A symmetric diophantine system concerning fifth powers

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

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A Public Key Cryptosystem Using Cyclotomic Matrices

10.5772/intechopen.101105 ◽

2021 ◽

Author(s):

Md. Helal Ahmed ◽

Jagmohan Tanti ◽

Sumant Pushp

Keyword(s):

Computational Complexity ◽

Communication Technology ◽

Public Key ◽

Public Key Cryptosystem ◽

Privileged Information ◽

Diophantine System ◽

Encryption And Decryption ◽

Information And Communication ◽

Asymptotic Complexity ◽

One Way Function

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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A Diophantine system

Number Theory and Algebraic Geometry ◽

10.1017/cbo9780511734946.005 ◽

2004 ◽

pp. 63-68

Author(s):

Andrew Bremner

Keyword(s):

Diophantine System

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EULER'S CRITERION FOR SEPTIC NONRESIDUES

International Journal of Number Theory ◽

10.1142/s1793042110003563 ◽

2010 ◽

Vol 06 (06) ◽

pp. 1329-1347 ◽

Cited By ~ 1

Author(s):

JAGMOHAN TANTI ◽

S. A. KATRE

Keyword(s):

Explicit Expression ◽

Root Of Unity ◽

Jacobi Sum ◽

Diophantine System ◽

Mod P

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity ( mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D ( mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

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