Lattice-based Cryptographic Schemes Research Articles

A Generalized Number-Theoretic Transform for Efficient Multiplication in Lattice Cryptography

The Number Theoretic Transform (NTT) has emerged as a powerful tool for efficiently computing convolutions of digital signals, due to its inherent advantages such as numerical stability, reliance on simple integer operations, and proven efficiency. Its applications have extended to accelerating polynomial multiplication in lattice-based cryptography. However, existing NTT multiplication algorithms impose restrictions on the underlying moduli, potentially affecting key and ciphertext sizes as well as computational overhead. Therefore, enabling NTT with small moduli holds significant potential for enhancing the overall system performance. This study introduces a novel reduction framework for NTT computation in cyclotomic rings employing field extensions. Our approach replaces the underlying polynomial ring with a two-dimensional isomorphic ring, effectively relaxing the restrictions imposed on the NTT moduli. The proposed framework is evaluated through two case studies relevant to the LAC and NTTRU lattice-based cryptographic schemes. Comprehensive theoretical analysis is provided, demonstrating the effectiveness of our approach in enabling NTT with small moduli and its potential to improve the efficiency of lattice-based cryptography.

Comparative analysis of lattice-based cryptographic schemes for secure IoT communications

This paper presents comparative analysis of lattice-based cryptographic schemes, focusing on their runtime and memory performance to evaluate their suitability for secure IoT communications. Specifically, analyzing FrodoKEM, Kyber, and sntrup761, assessing their key generation, encapsulation, and decapsulation operations across various IoT processors. Kyber schemes, particularly Kyber512 and Kyber1024, demonstrate the highest efficiency in terms of battery utilization and computational speed, making them ideal for highly constrained IoT devices. FrodoKEM variants, while providing robust security, show moderate performance, suitable for devices with more processing capacity. sntrup761, however, is less efficient, highlighting the need for careful selection of cryptographic schemes based on specific IoT requirements. The findings underscore the importance of balancing security, speed, and resource usage in PQC implementations, with Kyber schemes emerging as the most promising candidates for secure, resource-constrained IoT environments.

On Advances of Lattice-Based Cryptographic Schemes and Their Implementations

Lattice-based cryptography is centered around the hardness of problems on lattices. A lattice is a grid of points that stretches to infinity. With the development of quantum computers, existing cryptographic schemes are at risk because the underlying mathematical problems can, in theory, be easily solved by quantum computers. Since lattice-based mathematical problems are hard to be solved even by quantum computers, lattice-based cryptography is a promising foundation for future cryptographic schemes. In this paper, we focus on lattice-based public-key encryption schemes. This survey presents the current status of the lattice-based public-key encryption schemes and discusses the existing implementations. Our main focus is the learning with errors problem (LWE problem) and its implementations. In this paper, the plain lattice implementations and variants with special algebraic structures such as ring-based variants are discussed. Additionally, we describe a class of lattice-based functions called lattice trapdoors and their applications.

DPCrypto: Acceleration of Post-Quantum Cryptography Using Dot-Product Instructions on GPUs

Modern NVIDIA GPU architectures offer dot-product instructions (DP2A and DP4A), with the aim of accelerating machine learning and scientific computing applications. These dot-product instructions allow the computation of multiply-and-add instructions in a single clock cycle, effectively achieving higher throughput compared to conventional 32-bit integer units. In this paper, we show that the dot-product instruction can also be used to accelerate matrix-multiplication and polynomial convolution operations, which are widely used in post-quantum lattice-based cryptographic schemes. In particular, we propose a highly optimized implementation of FrodoKEM wherein the matrix-multiplication is accelerated by the dot-product instruction. We also present specially designed data structures that allow an efficient implementation of Saber key-encapsulation mechanism, utilizing the dot-product instruction to speed-up the polynomial convolution. The proposed FrodoKEM implementation achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4.37\times $ </tex-math></inline-formula> higher throughput than the state-of-the-art implementation on a V100 GPU. This paper also presents the first implementation of Saber on GPU platforms, achieving 124,418, 120,463, and 31,658 key exchanges per second on RTX3080, V100, and T4 GPUs, respectively. Since matrix-multiplication and polynomial convolution operations are the most time-consuming operations in lattice-based cryptographic schemes, we strongly believe that the proposed methods can be beneficial to other KEM and signatures schemes based on lattices.

Multi-Unit Serial Polynomial Multiplier to Accelerate NTRU-Based Cryptographic Schemes in IoT Embedded Systems.

Concern for the security of embedded systems that implement IoT devices has become a crucial issue, as these devices today support an increasing number of applications and services that store and exchange information whose integrity, privacy, and authenticity must be adequately guaranteed. Modern lattice-based cryptographic schemes have proven to be a good alternative, both to face the security threats that arise as a consequence of the development of quantum computing and to allow efficient implementations of cryptographic primitives in resource-limited embedded systems, such as those used in consumer and industrial applications of the IoT. This article describes the hardware implementation of parameterized multi-unit serial polynomial multipliers to speed up time-consuming operations in NTRU-based cryptographic schemes. The flexibility in selecting the design parameters and the interconnection protocol with a general-purpose processor allow them to be applied both to the standardized variants of NTRU and to the new proposals that are being considered in the post-quantum contest currently held by the National Institute of Standards and Technology, as well as to obtain an adequate cost/performance/security-level trade-off for a target application. The designs are provided as AXI4 bus-compliant intellectual property modules that can be easily incorporated into embedded systems developed with the Vivado design tools. The work provides an extensive set of implementation and characterization results in devices of the Xilinx Zynq-7000 and Zynq UltraScale+ families for the different sets of parameters defined in the NTRUEncrypt standard. It also includes details of their plug and play inclusion as hardware accelerators in the C implementation of this public-key encryption scheme codified in the LibNTRU library, showing that acceleration factors of up to 3.1 are achieved when compared to pure software implementations running on the processing systems included in the programmable devices.

Fast Implementation of Multiplication on Polynomial Rings

Multiplication on polynomial rings has been widely used in public-key cryptographic schemes based on ideal lattices. It is an important module that significantly affects the efficiency of the schemes. Improved Preprocess-then-NTT (IPtNTT) is an algorithm which can fast realize multiplication on polynomial rings. Compared with the Number Theoretic Transform (NTT), the IPtNTT weakens the parameter restriction of lattice-based public-key cryptographic schemes. By optimizing the IPtNTT with the AVX2 instruction set, we reduce the clock cycles consumed by multiplication on polynomial rings to 15%–22%. According to the experimental results, we give specific suggestions on using AVX2 optimized IPtNTT to realize multiplication on polynomial rings with different parameters chosen in lattice-based public-key cryptosystems.

CoHA-NTT: A Configurable Hardware Accelerator for NTT-based Polynomial Multiplication

In this paper, we introduce a configurable hardware architecture that can be used to generate unified and parametric NTT-based polynomial multipliers that support a wide range of parameters of lattice-based cryptographic schemes proposed for post-quantum cryptography. Both NTT and inverse NTT operations can be performed using the unified butterfly unit of our architecture, which constitutes the core building block in NTT operations. The multitude of this unit plays an essential role in achieving the performance goals of a specific application area or platform. To this end, the architecture takes the size of butterfly units as input and generates an efficient NTT-based polynomial multiplier hardware to achieve the desired throughput and area requirements. More specifically, the proposed hardware architecture provides run-time configurability for the scheme parameters and compile-time configurability for throughput and area requirements. This work presents the first architecture with both run-time and compile-time configurability for NTT-based polynomial multiplication operations to the best of our knowledge. The implementation results indicate that the advanced configurability has a negligible impact on the time and area of the proposed architecture and that its performance is on par with the state-of-the-art implementations in the literature, if not better. The proposed architecture comprises various sub-blocks such as modular multiplier and butterfly units, each of which can be of interest on its own for accelerating lattice-based cryptography. Thus, we provide the design rationale of each sub-block and compare it with those in the literature, including our earlier works in terms of configurability and performance.

Improved Reduction Between SIS Problems Over Structured Lattices

Many lattice-based cryptographic schemes are constructed based on hard problems on an algebraic structured lattice, such as the short integer solution (SIS) problems. These problems are called ring-SIS (R-SIS) and its generalized version, module-SIS (M-SIS). Generally, it has been considered that problems defined on the module lattice are more difficult than the problems defined on the ideal lattice. However, Koo, No, and Kim showed that R-SIS is more difficult than M-SIS under some norm constraints of R-SIS. However, this reduction has problems in that the rank of the module is limited to about half of the instances of R-SIS, and the comparison is not performed through the same modulus of R-SIS and M-SIS. In this paper, we propose that R-SIS is more difficult than M-SIS with the same modulus and ring dimension under some constraint of R-SIS. Also, we show that R-SIS with the modulus prime q is more difficult than M-SIS with the composite modulus c, such that c is divided by q. In particular, it shows that through the reduction from M-SIS to R-SIS with the same modulus, the rank of the module is extended as much as the number of instances of R-SIS from half of the number of instances of R-SIS. Finally, this paper shows that R-SIS is more difficult than M-SIS under some constraints, which is tighter than the M-SIS in the previous work.

Parameterized Hardware Accelerators for Lattice-Based Cryptography and Their Application to the HW/SW Co-Design of qTESLA

This paper presents a set of efficient and parameterized hardware accelerators that target post-quantum lattice-based cryptographic schemes, including a versatile cSHAKE core, a binary-search CDT-based Gaussian sampler, and a pipelined NTT-based polynomial multiplier, among others. Unlike much of prior work, the accelerators are fully open-sourced, are designed to be constant-time, and can be parameterized at compile-time to support different parameters without the need for re-writing the hardware implementation. These flexible, publicly-available accelerators are leveraged to demonstrate the first hardware-software co-design using RISC-V of the post-quantum lattice-based signature scheme qTESLA with provably secure parameters. In particular, this work demonstrates that the NIST’s Round 2 level 1 and level 3 qTESLA variants achieve over a 40-100x speedup for key generation, about a 10x speedup for signing, and about a 16x speedup for verification, compared to the baseline RISC-V software-only implementation. For instance, this corresponds to execution in 7.7, 34.4, and 7.8 milliseconds for key generation, signing, and verification, respectively, for qTESLA’s level 1 parameter set on an Artix-7 FPGA, demonstrating the feasibility of the scheme for embedded applications.

Improved Ring LWR-Based Key Encapsulation Mechanism Using Cyclotomic Trinomials

In the field of post-quantum cryptography, lattice-based cryptography has received the most noticeable attention. Most lattice-based cryptographic schemes are constructed based on the polynomial ring R q = Z q [x]/f (x), using a cyclotomic polynomial f (x). Until now, the most preferred cyclotomic polynomials have been x n + 1, where n is a power of two, and x n + · · · + x + 1, where n + 1 is a prime. The former results in the smallest decryption error size, but the choice of degree is limited. On the other hand, the latter gives rise to the largest decryption error size, but the choice of degree is very flexible. In this paper, we use a new polynomial ring Rq = Zq/f (x) with a cyclotomic trinomial f (x) = x n - x n/2 + 1 as an intermediate that combines the advantages of the other rings. Since the degree n is chosen freely as n = 2 a 3 b for positive integers a and b, the choice of the degree n is moderate. Furthermore, since the error propagation is small in the middle of polynomial multiplication in the new ring, if the middle part is truncated and used, the decryption error size can be reduced. Based on these observations, we propose a new, practical key encapsulation mechanism (KEM) that is constructed over a ring with a cyclotomic trinomial. The security of our KEM is based on the hardness of ring learning-with-rounding (LWR) problems. With appropriate parameterization for the current 128-bit security model, we show that our KEM obtains shorter secret keys and ciphertexts, especially compared to the previous Ring-LWR-based KEM, Round5, with no error correction code. We then implement our KEM and compare its performance with that of several KEMs that were presented in the second round of the NIST PQC conference.

Post-Quantum Lattice-Based Cryptography Implementations

The advent of quantum computing threatens to break many classical cryptographic schemes, leading to innovations in public key cryptography that focus on post-quantum cryptography primitives and protocols resistant to quantum computing threats. Lattice-based cryptography is a promising post-quantum cryptography family, both in terms of foundational properties as well as in its application to both traditional and emerging security problems such as encryption, digital signature, key exchange, and homomorphic encryption. While such techniques provide guarantees, in theory, their realization on contemporary computing platforms requires careful design choices and tradeoffs to manage both the diversity of computing platforms (e.g., high-performance to resource constrained), as well as the agility for deployment in the face of emerging and changing standards. In this work, we survey trends in lattice-based cryptographic schemes, some recent fundamental proposals for the use of lattices in computer security, challenges for their implementation in software and hardware, and emerging needs for their adoption. The survey means to be informative about the math to allow the reader to focus on the mechanics of the computation ultimately needed for mapping schemes on existing hardware or synthesizing part or all of a scheme on special-purpose har dware.

A detailed analysis of the hybrid lattice-reduction and meet-in-the-middle attack

Abstract Over the past decade, the hybrid lattice-reduction and meet-in-the middle attack (called hybrid attack) has been used to evaluate the security of many lattice-based cryptographic schemes such as NTRU, NTRU Prime, BLISS and more. However, unfortunately, none of the previous analyses of the hybrid attack is entirely satisfactory: They are based on simplifying assumptions that may distort the security estimates. Such simplifying assumptions include setting probabilities equal to 1, which, for the parameter sets we analyze in this work, are in fact as small as 2 - 80 2^{-80} . Many of these assumptions lead to underestimating the scheme’s security. However, some lead to security overestimates, and without further analysis, it is not clear which is the case. Therefore, the current security estimates against the hybrid attack are not reliable, and the actual security levels of many lattice-based schemes are unclear. In this work, we present an improved runtime analysis of the hybrid attack that is based on more reasonable assumptions. In addition, we reevaluate the security against the hybrid attack for the NTRU, NTRU Prime and R-BinLWEEnc encryption schemes as well as for the BLISS and GLP signature schemes. Our results show that there exist both security over- and underestimates in the literature.

STUDY OF LATTICE BASED FHE FOR CLOUD DATA SECURITY

Cloud Computing is an transpiring trend in the modern world. It is a way of holding the Internet to use software or other IT services on demand. Due to its fast growth and popularity, number of users deposit their data and applications on the cloud. The impressive growth in cloud computing has proved to be promising innovation and more suitable for storing data and applications remotely. But its uses improvement is hindered by the security issue. Cloud doesn’t provide more security for its services and storage purpose. The traditional security approach of encryption doesn’t make cloud fully secure. So there is a need to develop such a technique which increases the security level of cloud. In order to solve the problem of data security in cloud computing system, lattice-based cryptographic schemes implements the so called “Fully Homomorphic Encryption (FHE) scheme”, which allows processing directly on encrypted data and holds the promise eventually to solve the security problems with cloud computing. In this paper we survey on the existing lattice based FHE encryption techniques. Fully homomorphic encryption is a good solution to enhance security measures of cloud system that handles critical data. This makes cloud computing more stable and solid.

Sieving for shortest vectors in ideal lattices: a practical perspective

The security of many lattice-based cryptographic schemes relies on the hardness of finding short vectors in integral lattices. We propose a new variant of the parallel Gauss sieve algorithm to compute such short vectors. It combines favourable properties of previous approaches resulting in reduced run time and memory requirement per node. Our publicly available implementation outperforms all previous Gauss sieve approaches for dimensions 80, 88, and 96. When computing short vectors in ideal lattices, we show how to reduce the number of multiplications and comparisons by using a symbolic Fourier transform. We computed a short vector in a negacyclic ideal lattice of dimension 128 in less than nine days on 1,024 cores, more than twice as fast as the recent record computation for the same lattice on the same computer hardware.

Sieving for shortest vectors in ideal lattices: a practical perspective

The security of many lattice-based cryptographic schemes relies on the hardness of finding short vectors in integral lattices. We propose a new variant of the parallel Gauss sieve algorithm to compute such short vectors. It combines favorable properties of previous approaches resulting in reduced run time and memory requirement per node. Our publicly available implementation outperforms all previous Gauss sieve approaches for dimensions 80, 88, and 96. When computing short vectors in ideal lattices, we show how to reduce the number of multiplications and comparisons by using a symbolic Fourier transform. We computed a short vector in a negacyclic ideal lattice of dimension 128 in less than nine days on 1024 cores, more than twice as fast as the recent record computation for the same lattice on the same computer hardware.

On Constrained Implementation of Lattice-Based Cryptographic Primitives and Schemes on Smart Cards

Most lattice-based cryptographic schemes with a security proof suffer from large key sizes and heavy computations. This is also true for the simpler case of authentication protocols that are used on smart cards as a very-constrained computing environment. Recent progress on ideal lattices has significantly improved the efficiency and made it possible to implement practical lattice-based cryptography on constrained devices. However, to the best of our knowledge, no previous attempts have been made to implement lattice-based schemes on smart cards. In this article, we provide the results of our implementation of several state-of-the-art lattice-based authentication protocols on smart cards and a microcontroller widely used in smart cards. Our results show that only a few of the proposed lattice-based authentication protocols can be implemented using limited resources of such constrained devices; however, cutting-edge ones are suitably efficient to be used practically on smart cards. Moreover, we have implemented fast Fourier transform (FFT) and discrete Gaussian sampling with different typical parameter sets, as well as versatile lattice-based public-key encryptions. These results have noticeable points that help to design or optimize lattice-based schemes for constrained devices.

Worst-case to average-case reductions for module lattices

Most lattice-based cryptographic schemes are built upon the assumed hardness of the Short Integer Solution (SIS) and Learning With Errors (LWE) problems. Their efficiencies can be drastically improved by switching the hardness assumptions to the more compact Ring-SIS and Ring-LWE problems. However, this change of hardness assumptions comes along with a possible security weakening: SIS and LWE are known to be at least as hard as standard (worst-case) problems on euclidean lattices, whereas Ring-SIS and Ring-LWE are only known to be as hard as their restrictions to special classes of ideal lattices, corresponding to ideals of some polynomial rings. In this work, we define the Module-SIS and Module-LWE problems, which bridge SIS with Ring-SIS, and LWE with Ring-LWE, respectively. We prove that these average-case problems are at least as hard as standard lattice problems restricted to module lattices (which themselves bridge arbitrary and ideal lattices). As these new problems enlarge the toolbox of the lattice-based cryptographer, they could prove useful for designing new schemes. Importantly, the worst-case to average-case reductions for the module problems are (qualitatively) sharp, in the sense that there exist converse reductions. This property is not known to hold in the context of Ring-SIS/Ring-LWE: Ideal lattice problems could reveal easy without impacting the hardness of Ring-SIS/Ring-LWE.