On the Security of Lattice-Based Cryptography Against Lattice Reduction and Hybrid Attacks

Over the past decade, lattice-based cryptography has emerged as one of the most promising candidates for post-quantum public-key cryptography. For most current lattice-based schemes, one can recover the secret key by solving a corresponding instance of the unique Shortest Vector Problem (uSVP), the problem of finding a shortest non-zero vector in a lattice which is unusually short. This work is concerned with the concrete hardness of the uSVP. In particular, we study the uSVP in general as well as instances of the problem with particularly small or sparse short vectors, which are used in cryptographic constructions to increase their efficiency. We study solving the uSVP in general via lattice reduction, more precisely, the Block-wise Korkine-Zolotarev (BKZ) algorithm. In order to solve an instance of the uSVP via BKZ, the applied block size, which specifies the BKZ algorithm, needs to be sufficiently large. However, a larger block size results in higher runtimes of the algorithm. It is therefore of utmost interest to determine the minimal block size that guarantees the success of solving the uSVP via BKZ. In this thesis, we provide a theoretical and experimental validation of a success condition for BKZ when solving the uSVP which can be used to determine the minimal required block size. We further study the practical implications of using so-called sparsification techniques in combination with the above approach. With respect to uSVP instances with particularly small or sparse short vectors, we investigate so-called hybrid attacks. We first adapt the “hybrid lattice reduction and meet-in-the-middle attack” (or short: the hybrid attack) by Howgrave-Graham on the NTRU encryption scheme to the uSVP. Due to this adaption, the attack can be applied to a larger class of lattice-based cryptosystems. In addition, we enhance the runtime analysis of the attack, e.g., by an explicit calculation of the involved success probabilities. As a next step, we improve the hybrid attack in two directions as described in the following. To reflect the potential of a modern attacker on classical computers, we show how to parallelize the attack. We show that our parallel version of the hybrid attack scales well within realistic parameter ranges. Our theoretical analysis is supported by practical experiments, using our implementation of the parallel hybrid attack which employs Open Multi-Processing and the Message Passing Interface. To reflect the power of a potential future attacker who has access to a large-scale quantum computer, we develop a quantum version of the hybrid attack which replaces the classical meet-in-the-middle search by a quantum search. Not only is the quantum hybrid attack faster than its classical counterpart, but also applicable to a wider range of uSVP instances (and hence to a larger number of lattice-based schemes) as it uses a quantum search which is sensitive to the distribution on the search space. Finally, we demonstrate the practical relevance of our results by using the techniques developed in this thesis to evaluate the concrete security levels of the lattice-based schemes submitted to the US National Institute of Standards and Technology’s process of standardizing post-quantum public-key cryptography

Making Existential-Unforgeable Signatures Strongly Unforgeable in the Quantum Random-Oracle Model

Strongly unforgeable signature schemes provide a more stringent security guarantee than the standard existential unforgeability. It requires that not only forging a signature on a new message is hard, it is infeasible as well to produce a new signature on a message for which the adversary has seen valid signatures before. Strongly unforgeable signatures are useful both in practice and as a building block in many cryptographic constructions. This work investigates a generic transformation that compiles any existential-unforgeable scheme into a strongly unforgeable one, which was proposed by Teranishi et al. and was proven in the classical random-oracle model. Our main contribution is showing that the transformation also works against quantum adversaries in the quantum random-oracle model. We develop proof techniques such as adaptively programming a quantum random-oracle in a new setting, which could be of independent interest. Applying the transformation to an existential-unforgeable signature scheme due to Cash et al., which can be shown to be quantum-secure assuming certain lattice problems are hard for quantum computers, we get an efficient quantum-secure strongly unforgeable signature scheme in the quantum random-oracle model.Comment: 15 pages, to appear in Proceedings TQC 201

A Non-commutative Cryptosystem Based on Quaternion Algebras

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.Comment: Submitted for possible publicatio

Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory

The present survey reports on the state of the art of the different cryptographic functionalities built upon the ring learning with errors problem and its interplay with several classical problems in algebraic number theory. The survey is based to a certain extent on an invited course given by the author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other authors/ comment of the author: quotation has been added to Theorem 5.

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 nonideal 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 4.7×, 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 1.6×, and (even under heavy load) server throughput only decreases by factors of 1.5× and 1.2× 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

NewHope: A Mobile Implementation of a Post-Quantum Cryptographic Key Encapsulation Mechanism

NIST anticipates the appearance of large-scale quantum computers by 2036 [34], which will threaten widely used asymmetric algorithms, National Institute of Standards and Technology (NIST) launched a Post-Quantum Cryptography Standardization Project to find quantum-secure alternatives. NewHope post-quantum cryptography (PQC) key encapsulation mechanism (KEM) is the only Round 2 candidate to simultaneously achieve small key values through the use of a security problem with sufficient confidence its security, while mitigating any known vulnerabilities. This research contributes to NIST project’s overall goal by assessing the platform flexibility and resource requirements of NewHope KEMs on an Android mobile device. The resource requirements analyzed are transmission size as well as scheme runtime, central processing unit (CPU), memory, and energy usage. Results from each NewHope KEM instantiations are compared amongst each other, to a baseline application, and to results from previous work. NewHope PQC KEM was demonstrated to have sufficient flexibility for mobile implementation, competitive performance with other PQC KEMs, and to have competitive scheme runtime with current key exchange algorithms