NTRU Cryptosystem Research Articles

Secure computation protocol of Chebyshev distance under the malicious model

Secure multi-party computation of Chebyshev distance represents a crucial method for confidential distance measurement, holding significant theoretical and practical implications. Especially within electronic archival management systems, secure computation of Chebyshev distance is employed for similarity measurement, classification, and clustering of sensitive archival information, thereby enhancing the security of sensitive archival queries and sharing. This paper proposes a secure protocol for computing Chebyshev distance under a semi-honest model, leveraging the additive homomorphic properties of the NTRU cryptosystem and a vector encoding method. This protocol transforms the confidential computation of Chebyshev distance into the inner product of confidential computation vectors, as demonstrated through the model paradigm validating its security under the semi-honest model. Addressing potential malicious participant scenarios, a secure protocol for computing Chebyshev distance under a malicious model is introduced, utilizing cryptographic tools such as digital commitments and mutual decryption methods. The security of this protocol under the malicious model is affirmed using the real/ideal model paradigm. Theoretical analysis and experimental simulations demonstrate the efficiency and practical applicability of the proposed schemes.

Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem

We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived codes are good in that they exhibit constant rate and average distance scaling Δ∝n with high probability, where n is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance. The derived class of NTRU-GKP codes has the additional property that decoding for a stochastic displacement noise model is equivalent to decrypting the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. This construction highlights how the GKP code bridges aspects of classical error correction, quantum error correction as well as post-quantum cryptography. We underscore this connection by discussing the computational hardness of decoding GKP codes and propose, as a new application, a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem.

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): polynomial multiplication, key generation, encapsulation and decapsulation are sped up by 1.54 – 3.07 × , 1.08 – 1.33 × , 1.11 – 1.50 × and 1.20 – 1.98 × , respectively, over the previous state-of-the-art.

Secure Data Storage in Cloud Computing Using Code Based McEliece and NTRU Cryptosystems

Secure Data Storage in Cloud Computing Using Code Based McEliece and NTRU Cryptosystems

Lossless Data Hiding in NTRU Cryptosystem by Polynomial Encoding and Modulation

Lossless Data Hiding in NTRU Cryptosystem by Polynomial Encoding and Modulation

Securing IoT Networks with NTRU Cryptosystem: A Practical Approach on ARM-based Devices for Edge and Fog Layer Integration

Securing IoT Networks with NTRU Cryptosystem: A Practical Approach on ARM-based Devices for Edge and Fog Layer Integration

TRUHSH: Developing NTRU Cryptosystem in Terms of Security and Performance

TRUHSH: Developing NTRU Cryptosystem in Terms of Security and Performance

Timing-Attack-Resistant Acceleration of NTRU Round 3 Encryption on Resource-Constrained Embedded Systems

The advent of quantum computing with high processing capabilities will enable brute force attacks in short periods of time, threatening current secure communication channels. To mitigate this situation, post-quantum cryptography (PQC) algorithms have emerged. Among the algorithms evaluated by NIST in the third round of its PQC contest was the NTRU cryptosystem. The main drawback of this algorithm is the enormous amount of time required for the multiplication of polynomials in both the encryption and decryption processes. Therefore, the strategy of speeding up this algorithm using hardware/software co-design techniques where this operation is executed on specific hardware arises. Using these techniques, this work focuses on the acceleration of polynomial multiplication in the encryption process for resource-constrained devices. For this purpose, several hardware multiplications are analyzed following different strategies, taking into account the fact that there are no possible timing information leaks and that the available resources are optimized as much as possible. The designed multiplier is encapsulated as a fully reusable and parametrizable IP module with standard AXI4-Stream interconnection buses, which makes it easy to integrate into embedded systems implemented on programmable devices from different manufacturers. Depending on the resource constraints imposed, accelerations of up to 30–45 times with respect to the software-level multiplication runtime can be achieved using dedicated hardware, with a device occupancy of around 5%.

High-order masking of NTRU

The main protection against side-channel attacks consists in computing every function with multiple shares via the masking countermeasure. While the masking countermeasure was originally developed for securing block-ciphers such as AES, the protection of lattice-based cryptosystems is often more challenging, because of the diversity of the underlying algorithms. In this paper, we introduce new gadgets for the high-order masking of the NTRU cryptosystem, with security proofs in the classical ISW probing model. We then describe the first fully masked implementation of the NTRU Key Encapsulation Mechanism submitted to NIST, including the key generation. To assess the practicality of our countermeasures, we provide a concrete implementation on ARM Cortex-M3 architecture, and eventually a t-test leakage evaluation.

Revisiting lower dimension lattice attacks on NTRU

Vectors are called short vectors when the Euclidean norm of vectors is relatively small. By now, some researches have been done to find sufficient short vectors in certain variants of NTRU lattice to analyze the security of NTRU systems. These variants are all taking advantage of the property of NTRU cryptosystem to modify the standard NTRU lattice. Related variants include zero-run lattices with dimension 2N, zero-forced lattices with dimension 2(N−r), r<N, and IN-lattices with dimension N.Motivated by different constructions, we revisit related variants and develop some techniques to further reduce the dimension of IN-lattice, such that the expected breaking time can be decreased significantly. Firstly, by working with the cyclic matrix associated with the NTRU public key h, shifts of the private vector f can be accommodated. Hence, t randomly chosen coefficients in a shift of the private vector f can be forced to equal zero. This not only results in a new lattice with a lower dimension but also creates an environment to utilize parallel mechanisms to improve the probability of success. Finally, experiments have demonstrated that our approach outperforms the previous methods.

A Blockchain-Based Secure Multi-Party Computation Scheme with Multi-Key Fully Homomorphic Proxy Re-Encryption

At present, secure multi-party computing is an effective solution for organizations and institutions that want to derive greater value and benefit from the collaborative computing of their data. Most current secure multi-party computing solutions use encryption schemes that are not resistant to quantum attacks, which is a security risk in today’s quickly growing quantum computing, and, when obtaining results, the result querier needs to collect the private keys of multiple data owners to jointly decrypt them, or there needs to be an interaction between the data owner and the querier during the decryption process. Based on the NTRU cryptosystem, which is resistant to quantum computing attacks and has a simple and easy-to-implement structure, and combined with multi-key fully homomorphic encryption (MKFHE) and proxy re-encryption, this paper proposes a secure multi-party computing scheme based on NTRU-type multi-key fully homomorphic proxy re-encryption in the blockchain environment, using the blockchain as trusted storage and a trusted execution environment to provide data security for multi-party computing. The scheme meets the requirements of being verifiable, conspiracy-proof, individually decryptable by the querier, and resistant to quantum attacks.

NTRsh: A New Secure Variant of NTRUEncrypt Based on Tripternion Algebra

Since cryptography relied on mathematical theories and applications of computer science, mathematical algorithms are built with high difficulty. In this paper, we make an improvement and modification NTRU cryptosystem called NTRSH by using a new tripternion algebra and changing the mathematical structure for public and private keys, as well as for text encryption and decryption to obtain more and higher security.

Privacy-preserving hierarchical deterministic key generation based on a lattice of rings in public blockchain

Blockchain has revolutionized numerous fields, which include financial services, health care, the Internet of things, academia and supply chain management. Blockchain technology enables us to have an immutable, distributed ledger for managing the transactions of untrusted users. However, the technology has many open challenges, such as privacy leaks, scalability, and energy consumption. User identity can be easily tracked using network analysis, as transactions are accessible to everyone, which is a serious concern of blockchain. In this paper, we propose a new efficient, privacy-preserving, and quantum-resistant key generation algorithm, namely, lattice-based hierarchical deterministic key generation (LB-HDKG), for maintaining user privacy in the public blockchain. The LB-HDKG scheme generates many cryptographic keys in a tree-like structure from a single seed to hide the links between transactions of the same user. Our proposal uses the lattice NTRU cryptosystem, the security of which relies on the shortest vector problem (SVP) and closest vector problem (CVP) over the polynomial ring. Operations on the lattice NTRU cryptosystem are efficient and secure against classical computers and quantum computers. Security and performance analyses of our scheme show that the model is more secure and efficient and should replace current models to safeguard data from quantum computers.

Lattice Attacks on NTRU Revisited

NTRU cryptosystem was proposed by J. Hoffstein, J.Pipher and J.H. Silverman in 1996, whose security is related to the hardness of finding sufficient short vectors in NTRU lattice with dimension $2N$ . Many researchers conjecture that the private key vector is indeed the shortest vector in the lattice in most cases. However, no formal proof has been provided in the literature before to the best of our knowledge. In this paper, we revisit the lattice attack on NTRU and present a new dimension reduction attack on NTRU without considering the pattern of private polynomials. More precisely, we show that one can recover a group of equivalent private keys by solving shortest vector problem in a new dimension-reduced lattice with dimension $N+k, k , where $k$ is related to the specific parameters selected. As a corollary of our attack, we prove that the private key vector and its rotations are the shortest vectors of the original NTRU lattice with an overwhelming probability, which confirms the conjecture of the length of the shortest vector of the original NTRU lattice.

BKW-attack on NTRUCIPHER and NTRUCIPHER+ enctyption schemes

Due to the appearance of quantum computers, which will significantly reduce the time of solving certain problems, the security of many standardized cryptosystems is under threat. This prompted NIST to launch an open competition to create new post-quantum standards in 2016. In the summer of 2020, the NTRU algorithm, one of the fastest post-quantum algorithms based on lattices in Euclidean space (1996), was entered the seven finalists of this competition. However, only in 2017 was proposed an analog of this encryption scheme – symmetric encryption scheme NTRUCipher. Preliminary researches of this encryption scheme have been conducted but it’s security to chosen-plaintext attack, which consists of compiling a system of linear equations corrupted by noise (over a finite field of simple order) and solving it using a generalized BKW algorithm, have not been analyzed. For the first time, the NTRUCipher + cipher scheme is proposed in this article. Its main difference is the usage of an additional random polynomial when encrypting. The security of NTRUCipher cipher scheme and its modification NTRUCipher+ against BKW-attack is researched. Such an attack is possible for symmetric NTRU-like cipher schemes but it has not been considered before. Analytical (upper and lower) bounds of the BKW attack’s complexity on NTRUCipher and NTRUCipher + are obtained. The comparison of these cipher schemes on the encrypted messages’ length against BKW-attack at certain identical fixed parameters is carried out. It is shown that the security increase of the NTRUCipher cipher scheme against BKW-attack due to the usage of an additional additive in encryption is almost completely leveled by increasing the upper bound of the decryption failure probability. Research allows to compare these cipher schemes in terms of security and practicality and to conclude that it is inexpedient to use NTRUCipher+ to increase the security of the NTRUCipher cipher scheme to BKW attack. In the future, it is planned to develop methods for constructing symmetric analogs of the NTRU cryptosystem based on other general lattice-based structures.

Modified multi-key fully homomorphic encryption based on NTRU cryptosystem without key-switching

The Multi-Key Fully Homomorphic Encryption (MKFHE) based on the NTRU cryptosystem is an important alternative to the post-quantum cryptography due to its simple scheme form, high efficiency, and fewer ciphertexts and keys. In 2012, Lopez-Alt et al. proposed the first NTRU-type MKFHE scheme, the LTV12 scheme, using the key-switching and modulus-reduction techniques, whose security relies on two assumptions: the Ring Learning With Error (RLWE) assumption and the Decisional Small Polynomial Ratio (DSPR) assumption. However, the LTV12 and subsequent NTRU-type schemes are restricted to the family of power-of-2 cyclotomic rings, which may affect the security in the case of subfield attacks. Moreover, the key-switching technique of the LTV12 scheme requires a circular application of evaluation keys, which causes rapid growth of the error and thus affects the circuit depth. In this paper, an NTRU-type MKFHE scheme over prime cyclotomic rings without key-switching is proposed, which has the potential to resist the subfield attack and decrease the error exponentially during the homomorphic evaluating process. First, based on the RLWE and DSPR assumptions over the prime cyclotomic rings, a detailed analysis of the factors affecting the error during the homomorphic evaluations in the LTV12 scheme is provided. Next, a Low Bit Discarded & Dimension Expansion of Ciphertexts (LBD&DEC) technique is proposed, and the inherent homomorphic multiplication decryption structure of the NTRU is proposed, which can eliminate the key-switching operation in the LTV12 scheme. Finally, a leveled NTRU-type MKFHE scheme is developed using the LBD&DEC and modulus-reduction techniques. The analysis shows that the proposed scheme compared to the LTV12 scheme can decrease the magnitude of the error exponentially and minimize the dimension of ciphertexts.

Matrix Lie group as an algebraic structure for NTRU like cryptosystem

In this paper, we briefly describe the mathematical structure lattice and its cryptographic hard problems. The lattice based public key cryptosystem NTRU and its matrix variant Matrix NTRU are analyzed. Based on the analysis we propose the algebraic structure Matrix Lie Group as an alternative for the algebraic structure in Matrix NTRU in order to overcome some of the flaws in the Matrix NTRU cryptosystem.

A new NTRU cryptosystem outperforms three highly secured NTRU-analog systems through an innovational algebraic structure

NTRU public key cryptosystem relies on computations that are efficiently performed with insignificant storage and time complexity. Many researchers were motivated to improve NTRU performance by replacing the original polynomial ring and mathematical structure with new modified algebraic structures, such as QTRU, OTRU, BITRU, and many others. In this work, a new NTRU-analog cryptosystem called QOBTRU is proposed based on a newly designed algebraic structure called Carternion algebra. QOBTRU is at least quadratic higher than the original NTRU in terms of computational and space complexity. It is intended to have an alternative security and performance attribute upon three chosen highly performed multidimensional NTRU-like cryptosystems, which are QTRU, OTRU, and BITRU. Their comparison with QOBTRU highlighted its strength over them. Some statistical and security analysis is performed to demonstrate its efficiency.

A MULTI-DIMENSIONAL ALGEBRA FOR DESIGNING AN IMPROVED NTRU CRYPTOSYSTEM

The NTRU public key cryptosystem is essentially based on the Nth degree trun- cated polynomial ring Z[X]/XN−1 . By altering the Z[X]/XN−1 or its mathematical con- struction, many researchers have worked to improve NTRU. In this paper, we present a new multi-dimensional public key cryptosystem which depends on a commutative quaternion al- gebra with a new structure. The new structure is an alternative to NTRU called NTRTE. Some data are used for comparing the key security and message security of NTRTE with the original NTRU and QTRU cryptosystems. Based on a security assessment, the comparison reveals that NTRTE is more resistant to them.

Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems A la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC code-based cryptosystem) in a rank metric context, due to the similar form of the public keys. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to $\frac {2}{3}(n-k)$ when the previous decoding algorithm only decodes up to $\frac {n-k}{2}$ errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to $\frac {n-k}{2}$ . This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security.