Abstract

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn=FqX/Xn, where n≥1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves.

Highlights

  • Elliptic curves are especially important in number theory and constitute a major area of current research; for example, they were used in Andrew Wiles’s proof of Fermat’s Last Theorem

  • The history of cryptography has long been the history of secret codes and along all previous times, this has affected the fate of men and nations [4]

  • Cryptography used elliptic curves for more than 40 years the appearance of the DiffieHellman key exchange protocol and the ElGamal cryptogram [13–15]. These cryptographic protocols use in particular group structures, for by applying these methods to groups defined by elliptic curves, a new speciality was born at the end of the 1980: elliptic curve cryptography (ECC), Elliptic Curve Cryptography

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Summary

Introduction

Elliptic curves are especially important in number theory and constitute a major area of current research; for example, they were used in Andrew Wiles’s proof of Fermat’s Last Theorem. They find applications in elliptic curve cryptography (ECC), integer factorization, classical mechanics in the description of the movement of spinning tops, to produce efficient codes. For these reasons, the subject is well known, presented, and worth exploring. Let Rn 1⁄4 q1⁄2XŠ=ðXnÞ be a q-algebra of dimension n, with ð1, ε, ..., εnÀ1Þ as a qbasis, where ε 1⁄4 X, εn 1⁄4 0, q is the finite field of order q 1⁄4 pr, and p being a prime integer [27–29]

Internal laws in Rn
Primitive triples
The projective plane on a finite ring
A affine Weierstrass equation on Rn is an equation of the form
Projective coordinates and group law
The morphisms πk et θk
Main applications
Applications
Elliptical Diffie-Hellman cryptosystem
Coding example
Coding of elements of G
Conclusion

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