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### Orders of Gauss periods in finite fields [chapter]

Joachim von zur Gathen, Igor Shparlinski
1995 Lecture Notes in Computer Science
This can be considered as a step towards solving the celebrated problem of nding primitive roots in nite elds in polynomial time.  ...  The authors would like to thank Francesco Pappalardi for a useful discussion of the e ective computability issues related to Artin's conjecture.  ...  The rst version of this paper appeared in the Proceeding of the 6th International Symposium on Algorithms and Computation, Cairns, 1995 (Lect. Notes in Comp. Sci., 1995, vol. 1004, pp. 208{215).  ...

### Short effective intervals containing primes

Olivier Ramaré, Yannick Saouter
2003 Journal of Number Theory
We prove that every interval xð1 À D À1 Þ; x contains a prime number with D ¼ 28 314 000 and provided xX10 726 905 041: The proof combines analytical, sieve and algorithmical methods. r  ...  and Carl Pomerance for helpful discussions on this subject, and for their hospitality.  ...  This last thanks extends to the Institute for Advanced Study where a good part of this work was completed. Thanks are also due to the referee for his careful reading of the paper.  ...

### Upper bounds for least witnesses and generating sets

Ronald Burthe
1997 Acta Arithmetica
In this section various results concerning G(n) and w(n) are proved, including connections with Dirichlet characters mod n.  ...  Let r be a prime factor of m and say that r b q − 1. Then r b p − 1. As above there exists an rth power non-residue of q, say a, with a < C ε q 1/(4 √ e)+ε .  ...  We now show a connection between G(n) and primality tests. Let n be a positive odd number greater than 1 with n − 1 = 2 s t where t is odd.  ...

### On Distinguishing Prime Numbers from Composite Numbers

Leonard M. Adleman, Carl Pomerance, Robert S. Rumely
1983 Annals of Mathematics
The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their  ...  Acknowledgements: I would like to thank Matt Baker, Dan Bernstein, Michael Guy, Eric Pine, François Dorais, Felipe Voloch, the referee and particularly Manindra Agrawal and Carl Pomerance for reading a  ...  I am also indebted to Dan Bernstein's fine website http://cr.yp.to/primetests.html, which contains a wealth of information and ideas.  ...

### It is easy to determine whether a given integer is prime

Andrew Granville
2004 Bulletin of the American Mathematical Society
The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their  ...  Acknowledgements: I would like to thank Matt Baker, Dan Bernstein, Michael Guy, Eric Pine, François Dorais, Felipe Voloch, the referee and particularly Manindra Agrawal and Carl Pomerance for reading a  ...  I am also indebted to Dan Bernstein's fine website http://cr.yp.to/primetests.html, which contains a wealth of information and ideas.  ...

### Smarandache Function Journal, 9 [article]

Sabin Tabirca, Tatiana Tabirca
1998 Zenodo
This journal is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies.  ...  SNJ is a referred journal: reviewed, indexed, cited, concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc.  ...  Acknowledgements I would like to thank Henry Ibstedt for his support according me to improve the quality of this article and for the numerical computation that he provided me.  ...

### Smarandache Function Journal, 12 [article]

Sabin Tabirca, Tatiana Tabirca
2001 Zenodo
This journal is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies.  ...  SNJ is a referred journal: reviewed, indexed, cited, concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc.  ...  For example, Fleuren used Elliptic Curve Primality Proving or Adleman-Promerance-Rumely tests obtaining all prime factors up to 20 digits.  ...

### On Prime-Order Elliptic Curves with Embedding Degrees k = 3, 4, and 6 [chapter]

Koray Karabina, Edlyn Teske
2008 Lecture Notes in Computer Science
Therefore, we will focus on ordinary elliptic curves because of efficiency and security reasons. Now, let #E(F q ) = n = hr where r is a large prime factor of n and gcd(q, r) = 1.  ...  Let n be a prime and let q be an integer such that n | φ k (q) and n ∤ k. Then n ∤ q d − 1 for each 1 ≤ d ≤ k − 1. Proof.  ...  Acknowledgements I would like to thank my supervisor Edlyn Teske for her invaluable comments and help which motivated me during this work.  ...

### Parallel ART for image reconstruction in CT using processor arrays

Dan Gordon
2006 International Journal of Parallel, Emergent and Distributed Systems
Reconstructing an image of n pixels from Q(n) equations can be done on a linear array of p ¼ Oð ffiffi ffi n p Þ processors with optimal efficiency (linear speedup) and O(n/p) memory for each processor  ...  The main advantages of ART are its robustness, its cyclic convergence on inconsistent systems, and its relatively good initial convergence.  ...  Acknowledgements The author is indebted to the anonymous reviewers whose comments led to a much-improved presentation.  ...

### Mathematical foundations of modern cryptography: computational complexity perspective [article]

Shafi Goldwasser
2002 arXiv   pre-print
This new approach has led to a sequence of advances, in setting and solving new mathematical challenges as well as in harnessing discrete mathematics to the task of solving real-world problems.  ...  The approach has been to consider classical problems through the prism of computational complexity, where the number of basic computational steps taken to solve a problem is the crucial qualitative parameter  ...  Moreover, as the primes are abundant by the prime number theorem (≈ 1 k for k-bit primes) and there exist probabilistic expected polynomial time algorithms for primality testing [30, 3] , it is easy to  ...

### Topics In Normal Bases Of Finite Fields [article]

N. A. Carella
2013 arXiv   pre-print
The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given.  ...  The last few chapters cover the theory of guassian and period normal bases of finite fields of low degrees.  ...  In other simple cases the polynomial x n − 1 has only a few irreducible factors. Another simple period normal basis test is covered below.  ...

### A Taxonomy of Pairing-Friendly Elliptic Curves

David Freeman, Michael Scott, Edlyn Teske
2009 Journal of Cryptology
Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairingbased cryptographic systems.  ...  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.  ...  The CM method, originally devised for use in primality testing, constructs a curve with endomorphism ring isomorphic to a given order O in a quadratic imaginary field Q( √ −D), and can be used to construct  ...

### Algorithms for Polynomial Real Root Isolation [chapter]

J. R. Johnson
1998 Texts and Monographs in Symbolic Computation
A F P R ID S also uses the fast relative primality test A F P R P to first check if A (i) and A'(x) are relatively prime before computing their gcd.  ...  Past Test for R elatively Prim e Polynom ials Frequently two polynomials are relatively prime.  ...  For a € (-4, -3), this polynomial is contained in the interval polynomial roots th at started at -1 and -2 move together and eventually become a multiple root when 6 = 0.00967.  ...

### Reviews and Descriptions of Tables and Books

1960 Mathematics of Computation
Let P be the fraction of a population having a continuous but unknown distribution function that lies between the r-th smallest and the s-th largest values in a random sample of n drawn from that population  ...  A test criterion is obtained by dividing the sequence between the Rth and the (R + l)st event, where ri + r2 = 2R or 2R + 1, and then assigning ranks 1, 2, • • • to the events by order of position beginning  ...

### Abstracts of papers

1937 Bulletin of the American Mathematical Society
ABSTRACTS OF PAPERS [March, dimension 2 and are analytic in terms of admissible local coordinates, the critical points off, g form a 1-dimensional complex whose 1-cells are simple analytic arcs.  ...  Part of the results (for w = 2, but not for n>2) probably could be obtained from a recent work of  ...  Let A -A\Ao be any positive integer, where Ao is prime to A(D) and every prime factor of Ai divides A(D).  ...
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