Acceleration strategies for post-quantum cryptographic schemes

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Xavier Guitart Morales i Oriol Farràs Ventura[en] The aim of project is to study the quantum-resistant cryptosystems Classic McEliece and NTRU, revising some of their previous literature and proving some of the main results upon which these cryptosystems are built. We also study the implementation strategies for the acceleration of these schemes. Finally, we make a comparative study of the reference implementations, considering metrics such as performance and key size

RLWE and PLWE over cyclotomic fields are not equivalent

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $\mathbb{Q}(\zeta_n)$ are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in $n$. We do so by providing a lower bound, holding for infinitely many positive integers $n$, for the condition number of the Vandermonde matrix of the $n$th cyclotomic polynomial

Quantum Key Search for Ternary LWE

Ternary LWE, i.e., LWE with coefficients of the secret and the error vectors taken from $\{-1, 0, 1\}$, is a popular choice among NTRU-type cryptosystems and some signatures schemes like BLISS and GLP. In this work we consider quantum combinatorial attacks on ternary LWE. Our algorithms are based on the quantum walk framework of Magniez-Nayak-Roland-Santha. At the heart of our algorithms is a combinatorial tool called the representation technique that appears in algorithms for the subset sum problem. This technique can also be applied to ternary LWE resulting in faster attacks. The focus of this work is quantum speed-ups for such representation-based attacks on LWE. When expressed in terms of the search space $\mathcal{S}$ for LWE keys, the asymptotic complexity of the representation attack drops from $\mathcal{S}^{0.24}$ (classical) down to $\mathcal{S}^{0.19}$ (quantum). This translates into noticeable attack\u27s speed-ups for concrete NTRU instantiations like NTRU-HRSS and NTRU Prime. Our algorithms do not undermine current security claims for NTRU or other ternary LWE based schemes, yet they can lay ground for improvements of the combinatorial subroutines inside hybrid attacks on LWE

Efficiently Masking Polynomial Inversion at Arbitrary Order

Physical side-channel analysis poses a huge threat to post-quantum cryptographic schemes implemented on embedded devices. Still, secure implementations are missing for many schemes. In this paper, we present an efficient solution for masked polynomial inversion, a main component of the key generation of multiple post-quantum KEMs. For this, we introduce a polynomial-multiplicative masking scheme with efficient arbitrary order conversions from and to additive masking. Furthermore, we show how to integrate polynomial inversion and multiplication into the masking schemes to reduce costs considerably. We demonstrate the performance of our algorithms for two different post-quantum cryptographic schemes on the Cortex-M4. For NTRU, we measure an overhead of 35% for the first-order masked inversion compared to the unmasked inversion while for BIKE the overhead is as little as 11%. Lastly, we verify the security of our algorithms for the first masking order by measuring and performing a TVLA based side-channel analysis

CRYSTALS - Kyber: A CCA-secure Module-Lattice-Based KEM

Rapid advances in quantum computing, together with the announcement by the National Institute of Standards and Technology (NIST) to define new standards for digital-signature, encryption, and key-establishment protocols, have created significant interest in post-quantum cryptographic schemes. This paper introduces Kyber (part of CRYSTALS - Cryptographic Suite for Algebraic Lattices - a package submitted to NIST post-quantum standardization effort in November 2017), a portfolio of post-quantum cryptographic primitives built around a key-encapsulation mechanism (KEM), based on hardness assumptions over module lattices. Our KEM is most naturally seen as a successor to the NEWHOPE KEM (Usenix 2016). In particular, the key and ciphertext sizes of our new construction are about half the size, the KEM offers CCA instead of only passive security, the security is based on a more general (and flexible) lattice problem, and our optimized implementation results in essentially the same running time as the aforementioned scheme. We first introduce a CPA-secure public-key encryption scheme, apply a variant of the Fujisaki-Okamoto transform to create a CCA-secure KEM, and eventually construct, in a black-box manner, CCA-secure encryption, key exchange, and authenticated-key-exchange schemes. The security of our primitives is based on the hardness of Module-LWE in the classical and quantum random oracle models, and our concrete parameters conservatively target more than 128 bits of post-quantum security

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.European Union 952622Ministerio de Ciencia e Innovación PID2020-116664RB100, 10.13039/50110001103

Achieving secure and efficient lattice-based public-key encryption: the impact of the secret-key distribution

Lattice-based public-key encryption has a large number of design choices that can be combined in diverse ways to obtain different tradeoffs. One of these choices is the distribution from which secret keys are sampled. Numerous secret-key distributions exist in the state of the art, including (discrete) Gaussian, binomial, ternary, and fixed-weight ternary. Although the secret-key distribution impacts both the concrete security and the performance of the schemes, it has not been compared in a detailed way how the choice of secret-key distribution affects this tradeoff. In this paper, we compare different aspects of secret-key distributions from submissions to the NIST post-quantum standardization effort. We consider their impact on concrete security (influenced by the entropy and variance of the distribution), and on decryption failures and IND-CCA2 security (influenced by the probability of sampling keys with ``non average, large\u27\u27 norm). Next, we select concrete parameters of an encryption scheme instantiated with the above distributions %optimized for key sizes, to identify which distribution(s) offer the best tradeoffs between security and key sizes. The conclusions of the paper are: first, the above optimization shows that fixed-weight ternary secret keys result in the smallest key sizes in the analyzed scheme. The reason is that such secret keys reduce the decryption failure rate and hence allow for a higher noise-to-modulus ratio, alleviating the slight increase in lattice dimension required for countering specialized attacks that apply in this case. Second, compared to secret keys with independently sampled components, secret keys with a fixed composition (i.e., the number of secret key components equal to any possible value is fixed) result in the scheme becoming more secure against active attacks based on decryption failures

Implementing and Measuring KEMTLS

KEMTLS is a novel alternative to the Transport Layer Security (TLS) handshake that integrates post-quantum algorithms. It uses key encapsulation mechanisms (KEMs) for both confidentiality and authentication, achieving post-quantum security while obviating the need for expensive post-quantum signatures. The original KEMTLS paper presents a security analysis, Rust implementation, and benchmarks over emulated networks. In this work, we provide full Go implementations of KEMTLS and other post-quantum handshake alternatives, describe their integration into a distributed system, and provide performance evaluations over real network conditions. We compare the standard (nonquantum-resistant) TLS 1.3 handshake with three alternatives: one that uses post-quantum signatures in combination with post-quantum KEMs (PQTLS), one that uses KEMTLS, and one that is a reduced round trip version of KEMTLS (KEMTLS-PDK). In addition to the performance evaluations, we discuss how the design of these protocols impacts TLS from an implementation and configuration perspective

Cloud-assisted Asynchronous Key Transport with Post-Quantum Security

In cloud-based outsourced storage systems, many users wish to securely store their files for later retrieval, and additionally to share them with other users. These retrieving users may not be online at the point of the file upload, and in fact they may never come online at all. In this asynchoronous environment, key transport appears to be at odds with any demands for forward secrecy. Recently, Boyd et al. (ISC 2018) presented a protocol that allows an initiator to use a modified key encapsulation primitive, denoted a blinded KEM (BKEM), to transport a file encryption key to potentially many recipients via the (untrusted) storage server, in a way that gives some guarantees of forward secrecy. Until now all known constructions of BKEMs are built using RSA and DDH, and thus are only secure in the classical setting. We further the understanding of the use of blinding in post-quantum cryptography in two aspects. First, we show how to generically build blinded KEMs from homomorphic encryption schemes with certain properties. Second, we construct the first post-quantum secure blinded KEMs, and the security of our constructions are based on hard lattice problems

Fast polynomial multiplication using matrix multiplication accelerators with applications to NTRU on Apple M1/M3 SoCs

Efficient polynomial multiplication routines are critical to the performance of lattice-based post-quantum cryptography (PQC). As PQC standards only recently started to emerge, CPUs still lack specialized instructions to accelerate such routines. Meanwhile, deep learning has grown immeasurably in importance. Its workloads call for teraflops-level of processing power for linear algebra operations, mainly matrix multiplication. Computer architects have responded by introducing ISA extensions, coprocessors and special-purpose cores to accelerate such operations. In particular, Apple ships an undocumented matrix-multiplication coprocessor, AMX, in hundreds of millions of mobile phones, tablets and personal computers. Our work repurposes AMX to implement polynomial multiplication and applies it to the NTRU cryptosystem, setting new speed records on the Apple M1 and M3 systems-on-chip (SoCs)