Pairings for cryptographers

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Such “pairing-friendly” curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature. We also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairing-friendly curves to choose to best satisfy a variety of performance and security requirements.

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A Taxonomy of Pairing-Friendly Elliptic Curves

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

/pdf/the-riemann-zeta-function-3l59kypwcz.pdf

The Riemann zeta function

We find all positive integer solutions ( x , y , a , b , n ) of
 x 2 + 2 a ⋅ 3 b = y n with n ≥ 3 and coprime x and y .

/pdf/on-the-equation-x2-2a-3b-yn-2k1p53d8fb.pdf

On the equation x2+2a⋅3b=yn

For an integer k ≥ 2, we consider the k−generalized Fibonacci sequence (F (k) n )n which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. F. Luca [2] in 2000 and recently D. Marques [3] proved that 55 and 44 are the largest numbers with only one distinct digit (so called repdigits) in the sequences (F (2) n )n and (F (3) n )n, respectively. Further, Marques conjectured that there are no repdigits having at least 2 digits in a k−generalized Fibonacci sequence for any k > 3. In this talk, we report about some arithmetic properties of (F (k) n )n and confirm the conjecture raised by Marques. This is a joint work with Florian Luca.

/pdf/on-a-conjecture-about-repdigits-in-k-generalized-fibonacci-1qouczidii.pdf

On a conjecture about repdigits in k-generalized Fibonacci sequences

The authors assemble a fascinating collection of topics from analytic number theory that provides an introduction to the subject with a very clear and unique focus on the anatomy of integers, that is, on the study of the multiplicative structure of the integers. Some of the most important topics presented are the global and local behavior of arithmetic functions, an extensive study of smooth numbers, the Hardy-Ramanujan and Landau theorems, characters and the Dirichlet theorem, the $abc$ conjecture along with some of its applications, and sieve methods. The book concludes with a whole chapter on the index of composition of an integer. One of this book's best features is the collection of problems at the end of each chapter that have been chosen carefully to reinforce the material. The authors include solutions to the even-numbered problems, making this volume very appropriate for readers who want to test their understanding of the theory presented in the book.

Analytic Number Theory: Exploring the Anatomy of Integers

Fibonacci and Lucas numbers with only one distinct digit.

http://math.univ-lyon1.fr/~adamczew/AdBuLu.pdf

Florian Luca

Papers

On the equation x2+2a⋅3b=yn

On a conjecture about repdigits in k-generalized Fibonacci sequences

Analytic Number Theory: Exploring the Anatomy of Integers

Fibonacci and Lucas numbers with only one distinct digit.

Sur la complexité des nombres algébriques