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Orders of Gauss periods in finite fields
[chapter]

1995
*
Lecture Notes in Computer Science
*

This can be considered as

doi:10.1007/bfb0015425
fatcat:fducnezrtffvpkzg3h5bwbdyoy
*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). ...##
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Short effective intervals containing primes

2003
*
Journal of Number Theory
*

We prove that every interval xð1

doi:10.1016/s0022-314x(02)00029-x
fatcat:wh42s52olfbkbdxs5fpweq772y
*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. ...##
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Upper bounds for least witnesses and generating sets

1997
*
Acta Arithmetica
*

In this section various results concerning G(n)

doi:10.4064/aa-80-4-311-326
fatcat:w5wosj7cfre5jkchwypcdez4zy
*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. ...##
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On Distinguishing Prime Numbers from Composite Numbers

1983
*
Annals of Mathematics
*

The problem of distinguishing

doi:10.2307/2006975
fatcat:nk7d7fitvfcfrnmutck5cgqst4
*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. ...##
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It is easy to determine whether a given integer is prime

2004
*
Bulletin of the American Mathematical Society
*

The problem of distinguishing

doi:10.1090/s0273-0979-04-01037-7
fatcat:usykvuq3jbcb7lfe3ioqk6nsci
*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. ...##
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Smarandache Function Journal, 9
[article]

1998
*
Zenodo
*

This journal is yearly published (in the Spring or Fall) in

doi:10.5281/zenodo.8956
fatcat:7zwof2krgrhlvd4mqd2d4sbodm
*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. ...##
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Smarandache Function Journal, 12
[article]

2001
*
Zenodo
*

This journal is yearly published (in the Spring or Fall) in

doi:10.5281/zenodo.8965
fatcat:kvjt6qzblrhehoutm7n27hrnna
*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. ...##
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On Prime-Order Elliptic Curves with Embedding Degrees k = 3, 4, and 6
[chapter]

2008
*
Lecture Notes in Computer Science
*

Therefore, we will focus on ordinary elliptic curves because of efficiency

doi:10.1007/978-3-540-79456-1_6
fatcat:kqlc7atwpng2flveyfe23ew56q
*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. ...##
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Parallel ART for image reconstruction in CT using processor arrays

2006
*
International Journal of Parallel, Emergent and Distributed Systems
*

Reconstructing an image of n pixels from Q(n) equations can be done on

doi:10.1080/17445760600604157
fatcat:dghchbkbinf7tgtj5cdskpbvve
*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. ...##
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Mathematical foundations of modern cryptography: computational complexity perspective
[article]

2002
*
arXiv
*
pre-print

This new approach has led to

arXiv:cs/0212055v1
fatcat:3ps4ukwnwrat7avo5tw7ifksue
*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 ...##
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Topics In Normal Bases Of Finite Fields
[article]

2013
*
arXiv
*
pre-print

The first few chapters cover

arXiv:1304.0420v1
fatcat:wl7zzhud4naonc3cnehlau3aea
*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. ...##
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A Taxonomy of Pairing-Friendly Elliptic Curves

2009
*
Journal of Cryptology
*

Elliptic curves with small embedding degree

doi:10.1007/s00145-009-9048-z
fatcat:gwawq4j23refpe74ucw4oqjfsy
*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 ...##
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Algorithms for Polynomial Real Root Isolation
[chapter]

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. ...

##
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Reviews and Descriptions of Tables and Books

1960
*
Mathematics of Computation
*

Let P be the fraction of

doi:10.1090/s0025-5718-60-99227-9
fatcat:4iun3aqcd5ep5fm32pfbbzk52q
*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 ...##
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Abstracts of papers

1937
*
Bulletin of the American Mathematical Society
*

ABSTRACTS OF PAPERS [March, dimension 2

doi:10.1090/s0002-9904-1937-06496-2
fatcat:infbd2hiy5gg5iw5vjdtocgupu
*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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