Learning With Errors based schemes (one key exchange mechanism and one digital signature scheme) on the TLS protocol via the OpenSSL library as a way of checking their overall viability in real-world scenarios, by comparing them to classical schemes implementing the same functionalities.Com a possibilidade do surgimento dos primeiros computadores quânticos, possivelmente capazes de quebrar muitos dos cripto-sistemas bem difundidos e considerados seguros, tornou-se necessário tomar precauções com a criação de novas técnicas criptográficas que visam manter as suas propriedades de segurança mesmo contra adversários que usem computadores quânticos. Existem já muitas propostas de algoritmos baseados em problemas matemáticos distintos que são considerados difíceis de resolver recorrendo a computadores quânticos. Recentemente, foi introduzido um novo problema baseado em reticulados denominado de Learning With Errors (e a sua variante Ring - Learning With Errors), e consequentemente foram propostos vários cripto-sistemas baseados nesse problema, alguns dos quais começam já a ser utilizáveis ao ponto de poderem ser comparados com os esquemas clássicos usados há décadas. O foco principal neste trabalho é a implementação de dois esquemas baseados no problema Ring - Learning With Errors (mais precisamente, um esquema de troca de chaves e uma assinatura digital) no protocolo TLS através da sua integração no OpenSSL como forma de verificar a sua viabilidade em contextos reais, comparando-os com esquemas clássicos que implementem as mesmas funcionalidades

Constant-time discrete Gaussian sampling

© 2018 IEEE. Sampling from a discrete Gaussian distribution is an indispensable part of lattice-based cryptography. Several recent works have shown that the timing leakage from a non-constant-time implementation of the discrete Gaussian sampling algorithm could be exploited to recover the secret. In this paper, we propose a constant-time implementation of the Knuth-Yao random walk algorithm for performing constant-time discrete Gaussian sampling. Since the random walk is dictated by a set of input random bits, we can express the generated sample as a function of the input random bits. Hence, our constant-time implementation expresses the unique mapping of the input random-bits to the output sample-bits as a Boolean expression of the random-bits. We use bit-slicing to generate multiple samples in batches and thus increase the throughput of our constant-time sampling manifold. Our experiments on an Intel i7-Broadwell processor show that our method can be as much as 2.4 times faster than the constant-time implementation of cumulative distribution table based sampling and consumes exponentially less memory than the Knuth-Yao algorithm with shuffling for a similar level of security

Spherical Gaussian Leftover Hash Lemma via the Rényi Divergence

Agrawal et al. (Asiacrypt 2013) proved the discrete Gaussian leftover hash lemma, which states that the linear transformation of the discrete spherical Gaussian is statistically close to the discrete ellipsoid Gaussian. Showing that it is statistically close to the discrete spherical Gaussian, which we call the discrete spherical Gaussian leftover hash lemma (SGLHL), is an open problem posed by Agrawal et al. In this paper, we solve the problem in a weak sense: we show that the distribution of the linear transformation of the discrete spherical Gaussian and the discrete spherical Gaussian are close with respect to the Rényi divergence (RD), which we call the weak SGLHL (wSGLHL). As an application of wSGLHL, we construct a sharper self-reduction of the learning with errors problem (LWE) problem. Applebaum et al. (CRYPTO 2009) showed that linear sums of LWE samples are statistically close to (plain) LWE samples with some unknown error parameter. In contrast, we show that linear sums of LWE samples and (plain) LWE samples with a known error parameter are close with respect to RD. As another application, we weaken the independence heuristic required for the fully homomorphic encryption scheme TFHE

Towards Post-Quantum Blockchain: A Review on Blockchain Cryptography Resistant to Quantum Computing Attacks

[Abstract] Blockchain and other Distributed Ledger Technologies (DLTs) have evolved significantly in the last years and their use has been suggested for numerous applications due to their ability to provide transparency, redundancy and accountability. In the case of blockchain, such characteristics are provided through public-key cryptography and hash functions. However, the fast progress of quantum computing has opened the possibility of performing attacks based on Grover's and Shor's algorithms in the near future. Such algorithms threaten both public-key cryptography and hash functions, forcing to redesign blockchains to make use of cryptosystems that withstand quantum attacks, thus creating which are known as post-quantum, quantum-proof, quantum-safe or quantum-resistant cryptosystems. For such a purpose, this article first studies current state of the art on post-quantum cryptosystems and how they can be applied to blockchains and DLTs. Moreover, the most relevant post-quantum blockchain systems are studied, as well as their main challenges. Furthermore, extensive comparisons are provided on the characteristics and performance of the most promising post-quantum public-key encryption and digital signature schemes for blockchains. Thus, this article seeks to provide a broad view and useful guidelines on post-quantum blockchain security to future blockchain researchers and developers.10.13039/501100010801-Xunta de Galicia (Grant Number: ED431G2019/01) 10.13039/501100011033-Agencia Estatal de Investigación (Grant Number: TEC2016-75067-C4-1-R and RED2018-102668-T) 10.13039/501100008530-European Regional Development FundXunta de Galicia; ED431G2019/0

Compact Lattice Gadget and Its Applications to Hash-and-Sign Signatures

Lattice gadgets and the associated algorithms are the essential building blocks of lattice-based cryptography. In the past decade, they have been applied to build versatile and powerful cryptosystems. However, the practical optimizations and designs of gadget-based schemes generally lag their theoretical constructions. For example, the gadget-based signatures have elegant design and capability of extending to more advanced primitives, but they are far less efficient than other lattice-based signatures. This work aims to improve the practicality of gadget-based cryptosystems, with a focus on hash-and-sign signatures. To this end, we develop a compact gadget framework in which the used gadget is a square matrix instead of the short and fat one used in previous constructions. To work with this compact gadget, we devise a specialized gadget sampler, called semi-random sampler, to compute the approximate preimage. It first deterministically computes the error and then randomly samples the preimage. We show that for uniformly random targets, the preimage and error distributions are simulatable without knowing the trapdoor. This ensures the security of the signature applications. Compared to the Gaussian-distributed errors in previous algorithms, the deterministic errors have a smaller size, which lead to a substantial gain in security and enables a practically working instantiation. As the applications, we present two practically efficient gadget-based signature schemes based on NTRU and Ring-LWE respectively. The NTRU-based scheme offers comparable efficiency to Falcon and Mitaka and a simple implementation without the need of generating the NTRU trapdoor. The LWE-based scheme also achieves a desirable overall performance. It not only greatly outperforms the state-of-the-art LWE-based hash-and-sign signatures, but also has an even smaller size than the LWE-based Fiat-Shamir signature scheme Dilithium. These results fill the long-term gap in practical gadget-based signatures

A Framework to Select Parameters for Lattice-Based Cryptography

Selecting parameters in lattice-based cryptography is a challenging task, which is essentially accomplished using one of two approaches. The first (very common) approach is to derive parameters assuming that the desired security level is equivalent to the bit hardness of the underlying lattice problem, ignoring the gap implied by available security reductions. The second (barely used) approach takes the gap and thus the security reduction into account. In this work, we investigate how efficient lattice-based schemes are if they respect existing security reductions. Thus, we present a framework to systematically select parameters for any lattice-based scheme using either approaches. We apply our methodology to the schemes by Lindner and Peikert (LP), by El Bansarkhani (LARA), and by Ducas et al. (BLISS). We analyze their security reductions and derive a gap of 2, 3, and 63 bits, respectively. We show how parameters impact the schemes\u27 efficiency when involving these gaps

Revisiting Preimage Sampling for Lattices

Preimage Sampling is a fundamental process in lattice-based cryptography whose performance directly affects the one of the cryptographic mechanisms that rely on it. In 2012, Micciancio and Peikert proposed a new way of generating trapdoors (and an associated preimage sampling procedure) with very interesting features. Unfortunately, in some applications such as digital signatures, the performance may not be as competitive as other approaches like Fiat-Shamir with Aborts. We first revisit the Lyubashevsky-Wichs (LW) sampler for Micciancio-Peikert (MP) trapdoors which leverages rejection sampling but suffered from strong parameter requirements that hampered performance. We propose an improved analysis which yields much more compact parameters. This leads to gains on the preimage size of about 60% over the LW sampler, and up to 30% compared to the original MP sampling technique. It sheds a new light on the LW sampler hoping to open promising perspectives for the efficiency of advanced lattice-based constructions relying on such mechanisms. We then show that we can leverage the special shape of the resulting preimages to design the first lattice-based aggregate signature supporting public aggregation and that achieves relevant compression compared to the concatenation of individual signatures. Our scheme is proven secure in the aggregate chosen-key model coined by Boneh et al. in 2003, based on the well-studied assumptions Module Learning With Errors and Module Short Integer Solution

A New Trapdoor over Module-NTRU Lattice and its Application to ID-based Encryption

A trapdoor over NTRU lattice proposed by Ducas, Lyubashevsky and Prest~(ASIACRYPT 2014) has been widely used in various crytographic primitives such as identity-based encryption~(IBE) and digital signature, due to its high efficiency compared to previous lattice trapdoors. However, the most of applications use this trapdoor with the power-of-two cyclotomic rings, and hence to obtain higher security level one should double the ring dimension which results in a huge loss of efficiency. In this paper, we give a new way to overcome this problem by introducing a generalized notion of NTRU lattices which we call \emph{Module-NTRU}~(MNTRU) lattices, and show how to efficiently generate a trapdoor over MNTRU lattices. Moreover, beyond giving parameter flexibility, we further show that the Gram-Schmidt norm of the trapdoor can be reached to about $q^{1/d},$ where MNTRU covers $d \ge 2$ cases while including NTRU as $d = 2$ case. Since the efficiency of trapdoor-based IBE is closely related to the Gram-Schmidt norm of trapdoor, our trapdoor over MNTRU lattice brings more efficient IBE scheme than the previously best one of Ducas, Lyubashevsky and Prest, while providing the same security level

Analysis of Error-Correcting Codes for Lattice-Based Key Exchange

Lattice problems allow the construction of very efficient key exchange and public-key encryption schemes. When using the Learning with Errors (LWE) or Ring-LWE (RLWE) problem such schemes exhibit an interesting trade-off between decryption error rate and security. The reason is that secret and error distributions with a larger standard deviation lead to better security but also increase the chance of decryption failures. As a consequence, various message/key encoding or reconciliation techniques have been proposed that usually encode one payload bit into several coefficients. In this work, we analyze how error-correcting codes can be used to enhance the error resilience of protocols like NewHope, Frodo, or Kyber. For our case study, we focus on the recently introduced NewHope Simple and propose and analyze four different options for error correction: i) BCH code; ii) combination of BCH code and additive threshold encoding; iii) LDPC code; and iv) combination of BCH and LDPC code. We show that lattice-based cryptography can profit from classical and modern codes by combining BCH and LDPC codes. This way we achieve quasi-error-free communication and an increase of the estimated post-quantum bit-security level by 20.39% and a decrease of the communication overhead by 12.8%