An algorithm for fast multiplication of sedenions

Cariow, Aleksandr; Cariowa, Galina

doi:10.1016/j.ipl.2013.02.011

Cited by 17 publications

(9 citation statements)

References 4 publications

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“…In conclusion, it should be noted that the proposed approach allows the construction of sufficiently well algorithms for the multiplication of hypernumbers with reduced computational complexity. In our previous work [15][16][17], we have applied the unified approach proposed here for the synthesis of fast algorithms for the multiplication of quaternions, octonions and sedenions. However, if the specific properties of the matrix are used, even more interesting solutions may be found [18][19].…”

Section: Discussionmentioning

confidence: 99%

An unified approach for developing rationalized algorithms for hypercomplex number multiplication

Cariow

Cariowa

2015

ELECTROTECHNICAL REVIEW

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In this article we present a common approach for the development of algorithms for calculating products of hypercomplex numbers. The main idea of the proposed approach is based on the representation of hypernumbers multiplying via the matrix-vector products and further creative decomposition of the matrix, leading to the reduction of arithmetical complexity of calculations. The proposed approach allows the construction of sufficiently well algorithms for hypernumbers multiplication with reduced computational complexity. If the schoolbook method requires N 2 real multiplications and N(N-1) real additions, the proposed approach allows to develop algorithms, which take only [N(N-1)/2]+2 real multiplications and 3Nlog 2 N+[N(N-3)+4]/2 real additions. Streszczenie. W artykule zostało przedstawione uogólnione podejście do syntezy algorytmów wyznaczania iloczynów liczb hiperzespolonych. Główna idea proponowanego podejścia polega na reprezentacji operacji mnożenia liczb hiperzespolonych w formie iloczynu wektorowomacierzowego i dalszej możliwości kreatywnej dekompozycji czynnika macierzowego prowadzącej do redukcji złożoności obliczeniowej. Proponowane podejście pozwala zbudować algorytmy wyróżniające się w porównaniu do metody naiwnej zredukowaną złożonością obliczeniową. Jeśli metoda naiwna wymaga wykonania N 2 mnożeń oraz N(N-1) dodawań liczb rzeczywistych to proponowane podejście pozwala syntetyzować algorytmy wymagające tylko [N(N-1)/2]+2 mnożeń oraz 3Nlog 2 N+[N(N-3)+4]/2 dodawań. (Uogólnione podejście do konstruowania zracjonalizowanych algorytmów mnożenia liczb hiperzespolonych-tytuł polski artykułu).

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Section: Discussionmentioning

confidence: 99%

An unified approach for developing rationalized algorithms for hypercomplex number multiplication

Cariow

Cariowa

2015

ELECTROTECHNICAL REVIEW

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“…On that basis, the new scheme provides strong security with a negligible speed difference over OctoM compared to the previous [21] scheme. In addition, to increase the efficiency of the scheme, we use a quick [25] algorithm for sedenion multiplication, which saves 134real multiplication, requiring 15 % fewer arithmetic operations than direct evaluation. We conclude this section by adding some notations, such as finite rings Zq = Z/qZ, a totally isotropic subspace V with dimension k ≤ n/2, i.e.…”

Section: Introductionmentioning

confidence: 99%

Noise Free Fully Homomorphic Encryption Scheme Over Non-Associative Algebra

Mustafa

Azar

et al. 2020

IEEE Access

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Among several approaches to privacy-preserving cryptographic schemes, we have concentrated on noise-free homomorphic encryption. It is a symmetric key encryption that supports homomorphic operations on encrypted data. We present a fully homomorphic encryption (FHE) scheme based on sedenion algebra over finite Z n rings. The innovation of the scheme is the compression of a 16-dimensional vector for the application of Frobenius automorphism. For sedenion, we have p 16 different possibilities that create a significant bijective mapping over the chosen 16-dimensional vector that adds permutation to our scheme. The security of this scheme is based on the assumption of the hardness of solving a multivariate quadratic equation system over finite Z n rings. The scheme results in 256n multivariate polynomial equations with 256 + 16n unknown variables for n messages. For this reason, the proposed scheme serves as a security basis for potentially post-quantum cryptosystems. Moreover, after sedenion, no newly constructed algebra loses its properties. This scheme would therefore apply as a whole to the following algebras, such as 32dimensional trigintadunion.

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“…Therefore, reducing the computational complexity of the multiplication of Clifford numbers is an important scientific and engineering problem. Efficient algorithms for the multiplication of quaternions, octonions and sedenions already exist [12,13,14,15]. No such *Corresponding author.…”

Section: Introductionmentioning

confidence: 99%