NTRU Research Articles

A Survey of Polynomial Multiplications for Lattice-Based Cryptosystems

We survey various mathematical tools used in software works multiplying polynomials in ℤ q [ x ] ⟨ x n − α x − β ⟩ . In particular, we survey implementation works targeting polynomial multiplications in lattice-based cryptosystems Dilithium, Kyber, NTRU, NTRU Prime, and Saber with instruction set architectures/extensions Armv7-M, Armv7E-M, Armv8-A, and AVX2. There are three emphases in this paper: (i) modular arithmetic, (ii) homomorphisms, and (iii) vectorization. For modular arithmetic, we survey Montgomery, Barrett, and Plantard multiplications. For homomorphisms, we survey (a) various homomorphisms such as Cooley–Tukey FFT, Good–Thomas FFT, Bruun's FFT, Rader's FFT, Karatsuba, and Toom–Cook; (b) various algebraic techniques for adjoining nice properties to the coefficient rings, including localization, Schönhage's FFT, Nussbaumer's FFT, and coefficient ring switching; and (c) various algebraic techniques related to the polynomial moduli, including twisting, composed multiplication, evaluation at ∞ , truncation, incomplete transformation, striding, and Toeplitz matrix-vector product. For vectorization, we survey the relations between homomorphisms and vector arithmetic. We then go through several case studies: We compare the implementations of modular multiplications used in Dilithium and Kyber, explain how the matrix-to-vector structure was exploited in Saber, and review the design choices of transformations for NTRU and NTRU Prime with vectorization. Finally, we outline several interesting implementation projects.

Performance and Efficiency Exploration of Hardware Polynomial Multipliers for Post-Quantum Lattice-Based Cryptosystems

The significant effort in the research and design of large-scale quantum computers has spurred a transition to post-quantum cryptographic primitives worldwide. The post-quantum cryptographic primitive standardization effort led by the US NIST has recently selected the asymmetric encryption primitive Kyber as its candidate for standardization and indicated NTRU, as a valid alternative if intellectual property issues are not solved. Finally, a more conservative alternative to NTRU, NTRUPrime was also considered as an alternate candidate, due to its design choices that remove the possibility for a large set of attacks preemptively. All the aforementioned asymmetric primitives provide good performances, and are prime choices to provide IoT devices with post-quantum confidentiality services. In this work, we present a comprehensive exploration of hardware designs for the computation of polynomial multiplications, the workhorse operation in all the aforementioned cryptosystems, with a thorough analysis of performance, compactness and efficiency. The presented designs cope with the differences in the arithmetics of polynomial rings employed by distinct cryptosystems, benefiting from configurations and optimizations that are applicable at synthesis time and/or run time. In this context, we target a use case scenario where long-term key pairs are used, such as the ones for VPNs (e.g., over IPSec), secure shell protocols and instant messaging applications. Our high-performance design variants exhibit figures of latency comparable to the ones needed for the execution of the symmetric cryptographic primitives also included in the Post-Quantum schemes. Notably, the performance figures of the designs proposed for NTRU and NTRU Prime surpass the ones described in the related literature.

Gadget-based Masking of Streamlined NTRU Prime Decapsulation in Hardware

Gadget-based Masking of Streamlined NTRU Prime Decapsulation in Hardware

Multiple-Valued Plaintext-Checking Side-Channel Attacks on Post-Quantum KEMs

In this paper, we present a side-channel analysis (SCA) on key encapsulation mechanisms (KEMs) based on the Fujisaki–Okamoto (FO) transformation and its variants. Many post-quantum KEMs usually perform re-encryption during key decapsulation to achieve chosen-ciphertext attack (CCA) security. The side-channel leakage of re-encryption can be exploited to mount a key-recovery plaintext-checking attack (KR-PCA), even if the chosen-plaintext attack (CCA) secure decryption constructing the KEM is securely implemented. Herein, we propose an efficient side-channel-assisted KR-PCA on post-quantum KEMs, and achieve a key recovery with significantly fewer attack traces than existing ones in TCHES 2022 and 2023. The basic concept of the proposed attack is to introduce a new KR-PCA based on a multiple-valued (MV-)PC oracle and then implement a dedicated MV-PC oracle based on a multi-classification neural network (NN). The proposed attack is applicable to the NIST PQC selected algorithm Kyber and the similar lattice-based Saber, FrodoKEM and NTRU Prime, as well as SIKE. We also present how to realize a sufficiently reliable MV-PC oracle from NN model outputs that are not 100% accurate, and analyze the tradeoff between the key recovery success rate and the number of attack traces. We assess the feasibility of the proposed attack through attack experiments on three typical symmetric primitives to instantiate a random oracle (SHAKE, SHA3, and AES software). The proposed attack reduces the number of attack traces required for a reliable key recovery by up to 87% compared to the existing attacks against Kyber and other lattice-based KEMs, under the condition of 99.9999% success rate for key recovery. The proposed attack can also reduce the number of attack traces by 85% for SIKE.

Streamlined NTRU Prime on FPGA

We present a novel full hardware implementation of Streamlined NTRU Prime, with two variants: a high-speed, high-area implementation and a slower, low-area implementation. We introduce several new techniques that improve performance, including a batch inversion for key generation, a high-speed schoolbook polynomial multiplier, an NTT polynomial multiplier combined with a CRT map, a new DSP-free modular reduction method, a high-speed radix sorting module, and new encoders and decoders. With the high-speed design, we achieve the to-date fastest speeds for Streamlined NTRU Prime, with speeds of 5007, 10,989, and 64,026 cycles for encapsulation, decapsulation, and key generation, respectively, while running at 285 MHz on a Xilinx Zynq Ultrascale+. The entire design uses 40,060 LUT, 26,384 flip-flops, 36.5 Bram, and 31 DSP.

Multi-Parameter Support with NTTs for NTRU and NTRU Prime on Cortex-M4

We propose NTT implementations with each supporting at least one parameter of NTRU and one parameter of NTRU Prime. Our implementations are based on size-1440, size-1536, and size-1728 convolutions without algebraic assumptions on the target polynomial rings. We also propose several improvements for the NTT computation. Firstly, we introduce dedicated radix-(2, 3) butterflies combining Good–Thomas FFT and vector-radix FFT. In general, there are six dedicated radix-(2, 3) butterflies and they together support implicit permutations. Secondly, for odd prime radices, we show that the multiplications for one output can be replaced with additions/subtractions. We demonstrate the idea for radix-3 and show how to extend it to any odd prime. Our improvement also applies to radix-(2, 3) butterflies. Thirdly, we implement an incomplete version of Good–Thomas FFT for addressing potential code size issues. For NTRU, our polynomial multiplications outperform the state-of-the-art by 2.8%−10.3%. For NTRU Prime, our polynomial multiplications are slower than the state-of-the-art. However, the SotA exploits the specific structure of coefficient rings or polynomial moduli, while our NTT-based multiplications exploit neither and apply across different schemes. This reduces the engineering effort, including testing and verification.

Faster characteristic three polynomial multiplication and its application to NTRU Prime decapsulation

Faster characteristic three polynomial multiplication and its application to NTRU Prime decapsulation

Will You Cross the Threshold for Me?

In this work, we propose generic and novel side-channel assisted chosenciphertext attacks on NTRU-based key encapsulation mechanisms (KEMs). These KEMs are IND-CCA secure, that is, they are secure in the chosen-ciphertext model. Our attacks involve the construction of malformed ciphertexts. When decapsulated by the target device, these ciphertexts ensure that a targeted intermediate variable becomes very closely related to the secret key. An attacker, who can obtain information about the secret-dependent variable through side-channels, can subsequently recover the full secret key. We propose several novel CCAs which can be carried through by using side-channel leakage from the decapsulation procedure. The attacks instantiate three different types of oracles, namely a plaintext-checking oracle, a decryptionfailure oracle, and a full-decryption oracle, and are applicable to two NTRU-based schemes, which are NTRU and NTRU Prime. The two schemes are candidates in the ongoing NIST standardization process for post-quantum cryptography. We perform experimental validation of the attacks on optimized and unprotected implementations of NTRU-based schemes, taken from the open-source pqm4 library, using the EM-based side-channel on the 32-bit ARM Cortex-M4 microcontroller. All of our proposed attacks are capable of recovering the full secret key in only a few thousand chosen ciphertext queries on all parameter sets of NTRU and NTRU Prime. Our attacks, therefore, stress on the need for concrete side-channel protection strategies for NTRUbased KEMs.

Curse of Re-encryption: A Generic Power/EM Analysis on Post-Quantum KEMs

This paper presents a side-channel analysis (SCA) on key encapsulation mechanism (KEM) based on the Fujisaki–Okamoto (FO) transformation and its variants. The FO transformation has been widely used in actively securing KEMs from passively secure public key encryption (PKE), as it is employed in most of NIST post-quantum cryptography (PQC) candidates for KEM. The proposed attack exploits side-channel leakage during execution of a pseudorandom function (PRF) or pseudorandom number generator (PRG) in the re-encryption of KEM decapsulation as a plaintext-checking oracle that tells whether the PKE decryption result is equivalent to the reference plaintext. The generality and practicality of the plaintext-checking oracle allow the proposed attack to attain a full-key recovery of various KEMs when an active attack on the underlying PKE is known. This paper demonstrates that the proposed attack can be applied to most NIST PQC third-round KEM candidates, namely, Kyber, Saber, FrodoKEM, NTRU, NTRU Prime, HQC, BIKE, and SIKE (for BIKE, the proposed attack achieves a partial key recovery). The applicability to Classic McEliece is unclear because there is no known active attack on this cryptosystem. This paper also presents a side-channel distinguisher design based on deep learning (DL) for mounting the proposed attack on practical implementation without the use of a profiling device. The feasibility of the proposed attack is evaluated through experimental attacks on various PRF implementations (a SHAKE software, an AES software, an AES hardware, a bit-sliced masked AES software, and a masked AES hardware based on threshold implementation). Although it is difficult to implement the oracle using the leakage from the TI-based masked hardware, the success of the proposed attack against these implementations (even except for the masked hardware), which include masked software, confirms its practicality.

Polynomial Multiplication in NTRU Prime

This paper proposes two different methods to perform NTT-based polynomial multiplication in polynomial rings that do not naturally support such a multiplication. We demonstrate these methods on the NTRU Prime key-encapsulation mechanism (KEM) proposed by Bernstein, Chuengsatiansup, Lange, and Vredendaal, which uses a polynomial ring that is, by design, not amenable to use with NTT. One of our approaches is using Good’s trick and focuses on speed and supporting more than one parameter set with a single implementation. The other approach is using a mixed radix NTT and focuses on the use of smaller multipliers and less memory. On a ARM Cortex-M4 microcontroller, we show that our three NTT-based implementations, one based on Good’s trick and two mixed radix NTTs, provide between 32% and 17% faster polynomial multiplication. For the parameter-set ntrulpr761, this results in between 16% and 9% faster total operations (sum of key generation, encapsulation, and decapsulation) and requires between 15% and 39% less memory than the current state-of-the-art NTRU Prime implementation on this platform, which is using Toom-Cook-based polynomial multiplication.

Single-Trace Attacks on Message Encoding in Lattice-Based KEMs

In this article, we propose single-trace side-channel attacks against lattice-based key encapsulation mechanisms (KEMs) that are the third-round candidates of the national institute of standards and technology (NIST) standardization project. Specifically, we analyze the message encoding operation in the encapsulation phase of lattice-based KEMs to obtain an ephemeral session key. We conclude that a single-trace leakage implies a whole key recovery: the experimental results realized on a ChipWhisperer UFO STM32F3 target board achieve a success rate of 100% for $\mathsf {CRYSTALS-KYBER}$ and $\mathsf {SABER}$ regardless of an optimization level and those greater than 79% for $\mathsf {FrodoKEM}$ . We further demonstrate that the proposed attack methodologies are not restricted to the above algorithms but are widely applicable to other NIST post-quantum cryptography (PQC) candidates, including $\mathsf {NTRU Prime}$ and $\mathsf {NTRU}$ .

Power Analysis on NTRU Prime

This paper applies a variety of power analysis techniques to several implementations of NTRU Prime, a Round 2 submission to the NIST PQC Standardization Project. The techniques include vertical correlation power analysis, horizontal indepth correlation power analysis, online template attacks, and chosen-input simple power analysis. The implementations include the reference one, the one optimized using smladx, and three protected ones. Adversaries in this study can fully recover private keys with one single trace of short observation span, with few template traces from a fully controlled device similar to the target and no a priori power model, or sometimes even with the naked eye. The techniques target the constant-time generic polynomial multiplications in the product scanning method. Though in this work they focus on the decapsulation, they also work on the key generation and encapsulation of NTRU Prime. Moreover, they apply to the ideal-lattice-based cryptosystems where each private-key coefficient comes from a small set of possibilities.

METHODS OF BUILDING GENERAL PARAMETERS AND KEYS FOR NTRU PRIME UKRAINE OF 5TH – 7TH LEVELS OF STABILITY. PRODUCT FORM

The research was carried out and the development of an effective practical algorithm for the construction of system-wide parameters and keys of cryptographic transformations such as asymmetric ciphers for a special form of setting the Product Form polynomial was performed. The experimental confirmation of the built-in system-wide parameters for 5–7 stability levels NTRU PRIME UKRAINE, taking into account a combined attack is given.

Methods for constructing system-wide parameters and keys for NTRU PRIME UKRAINE 5 – 7 stabvility levels. Product form

The research was carried out and the development of an effective practical algorithm for the construction of system-wide parameters and keys of cryptographic transformations such as asymmetric ciphers for a special form of setting the Product Form polynomial was performed. The experimental confirmation of the built-in system-wide parameters for 5 – 7 stability levels NTRU PRIME UKRAINE, taking into account a combined attack. is given.

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.

NTRU based Security in Cloud Computing

The usage of Cloud Computing users keep their facts onto the third party owners and experience the online demand programs, services and garage from a shared pool of configurable computing resources, without the load of facts garage and protection and costs. The furnished security must guarantee no longer best on authentication but also on files over the cloud for the outsourced records, that is now maintained by way of third parties consisting of cloud carriers. Unluckily, the infrastructure of cloud computing is constructed on internet, and will stumble upon all of the risk of net. To lower the risks, an authentication mechanism is the viable answer. However, conventional alphanumeric password authentication mechanism isn't always relaxed sufficient. A cozy authentication mechanism the use of public-key encryption based totally password have to be proposed in this paper for enhancing traditional authentication mechanism and let users get entry to cloud services securely.

General statements and analysis of the end-to-end encryption algorithm NTRU Prime IIT Ukraine

The NTRU-like cryptosystem NTRU Prime IIT Ukraine, created on the basis of existing cryptographic transformations end-to-end encryption type, is considered. The description of this cryptosystem is given and its analysis is carried out. Also, features of its implementation, comparison of the main characteristics and indicators, as well as the definition of differences from existing NTRU-like cryptographic algorithms are presented. Conclusions are made and recommendations are given.

DTRU1: first generalization of NTRU using dual integers

DTRU1: first generalization of NTRU using dual integers

Optimization of NTRU Prime asymmetric encryption algorithm

The results of algorithms optimization for the post-quantum mechanism of asymmetric encryption NTRU Prime are given. Namely: module reduction, blinding polynomial calculation, encryption and decryption algorithms. A comparative analysis of different polynomials multiplications is also given.

Analysis of the end-to-end encryption algorithm NTRU PRIME IIT UKRAINE taking into account known attacks

Modern cryptographic transformations of the end-to-end encryption type, namely – NTRU-like cryptosystems are considered. A new cryptographic system NTRU PRIME IIT UKRAINE was created based on existing cryptographic transformations of this type. A brief description of this cryptosystem is presented and an analysis of its resistance to known attacks is carried out, conclusions are made and recommendations are given.