Article

Free access

Riemann's Hypothesis and tests for primality

Author:

Gary L. MillerAuthors Info & Claims

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

Pages 234 - 239

https://doi.org/10.1145/800116.803773

Published: 05 May 1975 Publication History

PDF eReader

Abstract

The purpose of this paper is to present new upper bounds on the complexity of algorithms for testing the primality of a number. The first upper bound is 0(n^1/7); it improves the previously best known bound of 0(n^1/4) due to Pollard [11].

The second upper bound is dependent on the Extended Riemann Hypothesis (ERH): assuming ERH, we produce an algorithm which tests primality and runs in time 0((log n)⁴) steps. Thus we show that primality is testable in time a polynomial in the length of the binary representation of a number.

Finally, we give a partial solution to the relationship between the complexity of computing the prime factorization of a number, computing the Euler phi function, and computing other related functions.

References

[1]

N.C. Ankeny, "The Least Quadratic Non-Residue," Annals of mathematics 55 (1952) 65-72.

Crossref

Google Scholar

[2]

D.A. Burgess, "The Distribution of Quadratic Residues and Non-Residues," Mathematika 4 (1957) 106-112.

Crossref

Google Scholar

[3]

D.A. Burgess, "On Character Sums and Primitive Roots," Proc. London Math. Soc. (3) 12 (1962) 179-192.

Crossref

Google Scholar

[4]

D.A. Burgess, "On Character Sums and L-series," Proc. London Math. Soc. (3) 12 (1962) 193-206.

Crossref

Google Scholar

[5]

R.D. Carmichael, "On Composite Numbers p Which Satisfy the Fermat Congruence ap−1&Xgr;p," American Math. Monthly 19 (1912) 22-27.

Google Scholar

[6]

S. Cook, "The Complexity of Theorem-proving Procedures," Conference Record of Third ACM Symposium of Theory of Computing (1970) 151-158.

Digital Library

Google Scholar

[7]

H. Davenport and P. Erdös, "The Distribution of Quadratic and Higher Residues," Publ. Math. Debreien 2 (1952) 252-265.

Google Scholar

[8]

R. Karp, "Reducibility Among Combinatorial Problems," |Complexity of Computer Computations,# R.E. Miller and J.W. Thatcher, eds., Plenum Press, New York (1972) 85-103.

Google Scholar

[9]

D. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading Massachusetts (1969).

Digital Library

Google Scholar

[10]

H. Montgomery, Topics in Multiplicative Number Theory, Springer-Verlag Lecture Notes #227, 120.

Google Scholar

[11]

J. Pollard, "An Algorithm for Testing the Primality of Any Integer," Bull. London Math. Soc. 3 (1971) 337-340.

Crossref

Google Scholar

[12]

A. Selberg, "Contributions to the Theory of Dirichlet L Functions," Avhandlinger utgett av Det Norske Videnskops, Akademi i Oslo (1934).

Google Scholar

[13]

D. Shanks, "Class Number, A Theory of Factorization, and Genera," Proceedings of Symposia in Pure Mathematics (20), 1969 Number Theory Institute, AMS (1971) 415-440.

Google Scholar

[14]

A. Shönhage and V. Strassen, "Schnelle Multiplikation Grosser Zahlen," Computing 7 (1971) 281-292.

Crossref

Google Scholar

Cited By

View all

Einsele SPaterson K(2024)Average case error estimates of the strong Lucas testDesigns, Codes and Cryptography10.1007/s10623-023-01347-w92:5(1341-1378)Online publication date: 18-Jan-2024
https://doi.org/10.1007/s10623-023-01347-w
(2024)Generation of Pseudorandom and Prime Numbers for Cryptographic ApplicationsCryptography10.1002/9781394207510.ch16(531-565)Online publication date: 16-Feb-2024
https://doi.org/10.1002/9781394207510.ch16
Ernst HSchmidt JBeneken GErnst HSchmidt JBeneken G(2023)Algorithmen – Berechenbarkeit und KomplexitätGrundkurs Informatik10.1007/978-3-658-41779-6_12(521-575)Online publication date: 29-Nov-2023
https://doi.org/10.1007/978-3-658-41779-6_12
Show More Cited By

Index Terms

Riemann's Hypothesis and tests for primality
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Number-theoretic computations
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness

Recommendations

Riemann's hypothesis and tests for primality

In this paper we present two algorithms for testing primality of an integer. The first algorithm runs in 0(n1/7) steps; while, the second runs in 0(log4n) step but assumes the Extended Riemann Hypothesis. We also show that a class of functions which ...
Riemann hypothesis and finding roots over finite fields
STOC '85: Proceedings of the seventeenth annual ACM symposium on Theory of computing

It is shown that assuming Generalized Riemann Hypothesis, the roots of ƒ(x) = O mod p, where p is a prime and f(x) is an integral Abilene polynomial can be found in deterministic polynomial time. The method developed for solving this problem is also ...
Tests for primality under the Riemann hypothesis

Assuming the extended Riemann hypothesis (ERH), G. Miller produced in a very interesting paper [1], an algorithm which tests primality and runs in O((log n)^4+ε) steps. We provide a short proof of this result.

Comments

Information & Contributors

Information

Published In

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

May 1975

265 pages

ISBN:9781450374194

DOI:10.1145/800116

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 May 1975

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Acceptance Rates

STOC '75 Paper Acceptance Rate 31 of 87 submissions, 36%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

99
Total Citations
View Citations
634
Total Downloads

Downloads (Last 12 months)309
Downloads (Last 6 weeks)53

Reflects downloads up to 26 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Einsele SPaterson K(2024)Average case error estimates of the strong Lucas testDesigns, Codes and Cryptography10.1007/s10623-023-01347-w92:5(1341-1378)Online publication date: 18-Jan-2024
https://doi.org/10.1007/s10623-023-01347-w
(2024)Generation of Pseudorandom and Prime Numbers for Cryptographic ApplicationsCryptography10.1002/9781394207510.ch16(531-565)Online publication date: 16-Feb-2024
https://doi.org/10.1002/9781394207510.ch16
Ernst HSchmidt JBeneken GErnst HSchmidt JBeneken G(2023)Algorithmen – Berechenbarkeit und KomplexitätGrundkurs Informatik10.1007/978-3-658-41779-6_12(521-575)Online publication date: 29-Nov-2023
https://doi.org/10.1007/978-3-658-41779-6_12
Knezevic K(2021)Generating Prime Numbers Using Genetic Algorithms2021 44th International Convention on Information, Communication and Electronic Technology (MIPRO)10.23919/MIPRO52101.2021.9597026(1224-1229)Online publication date: 27-Sep-2021
https://doi.org/10.23919/MIPRO52101.2021.9597026
Morain FRenault GSmith B(2021)Deterministic factoring with oraclesApplicable Algebra in Engineering, Communication and Computing10.1007/s00200-021-00521-834:4(663-690)Online publication date: 16-Sep-2021
https://doi.org/10.1007/s00200-021-00521-8
Fatmi SChen XDhamija YWildes MTang Qvan Breugel F(2021)Probabilistic Model Checking of Randomized Java CodeModel Checking Software10.1007/978-3-030-84629-9_9(157-174)Online publication date: 3-Aug-2021
https://doi.org/10.1007/978-3-030-84629-9_9
Massimo JPaterson KLigatti JOu XKatz JVigna G(2020)A Performant, Misuse-Resistant API for Primality TestingProceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security10.1145/3372297.3417264(195-210)Online publication date: 30-Oct-2020
https://dl.acm.org/doi/10.1145/3372297.3417264
Kim GLi SHwang H(2020)Fast rebalanced RSA signature scheme with typical prime generationTheoretical Computer Science10.1016/j.tcs.2020.04.024Online publication date: May-2020
https://doi.org/10.1016/j.tcs.2020.04.024
Ernst HSchmidt JBeneken GErnst HSchmidt JBeneken G(2020)Algorithmen – Berechenbarkeit und KomplexitätGrundkurs Informatik10.1007/978-3-658-30331-0_12(501-554)Online publication date: 10-Jul-2020
https://doi.org/10.1007/978-3-658-30331-0_12
Sedlacek VJancar JSvenda P(2020)Fooling Primality Tests on SmartcardsComputer Security – ESORICS 202010.1007/978-3-030-59013-0_11(209-229)Online publication date: 13-Sep-2020
https://doi.org/10.1007/978-3-030-59013-0_11
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations