scholarly journals Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 858
Author(s):  
Alberto Pedrouzo-Ulloa ◽  
Juan Ramón Troncoso-Pastoriza ◽  
Nicolas Gama ◽  
Mariya Georgieva ◽  
Fernando Pérez-González
Keyword(s):  
Cyclotomic Polynomials ◽  
Time Efficiency ◽  
Space And Time ◽  
Low Degree ◽  
Learning With Errors ◽  
Lattice Based Cryptography ◽  
Low Degree Polynomials ◽  
Learning With Errors Problem ◽  
Recent Attack

The “Multivariate Ring Learning with Errors” problem was presented as a generalization of Ring Learning with Errors (RLWE), introducing efficiency improvements with respect to the RLWE counterpart thanks to its multivariate structure. Nevertheless, the recent attack presented by Bootland, Castryck and Vercauteren has some important consequences on the security of the multivariate RLWE problem with “non-coprime” cyclotomics; this attack transforms instances of m-RLWE with power-of-two cyclotomic polynomials of degree n=∏ini into a set of RLWE samples with dimension maxi{ni}. This is especially devastating for low-degree cyclotomics (e.g., Φ4(x)=1+x2). In this work, we revisit the security of multivariate RLWE and propose new alternative instantiations of the problem that avoid the attack while still preserving the advantages of the multivariate structure, especially when using low-degree polynomials. Additionally, we show how to parameterize these instances in a secure and practical way, therefore enabling constructions and strategies based on m-RLWE that bring notable space and time efficiency improvements over current RLWE-based constructions.

scholarly journals Time-memory trade-off in Toom-Cook multiplication: an application to module-lattice based cryptography

2020 ◽  
pp. 222-244
Author(s):  
Jose Maria Bermudo Mera ◽  
Angshuman Karmakar ◽  
Ingrid Verbauwhede
Keyword(s):  
Linear Transformations ◽  
Polynomial Multiplication ◽  
Matrix Vector Multiplication ◽  
Long Time ◽  
Key Encapsulation Mechanism ◽  
Learning With Errors ◽  
Lattice Based Cryptography ◽  
Learning With Errors Problem ◽  
Matrix Vector ◽  
Processing Steps

Since the introduction of the ring-learning with errors problem, the number theoretic transform (NTT) based polynomial multiplication algorithm has been studied extensively. Due to its faster quasilinear time complexity, it has been the preferred choice of cryptographers to realize ring-learning with errors cryptographic schemes. Compared to NTT, Toom-Cook or Karatsuba based polynomial multiplication algorithms, though being known for a long time, still have a fledgling presence in the context of post-quantum cryptography.In this work, we observe that the pre- and post-processing steps in Toom-Cook based multiplications can be expressed as linear transformations. Based on this observation we propose two novel techniques that can increase the efficiency of Toom-Cook based polynomial multiplications. Evaluation is reduced by a factor of 2, and we call this method precomputation, and interpolation is reduced from quadratic to linear, and we call this method lazy interpolation.As a practical application, we applied our algorithms to the Saber post-quantum key-encapsulation mechanism. We discuss in detail the various implementation aspects of applying our algorithms to Saber. We show that our algorithm can improve the efficiency of the computationally costly matrix-vector multiplication by 12−37% compared to previous methods on their respective platforms. Secondly, we propose different methods to reduce the memory footprint of Saber for Cortex-M4 microcontrollers. Our implementation shows between 2.6 and 5.7 KB reduction in the memory usage with respect to the smallest implementation in the literature.

scholarly journals Proxy Re-Encryption in cloud using ALBC (adaptive lattice based cryptography)

2019 ◽  
Vol 16 (3) ◽  
pp. 1455
Author(s):  
Chandrakala B M ◽  
S C Lingareddy
Keyword(s):  
Comparative Analysis ◽  
Data Sharing ◽  
Data Security ◽  
Key Generation ◽  
Encryption Scheme ◽  
Two Phase ◽  
Data Store ◽  
Learning With Errors ◽  
Lattice Based Cryptography

<p>In recent days, data sharing has provided the flexibility to share the data, store the data, and perform operation on data virtually as well as cost effectively. Data sharing in cloud is one of the feature, which is being popular and widely accepted. However, the concern here is to ensure the data security and this has led the researcher to research in this area. To provide the security several Proxy re-encryption scheme has been introduced, however all these method lacks of efficiency. Hence In this paper, we propose a scheme known as ALBC (Adaptive Lattice Based Cryptography), this scheme follows the two phase i.e. encryption and Re-encryption. Encryption phase has few algorithms such as Key_Gen, Enc, Dec. Similarly ALBC Re-Enc has five algorithm i.e. Key_Gen, Key_ReGen,  Enc, Re-Enc, Dec. our algorithm not only provides the security but also solves the problem of RL(Ring-learning) with errors problems. In order to evaluate, our algorithm is compared with the existing model in terms of encryption time, decryption time, re-encryption time, key generation  and key regeneration by varying the various key size. When we observe the comparative analysis, it is observed that our algorithm outperforms the existing algorithm.</p>

scholarly journals Irreducible skew polynomials over domains

2021 ◽  
Vol 29 (3) ◽  
pp. 75-89
Author(s):  
C. Brown ◽  
S. Pumplün
Keyword(s):  
Polynomial Ring ◽  
Skew Polynomial Ring ◽  
Weyl Algebras ◽  
Skew Polynomials ◽  
Low Degree ◽  
Skew Polynomial ◽  
Low Degree Polynomials ◽  
Injective Endomorphism

Abstract Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = tm − a ∈ R.

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