Fast algorithms under the extended Riemann hypothesis: A
concrete estimate
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Annual ACM Symposium on Theory of Computing archive
Proceedings of the fourteenth annual ACM symposium on
Theory of computing table of contents
San Francisco, California, United States
Pages: 290 - 295
Year of Publication: 1982
ISBN:0-89791-070-2
Author
Eric Bach
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SIGACT: ACM Special Interest Group on Algorithms and
Computation Theory
Publisher ACM Press
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ABSTRACT
Several results in theoretical computer science use the following
theorem: For a positive integer q, let Z-&-bull;q denote the
multiplicative group of all integers x, 0-&-lt;x-&-lt;q, that are relatively
prime to q. Let G be a proper subgroup of Z-&-bull;q. Then, assuming
the Extended Riemann Hypothesis, there is a constant C such that if q
is sufficiently large, Z-&-bull;q-&-minus;G contains a positive integer
N-&-le;C (logeq)2. We show that for q-&-ge;106, one may take C-&equil;60. As an application, we discuss a deterministic polynomial-time
primality test. Miller proved that such algorithms must exist if the ERH
is true, but we are unable to specify one without the concrete
information given above. We eliminate this difficulty, and show how to
implement a fast primality test.
REFERENCES
Note: OCR errors may be found in this Reference List extracted from
the full text article. ACM has opted to expose the complete List rather
than only correct and linked references.
1
Lars Ahlfors, Complex Analysis, New York: McGraw-Hill (1966).
2
Nesmith Ankeny, The Least Quadratic Non Residue, Annals of
Mathematics 55, pp. 65-72 (1952).
3 Andr-&-eacute; Blanchard, Initiation -&-agrave; la Th-&-eacute;orie
Analytique des Nombres Premiers, Paris: Dunod (1957).
4 Harold Davenport, Multiplicative Number Theory, Berlin: Springer
(1980).
5 Albert Ingham, The Distribution of Prime Numbers, New York:
Stechert-Hafner (1964).
6 Eugene Jahnke, Fritz Emde, and Friedrich L-&-ouml;sch, Tables of
Higher Functions, New York: McGraw-Hill (1960).
7 Gary Miller, Riemann's Hypothesis and Tests for Primality, Journal of
Computer and System Sciences 13, pp. 300-317 (1976).
8 Hugh Montgomery, Topics in Multiplicative Number Theory, Berlin:
Springer (1971).
9
Karl Prachar, Primzahlverteilung, Berlin: Springer (1957).
10 Hans Rademacher, On the Phragm-&-eacute;n-Lindel-&-ouml;f
Theorem and Some Applications, Mathematische Zeitschrift 72, pp.
192-204 (1959).
11 Barkley Rosser and Lowell Schoenfield, Approximate Formulas for
Some Functions of Prime Numbers, Illinois Journal of Mathematics 6,
pp. 64-94 (1962).
12 Edward Titchmarsh, The Theory of Functions, Oxford: Oxford
(1939).
13 Jacques V-&-eacute;lu, Tests for Primality under the Riemann
Hypothesis, SIGACT News 10, pp. 58-59 (1978).
14
Peter Weinberger, personal communication (1981).
CITINGS 4
L. Ronyai, Simple algebras are difficult, Proceedings of the nineteenth annual ACM
conference on Theory of computing, p.398-408, January 1987, New York, New York,
United States
Manuel Blum, How to exchange (secret) keys, Proceedings of the fifteenth annual
ACM symposium on Theory of computing, p.440-447, December 1983
Manuel Blum, How to exchange (secret) keys, ACM Transactions on Computer
Systems (TOCS), v.1 n.2, p.175-193, May 1983
M-D A Huang, Riemann hypothesis and finding roots over finite fields, Proceedings of
the seventeenth annual ACM symposium on Theory of computing, p.121-130, May
06-08, 1985, Providence, Rhode Island, United States
Collaborative Colleagues:
Eric Bach: Andris Ambainis
Joan Boyar
Jin-Yi Cai
Jin-yi Cai
Anne Condon
James Driscoll
Leah Epstein
Lene M. Favrholdt
Elton Glaser
Tao Jiang
Marcos Kiwi
Kim S. Larsen
Gary Lewandowski
Guo Lin
Guo-Hui Lin
Richard Lukes
Gary Miller
Ashwin Nayak
René Peralta
Jeffrey Shallit
Victor Shoup
Jonathan Sorenson
Celena Tanguay
Ashvin Vishwanath
John Watrous
John Harrison Watrous
H. C. Williams
Marty Joseph Wolf
Rob van Stee
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http://www.encyclopedia4u.com/g/generalized-riemann-hypothesis.html
Generalized Riemann
hypothesis
The Riemann hypothesis is one of the most important conjectures in
mathematics. It is a statement about the zeros of the Riemann zeta function.
Various geometrical and arithmetical objects can be described by so-called global
L-functions, which are formally similar to the Riemann zeta function. One can
then ask the same question about the zeros of these L-functions, yielding various
generalizations of the Riemann hypothesis. None of these conjectures have been
proven or disproven, but many mathematicians believe them to be true.
Global L-functions can be associated to elliptic curves, number fields (in which
case they are called Dedekind zeta functions), Maass waveforms, and Dirichlet
characters (in which case they are called Dirichlet L-functions). When the
Riemann hypothesis is formulated for Dedekind zeta functions, it is known as the
extended Riemann hypothesis and when it is formulated for Dirichlet Lfunctions, it is known as the generalized Riemann hypothesis. These two
statements will be discussed in more detail below.
Table of contents
1 Generalized Riemann Hypothesis (GRH)
1.1 Consequences of GRH
2 Extended Riemann Hypothesis (ERH)
Generalized Riemann Hypothesis (GRH)
The generalized Riemann hypothesis was probably formulated for the first time by
Piltz in 1884. Like the original Riemann hypothesis, it has far reaching
consequences about the distribution of prime numbers.
The formal statement of the hypothesis follows. A Dirichlet character is a
completely multiplicative arithmetic function χ such that there exists a positive
integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If
such a character is given, we define the corresponding Dirichlet L-function by
for every complex number s with real part > 1. By analytic continuation, this
function can be extended to a meromorphic function defined on the whole
complex plane. The generalized Riemann hypothesis asserts that for every
Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part
of s is between 0 and 1, then it is actually 1/2.
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
Consequences of GRH
An arithmetic progression in the natural numbers is a set of numbers of the form
a, a+d, a+2d, a+3d, ... where a and d are natural numbers and d is non-zero.
Dirichlet's theorem states that if a and d are coprime, then such an arithmetic
progression contains infinitely many prime numbers. Let π(x,a,d) denote the
number of prime numbers in this progression which are less than or equal to x. If
the generalized Riemann hypothesis is true, then for every a and d and for every
ε>0
where φ(d) denotes Euler's phi function and O is the Landau symbol. This is a
considerable strengthening of the prime number theorem.
If GRH is true, then for every prime p there exists a primitive root modulo p (a
generator of the multiplicative group of integers modulo p) which is less than 70
(ln(p))2; this is often used in proofs.
Goldbach's weak conjecture also follows from the generalized Riemann
hypothesis.
If GRH is true, then the Miller-Rabin primality test is guaranteed to run in
polynomial time. (A polynomial-time primality test which doesn't require GRH has
recently been published; see prime number.)
Extended Riemann Hypothesis (ERH)
Suppose K is a number field (a finite-dimensional field extension of the rationals
Q) with ring of integers OK (this ring is the integral closure of the integers Z in K).
If a is an integral ideal of OK, we denote its norm with Na. The Dedekind zeta
function of K is then defined by
for every complex number s with real part > 1. The sum extends over all integral
ideals a of OK.
The Dedekind zeta function satisfies a functional equation and can be extended
by analytic continuation to the whole complex plane. The resulting function
encodes important information about the number field K. The extended Riemann
hypothesis asserts that for every number field K and every complex number s
with ζK(s) = 0: if the real part of s is between 0 and 1, then it is in fact 1/2.
The ordinary Riemann hypothesis follows from the extended one if one takes the
number field to be Q, with ring of integers Z.
http://citeseer.nj.nec.com/cs?q=Extended+Riemann+Hypothesis&submit=Search+Do
cuments&cs=1
Searching for PHRASE extended riemann hypothesis.
Restrict to: Header Title Order by: Citations Hubs Usage Date Try:
Amazon B&N Google (RI) Google (Web) CSB DBLP
56 documents found. Order: citations weighted by year.
PRIMES is in P - Agrawal, Kayal, Saxena (2002) (Correct) (2 citations)
algorithm for primality testing assuming Extended Riemann Hypothesis (ERH)His
test was modified by Rabin
www.cse.iitk.ac.in/news/primality.pdf
A General Framework for Subexponential Discrete Logarithm.. - Enge, Gaudry
(2000) (Correct) (4 citations)
L q g (p 2 o(1)Assuming the extended Riemann hypothesis, there exists a probabilistic
www.math.uni-augsburg.de/~enge/vorabdrucke/subexp.ps.gz
Practical Zero-Knowledge Proofs: Giving Hints and Using.. - Boyar, Friedl, Lund
(1994) (Correct) (15 citations)
proof is efficient, assuming the Extended Riemann Hypothesis. We also give practical
zero-knowledge
ftp.cs.uchicago.edu/pub/publications/tech-reports/TR-88-22.ps
Fast Generation of Prime Numbers and Secure Public-Key.. - Maurer (1994)
(Correct) (13 citations)
a proof would follow from the unproven extended Riemann hypothesis (see
[62]Alford, Granville and
ftp.inf.ethz.ch/pub/publications/papers/ti/isc/Prime_Generation.ps.gz
New Algorithms for Finding Irreducible Polynomials over Finite.. - Shoup (1990)
(Correct) (26 citations)
runs in polynomial time assuming the Extended Riemann Hypothesis (ERH)They also
give a deterministic
www.shoup.net/papers/detirred.ps.Z
Searching for Primitive Roots in Finite Fields - Shoup (1992) (Correct) (17
citations)
this problem under the assumption of the Extended Riemann Hypothesis (ERH) for
the case where p is large
www.cs.wisc.edu/~shoup/papers/primroots.ps.Z
Elliptic Curve Normalization - Ciet, al. (2001) (Correct) (1 citation)
choice of B depends conditionally on the extended Riemann hypothesis and would
give B =log 6 p. This
www.dice.ucl.ac.be./crypto/tech_reports/CG2001_2.ps.gz
Applying Sieving To The Computation Of Quadratic Class Groups - Jacobson, Jr.
(1999) (Correct) (3 citations)
( p 2) under the assumption of the Extended Riemann Hypothesis (ERH)where L
\Delta (fi) exp i
ftp.informatik.tu-darmstadt.de/pub/TI/TR/TI-97-19.MPQS_QO.ps.gz
Fast Construction of Irreducible Polynomials over Finite Fields - Shoup (1993)
(Correct) (10 citations)
algorithm for this problem, unless the Extended Riemann Hypothesis is true
(Adleman &Lenstra 1986,
www.cs.wisc.edu/~shoup/papers/fastirred.ps.Z
On the complexity of intersecting finite state automata - Karakostas, Lipton, Viglas
(2000) (Correct) (1 citation)
runs in time 2 1 4 n With the Extended Riemann Hypothesis this bound only improves
to 2 1 5 n
www.cs.princeton.edu/~viglas/pub/fsa-inter.ps
Sorting out zero-knowledge - Brassard, CREPEAU (1990) (Correct) (5 citations)
or even physical nature, such as the Extended Riemann Hypothesis or the principles
of quantum physics
ftp.cs.mcgill.ca/pub/theorique/papers/crepeau/GZIP/BC90.ps.gz
Old and new deterministic factoring algorithms - McKee, Pinch (1996) (Correct) (2
citations)
given in [Sha]and if one assumes the extended Riemann Hypothesis, then there is no
difficulty (Sch]
ftp.dpmms.cam.ac.uk/pub/rgep/Papers/p52x.ps
An Investigation of Bounds for the Regulator of.. - Jacobson, Jr., Lukes.. (1995)
(Correct) (2 citations)
numerical experiments, involving the Extended Riemann Hypothesis and the CohenLenstra class number
www.expmath.com/restricted/4/4.3/jacobson.ps.gz
Computational Techniques in Quadratic Fields - Jacobson, Jr. (1995) (Correct) (2
citations)
and is conditional on the truth of the Extended Riemann Hypothesis. Our algorithm
makes use of some
www.informatik.th-darmstadt.de/TI/Mitarbeiter/jacobs/PS/jacobs.mthesis.ps.gz
Results And Estimates On Pseudopowers - Bach, Lukes, Shallit, Williams (1996)
(Correct) (2 citations)
It is possible to show, assuming the Extended Riemann Hypothesis (ERH)that the
least x-pseudosquare
math.uwaterloo.ca/~shallit/Papers/pseudo.ps
Constructing normal bases in finite fields - Gathen, Giesbrecht (1990) (Correct) (4
citations)
polynomial time assuming the Extended Riemann Hypothesis (ERH)or
deterministically in time
ftp.csd.uwo.ca/pub/mwg/norr.ps.Z
Open Problems in Number Theoretic Complexity, II - Adleman, McCurley (Correct)
(4 citations)
set of rationals. ERH refers to the extended Riemann hypothesis. For a b 2 Z, we write
a j b if
www.cs.sandia.gov/pub/papers/mccurley/open.ps
Constructing Nonresidues in Finite Fields and the Extended.. - Buchmann, Shoup
(1991) (Correct) (2 citations)
Nonresidues in Finite Fields and the Extended Riemann Hypothesis February 1, 1990
Johannes Buchmann
ftp.informatik.tu-darmstadt.de/pub/TI/reports/nonresidues.ps.gz
Specific Irreducible Polynomials with Linearly Independent.. - December Ian Blake
(Correct)
are known [11, 13, 24]Also if the extended Riemann hypothesis is true, then one can
construct in
www.math.clemson.edu/faculty/Gao/papers/BGM97.pdf
Attack on A New Public Key Cryptosystem from ISC'02 (LNCS 2433) - Zhang, Liu,
Kim (Correct)
polynomial over GF (p)under the extended Riemann hypothesis (ERH)it can be
factored
eprint.iacr.org/2002/178.ps.gz
New computations concerning the Cohen-Lenstra heuristics - Williams, Riele
(Correct)
We will also assume the truth of the extended Riemann hypothesis (ERH) for L(s,
Xp)Broadly speaking
www.cwi.nl/ftp/CWIreports/MAS/MAS-R0215.ps.Z
Towards a deterministic polynomial-time Primality Test - Kayal, Saxena (2002)
(Correct)
on a widely believed conjecture, the Extended Riemann Hypothesis. We also show
that any n which is
www.cse.iitk.ac.in/research/btp2002/primality.ps.gz
A Remark on Plotkin's Bound - Warwick De Launey (Correct)
sequence. This suggests assuming the Extended Riemann Hypothesis (ERH)and
seeing what can be proved.
www.ccrwest.org/gordon/plotkin.ps
Deciding Properties of Polynomials without Factoring - Sander, Shokrollahi (1997)
(Correct)
equations. Assuming the validity of the Extended Riemann Hypothesis, our
algorithms run in time O(n 6
www.star-lab.com/sander/publications/body2.ps
Gauß Periods in Finite Fields - Gathen, Shparlinski (Correct)
x :Hooley (1967) proved this under the Extended Riemann Hypothesis, and also
determined c(a) explicitly.
www.uni-paderborn.de/sfb376/projects/a4/fq5-gatshp00.ps
Primality Testing, Integer Factorization, and Discrete Logarithms - Garefalakis (2000)
(Correct)
widely believed) conjectures, namely the Extended Riemann Hypothesis (ERH)In
both cases those algorithms
www.cs.utoronto.ca/~theo/report.ps
An Upper Bound on the Least Inert Prime in a Real Quadratic .. - Andrew Granville
Mollin (Correct)
conjecture under the assumption of the Extended Riemann Hypothesis (ERH)as
follows. Theorem 3 of Bach
www.math.uga.edu/~andrew/Postscript/HCW4.ps
Some Arithmetic Properties Of Shanks's Generalized Euler And.. - Teske, Williams
(Correct)
very significant role in a proof of the Extended Riemann Hypothesis that he felt might
be produced from a
cacr.math.uwaterloo.ca/~eteske/teske/gen_classno.ps
Short Proofs for Nondivisibility of Sparse Polynomials .. - Grigoriev, Karpinski, ..
(1991) (Correct)
of Sparse Polynomials under the Extended Riemann Hypothesis Dima Grigoriev 1
Marek Karpinski 2
ftp.icsi.berkeley.edu/pub/techreports/1991/tr-91-013.ps.gz
New Quadratic Polynomials With High Densities Of Prime Values - Jacobson, Jr.,
Williams (Correct)
values of jj up to 70 digits (under the Extended Riemann Hypothesis |ERH) and to
provide some new values
www.cacr.math.uwaterloo.ca/~mjjacobs/jacobs/PS/PrimePoly.ps
A Problem Concerning a Character Sum - Teske, Williams (1999) (Correct)
of complexity O(jdj 1=5under the Extended Riemann Hypothesis (ERH)This was a
considerable
cacr.math.uwaterloo.ca/~eteske/teske/character_sums.ps
.1 Primality testing cont'd. - The Miller-Rabin Primality (Correct)
number theoretic conjecture known as the extended Riemann hypothesis, ERH. See
[5] for details.Claim 2
www.nada.kth.se/theory/aalg/lecturenotes/lecture2/lecture2.ps
Constructing Elements Of Large Order In Finite Fields - Gathen, Shparlinski (1999)
(Correct)
this is the best known result. Even the Extended Riemann Hypothesis (ERH) does not
imply any essentially
www.comp.mq.edu.au/~igor/GaussPer-2.ps
On The Computational Hardness Of Testing Square-Freeness.. - Karpinski,
Shparlinski (1999) (Correct)
recently been proved [11] that under the Extended Riemann Hypothesis this problem
belongs to the class
www.comp.mq.edu.au/~igor/SprPol-SqrFree.ps
Computing Jacobi Symbols Modulo Sparse Integers And Polynomials .. - Shparlinski
(Correct)
type is a perfect square (assuming the Extended Riemann Hypothesis)We also obtain
analogues of these
www.comp.mq.edu.au/~igor/SparseArithm.ps
Orders of Gauß Periods in Finite Fields - Gathen, Shparlinski (1998) (Correct)
stage, especially if one assumes the Extended Riemann Hypothesis (ERH)see Chapter
3 of [21]
www.comp.mq.edu.au/~igor/GaussPer-1.ps
Finding normal integral bases of cyclic number fields of.. - Acciaro, Fieker (1999)
(Correct)
group of L over Q. If one assumes the Extended Riemann Hypothesis, then it is
possible to compute in
www.math.tu-berlin.de/~kant/publications/papers/normal.ps.gz
Short Representation of Quadratic Integers - Buchmann, Thiel, Williams (1992)
(Correct)
big height. For example, assuming the extended Riemann hypothesis and that \Delta is
sufficiently large,
ftp.informatik.tu-darmstadt.de/pub/TI/reports/short_rep.ps.gz
Deciding Properties of Polynomials without Factoring - Sander, Shokrollahi (1997)
(Correct)
equations. Assuming the validity of the Extended Riemann Hypothesis, our
algorithms run in time O(n 6
netlib.lucent.com/cm/ms/who/amin/publication/NT/FOCS97.ps
On the deterministic complexity of factoring polynomials - Gao (1999) (Correct)
over finite fields assuming the extended Riemann hypothesis (ERH)By the works of
Berlekamp
www.math.clemson.edu/faculty/Gao/papers/FacPoly.ps.gz
New Quadratic Polynomials With High Densities Of Prime Values - Jacobson, Jr.,
Williams (1999) (Correct)
of j\Deltaj up to 70 digits (under the Extended Riemann Hypothesis -ERH) and to
provide some new
cacr.uwaterloo.ca/techreports/1999/corr99-32.ps
The Size Of The Fundamental Solutions Of Consecutive Pell.. - Jacobson, Jr.,
Williams (1999) (Correct)
is the best possible result under the Extended Riemann Hypothesis. Finally, we
present some numerical
cacr.uwaterloo.ca/techreports/1999/corr99-18.ps
Decision Problems in Quadratic Function Fields of High Genus - Scheidler (1999)
(Correct)
NP and co-NP under the assumption of the extended Riemann hypothesis (5, 6, 9]We
investigate the
www.math.udel.edu/~scheidle/Papers/dp.ps
A Problem Concerning A Character Sum - Teske, Williams (1998) (Correct)
of complexity O(jdj 1=5under the Extended Riemann Hypothesis (ERH)This
represented a considerable
ftp.informatik.th-darmstadt.de/pub/TI/reports/teske.shanks.ps.gz
Orders of Gauß Periods in Finite Fields - Gathen, Shparlinski (1996) (Correct)
stage, especially if one assumes the Extended Riemann Hypothesis (ERH)see Chapter
3 of [16]
math-www.uni-paderborn.de/~aggathen/Publications/orders.ps
Smoothness and Factoring Polynomials over Finite Fields - Shoup (1996) (Correct)
log p) O(1) under the assumption of the Extended Riemann Hypothesis (ERH)The
algorithm we describe is a
www.cs.wisc.edu/~shoup/papers/smooth.ps.Z
Approximate Constructions In Finite Fields - Shparlinski (Correct)
O(1) Hereafter, the ERH denotes the Extended Riemann Hypothesis. 2 Problem P1
First of all we consider
www.comp.mq.edu.au/~igor/ApprConstr.ps
Least primes in arithmetic progressions - Granville (Correct)
[20] showed, under the assumption of the Extended Riemann hypothesis, that p(q) q 2
log q) 4
www.math.uga.edu/~andrew/Postscript/laval.ps
On The Computational Hardness Of Testing Square-Freeness.. - Karpinski,
Shparlinski (1999) (Correct)
recently been proved [11] that under the Extended Riemann Hypothesis this problem
belongs to the class
theory.informatik.uni-bonn.de/~marek/publications/85201-CS.ps.Z
Primality Testing, Integer Factorization, and Discrete Logarithms - Garefalakis (1998)
(Correct)
widely believed) conjectures, namely the Extended Riemann Hypothesis (ERH)In
both cases those algorithms
www.cs.utoronto.ca/~theo/depth.ps.Z
Efficient Algorithms for Computing the Jacobi Symbol.. - Meyer, Sorenson
(Correct)
= Gamma1. Under the assumption of the Extended Riemann Hypothesis (ERH)the
Ankeny-Bach theorem states
abyss.snu.ac.kr/~parkmj/math_paper/kjac.ps
Short Proofs for Nondivisibility of Sparse Polynomials.. - Grigoriev, Karpinski..
(1996) (Correct)
of Sparse Polynomials under the Extended Riemann Hypothesis Dima Yu. Grigoriev
Max Planck
theory.cs.uni-bonn.de/pub/reports/cs-reports/1991/8562-cs.ps.gz
Gary L. Miller: Riemann's Hypothesis and Tests for Primality. STOC 1975: 234-239
Gary L. Miller: Riemann's Hypothesis and Tests for Primality. J. Comput. Syst. Sci.
13(3): 300-317 (1976)
1975
Rūsiņš Freivalds. Models of computation, Riemann Hypothesis, and classical
mathematics. "Lecture Notes in Computer Science ", Springer, 1998, v.1521, p. 89-106.