Security Analysis of DBTRU Cryptosystem

This note contains a linear algebra attack on elementary complex analysis which leads to a three‐fold unification between linear algebra, multivariable calculus and complex analysis.

CTRU, a public key cryptosystem was proposed by Gaborit, Ohler and Sole. It is analogue of NTRU, the ring of integers replaced by the ring of polynomials \(\mathbb{F}_2[T]\). It attracted attention as the attacks based on either LLL algorithm or the Chinese Remainder Theorem are avoided on it, which is most common on NTRU. In this paper we presents a polynomial-time algorithm that breaks CTRU for all recommended parameter choices that were derived to make CTRU secure against popov normal form attack. The paper shows if we ascertain the constraints for perfect decryption then either plaintext or private key can be achieved by polynomial time linear algebra attack.KeywordsQuotient RingChinese Remainder TheoremElementary Linear AlgebraLattice Basis ReductionRecommended ParameterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents the security analysis of symmetric homomorphic cryptosystems based on polynomial ring homomorphisms.First we describe an approach to carry out a known plaintexts attack (KPA) on such cryptosystems in the case when the plaintexts space is a finite field. The approach uses Euclidean algorithm and Berlekamp algorithm for polynomial factorization. We give theoretical estimations of probability to recover a secret key and provide experimental results. We analyse the complexity of our attack and give recommendations how it can be implemented in parallel.Second we discuss how introduced approach can be adapted for the case when the plaintexts space is a ring of integers modulo composite number. Here a particular attention is given to modules that are products of two primes. Results of computer experiment are given.The presented known plaintexts attack on cryptosystems allows to recover secret key with overwhelming probability if the number of pairs (ciphertext, plaintext) is more than five.

For the first time to our knowledge we demonstrate that simply using all free parameters of the single recirculating PC PMD emulator leads to near ideal statistical distribution, making it an instrument of choice.

The question of whether all projective modules (or all finitely generated projective modules) over a given ring are free or extended is solved positively only for relatively small class of rings. In 1976, a significant progress was achieved in this area of research: D. Quillen and A. A. Suslin simultaneously and independently of one another proved that all projective modules over algebras of polynomials over fields are free modules, which was a solution to the well-known J.-P. Serre’s problem. Later similar results were obtained by various researchers for projectivemodules over ringsmore general then algebras of polynomials over fields. For example, V. A. Artamonov solved this problem for rings of quantum polynomials. The current state of this line of research is described in detail in the survey paper [1] containing an extensive list of references. Inmany cases, it is possible to prove that all projectivemodules of sufficiently large ranks over a given ring are free (or extended), but for projectivemodules of small ranks (for example, of rank one) this result does not hold. In this paper, we discuss problems of more special character: The problem of finding in an explicit form isomorphisms between a given projective module and a free (or extended) module, and, for an explicitly given projective module, the problem of efficient determination whether it is a free (or extended) module. By definition, a finitely generated projective module is a direct summand of a free module. Therefore, the most general way to define such a module is to indicate the idempotent matrix corresponding to the projection onto the direct summand. We show that, for modules over graded rings, the fact that a module is extended can be established if the sequence of powers of nonzero homogeneous components of the given idempotent matrix is known (Theorem 1), or under condition that certain blocks of positive homogeneous components of the same matrix vanish (Theorem 2). In the above described cases, there is a method of explicit constructing of the desired isomorphism. Statements of such kind are derived as consequences of the results of [2], where the author considered a situation which is so general that one can take as an “algebra” for which the results of [2] are valid not only linear multioperator algebras but modules over a certain class of rings. In this paper, we use the basic constructions, definitions, and notation from [2]. We make only insignificant changes in notation in order that it correspond to the generally adopted notation in the theory of rings and modules. So, the notation f1 used in [2] (for a polynomial, it corresponds to the homogeneous component of the first degree) is changed here by f0 since it corresponds to the zero component of the grading of the ring. Correspondingly, the other indices are also shifted by one.

This paper continues our research of the so-called matrix power function and its application to cryptography. We consider the simplest case of the improved matrix power asymmetric cypher (MPAC). We show that our protocol is resistant to linear algebra attack, which can be applied to the initial version of MPAC to break it in polynomial time. Our main goal is to present the general idea for the choice of public parameters of MPAC protocol to avoid this attack while also avoiding exhaustive search attack.

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Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC, and ElGamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found to be able to solve certain mathematical problems on non-commutative algebraic structures. Against this background, Raulynaitis et al. have proposed a novel asymmetric cipher protocol using a matrix decomposition problem. Their proposed scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we show that the asymmetric cipher of the non-commutative cryptography scheme is vulnerable to a linear algebra attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. We also propose an improvement to enhance the scheme of Raulynaitis et al.

At the current state of the art, algebraic attacks are the most efficient method for finding preimages and collisions for arithmetization-oriented hash functions, such as the closely related primitives Poseidon/Poseidon2 and Neptune. In this paper, we revisit Gröbner basis (GB) attacks that exploit subspace trails to linearize some partial rounds, considering both sponge and compression modes.Starting from Poseidon’s original security evaluation, we identified some inaccuracies in the model description that may lead to misestimated round requirements. Consequently, we reevaluate and improve the proposed attack strategy. We find that depending on the concrete instantiation, the original security analysis of Poseidon under- or overestimates the number of rounds needed for security. Moreover, we demonstrate that GB attacks leveraging subspace trails can outperform basic GB attacks for Poseidon/Poseidon2 and Neptune.We propose a variant of the previous attack strategy that exploits a crucial difference between Poseidon/Poseidon2 and Neptune: while Poseidon’s inverse round functions have a high degree, Neptune’s inverse external rounds maintain the same degree as the forward rounds. Using this new model, we demonstrate that Neptune’s security in compression mode cannot be reduced to its security against the Constrained-Input-Constrained-Output (CICO) problem. To the best of our knowledge, this is the first time a concrete example has been provided where finding preimages is easier than solving the corresponding CICO problem.Our results emphasize the importance of considering the mode of operation in security analysis while confirming the overall security of Poseidon/Poseidon2 and Neptune against the presented algebraic attacks.

In this paper we introduce a new chaotic stream cipher Mmohocc which utilizes the fundamental chaos characteristics. The designs of the major components of the cipher are given. Its cryptographic properties of period, auto- and cross-correlations, and the mixture of Markov processes and spatiotemporal effects are investigated. The cipher is resistant to the related-key-IV attacks, Time/Memory/Data tradeoff attacks, algebraic attacks, and chosen-text attacks. The keystreams successfully passed two batteries of statistical tests and the encryption speed is comparable with RC4.

Numerical simulations related to human hearing, architectural and underwater acoustics, and multiple scattering require computational solution of the Helmholtz equation in three dimensions with complex shaped boundaries. Boundary element methods (BEM) are among the most accurate and efficient methods used for this purpose. However, solution of high frequency/large domain problems is challenging due to poor scaling of conventional solvers with the frequency and domain size. The use of the fast multipole methods (FMM) resolves many problems related to the scalability [N.A. Gumerov and R. Duraiswami, J. Acoust. Soc. Am. 125(1), 191205, 2009]. Additional accelerations are needed to be able to solve practical problems over the entire range of human audible frequencies and can be provided using graphics processors (GPUs) with multicore CPUs. In this work, we report development and demonstration of the FMM/GPU accelerated BEM for the Helmholtz equation in 3D designed for hybrid CPU/GPU architectures. Innovations related to choices of preconditioners, parallelization strategies, and choice of optimal parameters will be presented. A single PC version of the algorithm shows accelerations of the order of 10 times compared to the BEM accelerated with the FMM alone. Results for computing head related transfer functions and other standard calculations will be provided.

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.

Stable numerical methods for solving simultaneous systems of linear algebraic equations

In recent years, almost all systems using linear feedback shift storage device (LFSR) design of stream ciphers have been cracked by Algebraic attack. In light of this, finding a modern cipher that avoids old security issues with design of both security and performance becomes an important issue. In 2004, European Union initiated the selection of a new streaming secret (eSTREAM), which is analogous to United States’ formulation of AES block density standards. Forty eight candidates were chosen after public screening, review and final selection process. Details were published in May 2008 results report. This article is to discuss eSTREAM stream cipher. eSTREAM was part of group of eight surviving third phase of software selection process focusing on HC stream cipher for research. In recent years, stream cipher attack studies have been very popular, especially employing Distinguishing Attack. This article also performed security analysis for HC stream cipher, and proposes a design for a reduced version of HC-256. The reduced version of HC-256 is comparable to original HC-256 in terms of resistance to distinguishing attacks. This new generation of core stream cipher algorithm design can help understand implications of theory and application of mathematical practices towards research and analysis of stream ciphers.

The Internet of Things (IoT) is an emerging technology, which comprises wireless smart sensors and actuators. Nowadays, IoT is implemented in different areas such as Smart Homes, Smart Cities, Smart Industries, Military, eHealth, and several real-world applications by connecting domain-specific sensors. Designing a security model for these applications is challenging for researchers since attacks (for example, zero-day) are increasing tremendously. Several security methods have been developed to ensure the CIA (Confidentiality, Integrity, and Availability) for Industrial IoT (IIoT). Though these methods have shown promising results, there are still some security issues that are open. Thus, the security and authentication of IoT based applications become quite significant. In this paper, we propose TD2SecIoT (Temporal, Data-Driven and Dynamic Network Layer Based Security Architecture for Industrial IoT), which incorporates Elliptic Curve Cryptography (ECC) and Nth-degree Truncated Polynomial Ring Units (NTRU) methods to ensure confidentiality and integrity. The proposed method has been evaluated against different attacks and performance measures (quantitative and qualitative) using the Cooja network simulator with Contiki-OS. The TD2SecIoT has shown a higher security level with reduced computational cost and time.