Computation of jacobi sums and cyclotomic numbers with reduced complexity

Ahmed, Helal; Tanti, Jagmohan

doi:10.5958/2320-3226.2019.00050.x

Cited by 3 publications

(8 citation statements)

References 5 publications

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“…Thus, calculation of cyclotomic numbers, so called to as cyclotomic number problems, of various orders is one of the primary problems in Number Theory. Complete answers for cyclotomic number problem for e =2À 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, l,2l, l 2 ,2l 2 with l an odd prime had been investigated by many authors see ( [35][36][37][38][39][40] and the references there in). The section contains the generalized definition of cyclotomic numbers of order e, useful notations followed by properties of cyclotomic numbers of order 2l 2 .…”

Section: Cyclotomic Numbersmentioning

confidence: 99%

“…For the construction of cyclotomic matrix, it does not require to determine all the cyclotomic numbers of a cyclotomic matrix which is shown in Table 1 [36]. By well-known properties of cyclotomic numbers of order 2l 2 , cyclotomic numbers are divided into various classes, therefore there are a pair of the relation between the entries of initial table (Table 1) of a cyclotomic matrix.…”

Section: Determine the Generators Of Chosen P (Ie Generators Of Fmentioning

confidence: 99%

“…This subsection presents the procedure to determine the equality relation of cyclotomic numbers (i.e. the relation of pair of a, b ðÞ 2l 2 ), which reduces the complexity of solutions of pair of a, b ðÞ 2l 2 (see also [36]). For the determination of cyclotomic matrices, it is not necessary to obtain all 4l 4 cyclotomic numbers of order 2l 2 .…”

Section: Determination Of Equality Relation Of Cyclotomic Numbersmentioning

confidence: 99%

“…Thus determination of cyclotomic numbers, so called cyclotomic number problem, of different orders is one of basic problems in number theory. Complete solutions for cyclotomic number problem for e = 2 − 6, 7, 8, 9, 10,11,12,14,15,16,18,20,22, l, 2l, l 2 , 2l 2 with l an odd prime have been investigated by many authors see ( [10,11,12,13] and the references there in).…”

Section: Introductionmentioning

confidence: 99%

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A Public Key Cryptosystem Using Cyclotomic Matrices

Helal¹,

Tanti²,

Pushp³

2022

Coding Theory - Recent Advances, New Perspectives and Applications

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Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this chapter, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to a diophantine system. The strategy uses in cyclotomic matrices to design a one-way function. The result of a one-way function that is efficient to compute, however, is hard to process its inverse except if privileged information about the hidden entry is known. Also, we demonstrate that encryption and decryption can be efficiently performed with the asymptotic complexity of Oe2.373. Finally, we study the computational complexity of the cryptosystem.

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Section: Cyclotomic Numbersmentioning

confidence: 99%

Section: Determine the Generators Of Chosen P (Ie Generators Of Fmentioning

confidence: 99%

Section: Determination Of Equality Relation Of Cyclotomic Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Public Key Cryptosystem Using Cyclotomic Matrices

Helal¹,

Tanti²,

Pushp³

2022

Coding Theory - Recent Advances, New Perspectives and Applications

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“…Solution of cyclotomy of order e, does not require to determine all the cyclotomic numbers of order e [3]. As illustrated in [3], cyclotomic numbers are divided into various classes.…”

Section: Algorithms For Equality Of Cyclotomic Numbersmentioning

confidence: 99%

Computation of Jacobi sums of order l^2 and 2l^2 with prime l

Helal¹,

Tanti²,

Pushp³

2019

Preprint

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In this article, we present fast computational algorithms for Jacobi sums of orders l 2 and 2l 2 with odd prime l. Cyclotomic numbers play a crucial rule in implementing these algorithms. To increase efficiency, we present explicit expressions for Jacobi sums of orders l 2 and 2l 2 in terms of the minimum number of cyclotomic numbers of order l 2 and 2l 2 respectively, which will be utilized in implementing the algorithms. To validate these expressions, we implement two additional algorithms which demonstrate the minimality of cyclotomic numbers required for the determination of all Jacobi sums of order l 2 and 2l 2 respectively.

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