. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter

Generalized Mersenne Numbers in Pairing-Based Cryptography

Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85

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The Weil Pairing, first introduced by Andre Weil in 1940, plays an important role in the theoretical study of the arithmetic of elliptic curves and Abelian varieties. It has also recently become extremely useful in cryptologic constructions related to those objects. This paper gives the definition of the Weil Pairing, describes efficient algorithms to calculate it, gives two applications, and describes the motivation to considering it.

/pdf/the-weil-pairing-and-its-efficient-calculation-221mqsj27f.pdf

The Weil Pairing, and Its Efficient Calculation

The Tate pairing has found several new applications in cryptography. This paper provides methods to quickly compute the Tate pairing, and hence enables efficient implementation of these cryptosystems. We also give division-free formulae for point tripling on a family of elliptic curves in characteristic three. Examples of the running time for these methods are given.

http://www.hpl.hp.com/techreports/2002/HPL-2002-23.pdf

Implementing the Tate Pairing

A Few Preliminaries. Mathematics and Proofs.Sets and Relations.Mathematical Induction.Complex and Matrix Algebra. 1. Groups and Subgroups. Binary Operations.Finite-State Machines (Automata).Isomorphic Binary Structures.Groups.Subgroups.Cyclic Groups and Generators.Cayley Digraphs. 2. More Groups and Cosets. Groups of Permutations.Automata.Orbits, Cycles, and the Alternating Groups.Plane Isometries.Cosets and the Theorem of Lagrange.Direct Products and Finitely Generated Abelian Groups.Periodic Functions, Plane Isometries.Binary Linear Codes. 3. Homomorphisms and Factor Groups. Homomorphisms.Factor Groups.Factor-group Computations and Simple Groups.Series of Groups.Group Action on a Set.Applications of G-sets to Counting. 4. Advanced Group Theory. Isomorphism Theorems Proof of the Jordan-Holder Theorem.Sylow Theorem.Applications of the Sylow Theory.Free Abelian Groups.Free Groups.Group Presentations. 5. Introduction to Rings and Fields. Rings and Fields.Integral Domains.Fermat's and Euler's Theorems.The Field of Quotients of an Integral Domain.Rings of Polynomials.Factorization of Polynomials over a Field.Noncommutative Examples.Ordered Rings and Fields. 6. Factor Rings and Ideals. Homomorphisms and Factor Rings.Prime and Maximal Ideals.Grobner Bases for Ideals. 7. Factorization*. Unique Factorization Domains.Euclidean Domains.Gaussian Integers and Norms. 8. Extension Fields. Introduction to Extension Fields.Vector Spaces.Algebraic Extensions.Geometric Constructions.Finite Fields.Additional Algebraic Structures. 9. Automorphisms and Galois Theory. Automorphisms of Fields.The Isomorphism Extension Theorem.Splitting Fields.Separable Extensions.Totally Inseparable Extensions.Galois Theory.Illustrations of Galois Theory.Cyclotomic Extensions.Insolvability of the Quintic. Bibliography. Notations. Answers To Odd-Numbered Exercises Not Requiring Proofs. Index. *Not required for the remainder of the text.

A first course in abstract algebra

Montgomery multiplication methods constitute the core of modular exponentiation, the most popular operation for encrypting and signing digital data in public-key cryptography. In this article, we study the operations involved in computing the Montgomery product, describe several high-speed, space-efficient algorithms for computing MonPro(a, b), and analyze their time and space requirements. Our focus is to collect several alternatives for Montgomery multiplication, three of which are new. However, we do not compare the Montgomery techniques to other modular multiplication approaches.

/pdf/analyzing-and-comparing-montgomery-multiplication-algorithms-2v3u095cel.pdf

Analyzing and comparing Montgomery multiplication algorithms

This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

/pdf/a-computational-introduction-to-number-theory-and-algebra-4hic91lvvh.pdf

Generalized Mersenne Numbers in Pairing-Based Cryptography

Citations

Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85

References

The Weil Pairing, and Its Efficient Calculation

Implementing the Tate Pairing

A first course in abstract algebra

Analyzing and comparing Montgomery multiplication algorithms

A Computational Introduction to Number Theory and Algebra

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On a generalized sum of the Mersenne Numbers

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