Access the full text.
Sign up today, get DeepDyve free for 14 days.
A. Lenstra, M. Manasse (1990)
Factoring by Electronic Mail
Faith Ellen, M. Tompa (1988)
The parallel complexity of exponentiating polynomials over finite fieldsJ. ACM, 35
S. Wagstaff (1979)
Greatest of the least primes in arithmetic progressions having a given modulusMathematics of Computation, 33
E. Allender (1986)
Characterizations of PUNC and precomputation
J. Calmet, R. Loos (1980)
An Improvement of Rabin's Probabilistic Algorithm for Generating Irreducible Polynomials over GF(p)Inf. Process. Lett., 11
L. Carlitz (1932)
The Arithmetic of Polynomials in a Galois FieldAmerican Journal of Mathematics, 54
J. Stein (1967)
Computational problems associated with Racah algebraJournal of Computational Physics, 1
L. Adleman, K. Kompella (1988)
Using smoothness to achieve parallelism
J. Lagarias, H. Montgomery, A. Odlyzko (1979)
A bound for the least prime ideal in the Chebotarev Density TheoremInventiones mathematicae, 54
J. Young, D. Buell (1988)
The twentieth Fermat number is compositeMathematics of Computation, 50
R. Solovay, V. Strassen (1977)
A Fast Monte-Carlo Test for PrimalitySIAM J. Comput., 6
Frobenius, Stickelberger
Ueber Gruppen von vertauschbaren Elementen.Journal für die reine und angewandte Mathematik (Crelles Journal), 1879
K. Whipple (1982)
On Computing LogarithmsSchool Science and Mathematics, 82
A. Lenstra, H. Lenstra (1991)
Algorithms in Number Theory
R. Brent (1973)
The First Occurrence of Large Gaps Between Successive PrimesMathematics of Computation, 27
J. Gathen (1983)
Parallel algorithms for algebraic problemsProceedings of the fifteenth annual ACM symposium on Theory of computing
D. Heath-Brown (1986)
ARTIN'S CONJECTURE FOR PRIMITIVE ROOTSQuarterly Journal of Mathematics, 37
H. Zassenhaus (1969)
On Hensel factorization, IJournal of Number Theory, 1
N. Tschebotareff (1926)
Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehörenMathematische Annalen, 95
C. Schnorr, H. Lenstra (1984)
A Monte Carlo factoring algorithm with linear storageMathematics of Computation, 43
W. Feller (1959)
An Introduction to Probability Theory and Its Applications
C. Pomerance (1987)
Very short primality proofsMathematics of Computation, 48
J. Gerver (1983)
Factoring large numbers with a quadratic sieveMathematics of Computation, 41
R. Boppana, R. Hirschfeld (1989)
Pseudorandom Generators and Complexity ClassesAdv. Comput. Res., 5
R. Peralta (1986)
A simple and fast probabilistic algorithm for computing square roots modulo a prime number
S. Goldwasser, J. Kilian (1986)
Almost all primes can be quickly certified
K. Ireland, M. Rosen (1982)
A classical introduction to modern number theory, 84
J. Shallit (1990)
On the Worst Case of Three Algorithms for Computing the Jacobi SymbolJ. Symb. Comput., 10
C. Hooley (1967)
On Artin's conjecture.Journal für die reine und angewandte Mathematik (Crelles Journal), 1967
L. Hua (1982)
Introduction to number theory
F. Hirzebruch (1966)
Topological methods in algebraic geometry
V. Pratt (1975)
Every Prime has a Succinct CertificateSIAM J. Comput., 4
R. Moenck (1973)
Fast computation of GCDsProceedings of the fifth annual ACM symposium on Theory of computing
J. Tate (1965)
Algebraic cycles and poles of zeta functions
N. Ankeny (1952)
The least quadratic non residueAnnals of Mathematics, 55
E. Titchmarsh (1930)
A divisor problemRendiconti del Circolo Matematico di Palermo (1884-1940), 54
D. Cantor (1987)
Computing in the Jacobian of a hyperelliptic curveMathematics of Computation, 48
L. Adleman, D. Estes, K. McCurley (1987)
Solving bivariate quadratic congruences in random polynomial timeMathematics of Computation, 48
C. Jacobi
Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie.Journal für die reine und angewandte Mathematik (Crelles Journal), 1846
E. Bach, James Driscoll, J. Shallit (1993)
Factor refinementJ. Algorithms, 15
Ming-Deh Huang (1985)
Riemann hypothesis and finding roots over finite fields
D. Cantor, H. Zassenhaus (1981)
A new algorithm for factoring polynomials over finite fieldsMathematics of Computation, 36
J. Davis, D. Holdridge, G. Simmons (1985)
Status report on factoring (at the Sandia National Labs)
A. Atkin, R. Larson (1982)
On a Primality Test of Solovay and StrassenSIAM J. Comput., 11
Jacques Vélu (1978)
Tests for primality under the Riemann hypothesisSIGACT News, 10
H. Lenstra (1987)
Factoring integers with elliptic curvesAnnals of Mathematics, 126
L. Schoenfeld (1976)
Corrigendum: “Sharper bounds for the Chebyshev functions () and (). II” (Math. Comput. 30 (1976), no. 134, 337–360)Mathematics of Computation, 30
M. Rabin (1980)
Probabilistic algorithm for testing primalityJournal of Number Theory, 12
M. Blum, S. Micali (1982)
How to generate cryptographically strong sequences of pseudo random bits23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
R. Brent (1980)
An improved Monte Carlo factorization algorithmBIT Numerical Mathematics, 20
Volker Strassen (1973)
Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von InterpolationskoeffizientenNumerische Mathematik, 20
E. Bach (1987)
Realistic analysis of some randomized algorithmsProceedings of the nineteenth annual ACM symposium on Theory of computing
E. Fogels (1962)
On the distribution of prime idealsActa Arithmetica, 7
L. Adleman, Kenneth Manders, G. Miller (1977)
On taking roots in finite fields18th Annual Symposium on Foundations of Computer Science (sfcs 1977)
Taher Gamal (1985)
On Computing Logarithms Over Finite Fields
H. Cramér (1936)
On the order of magnitude of the difference between consecutive prime numbers, 45
J. Hafner, K. McCurley (1989)
On the Distribution of Running Times of Certain Integer Factoring AlgorithmsJ. Algorithms, 10
D. Chudnovsky, G. Chudnovsky (1986)
Sequences of numbers generated by addition in formal groups and new primality and factorization testsAdvances in Applied Mathematics, 7
Douglas Wiedemann (1986)
Solving sparse linear equations over finite fieldsIEEE Trans. Inf. Theory, 32
K. Dickman
On the frequency of numbers containing prime factors of a certain relative magnitudeArkiv för matematik, astronomi och fysik
Andrew Odlyzko, Te Riele (1984)
Disproof of the Mertens conjecture.Journal für die reine und angewandte Mathematik (Crelles Journal), 1985
S. Holmes, D. Hunt, T. Lake, P. Marron, S. Reddaway, N. Westbury, G. Haworth (1983)
Primality-testing Mersenne Numbers (II)
J. Lune, E. Wattel (1969)
On the numerical solution of a differential-difference equation arising in analytic number theory
R. Schoof (1985)
Elliptic Curves Over Finite Fields and the Computation of Square Roots mod pMathematics of Computation, 44
A. Weil (1948)
Sur les courbes algébriques et les variétés qui s'en déduisent
S. Evdokimov (1992)
Factorization of solvable polynomials over finite fields and the generalized Riemann hypothesisJournal of Soviet Mathematics, 59
John Gill (1974)
Computational complexity of probabilistic Turing machines
A. Weil (1967)
Basic number theory
P. Bateman, W. Leveque (1976)
Reviews in Number TheoryMathematics of Computation, 30
H. Chernoff (1952)
A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of ObservationsAnnals of Mathematical Statistics, 23
R. Karp (1988)
A Survey of Parallel Algorithms for Shared-Memory Machines
R. Spira (1969)
Calculation of Dirichlet -functionsMathematics of Computation, 23
D. Lehmer (1933)
A Photo-Electric Number SieveAmerican Mathematical Monthly, 40
M. Ben-Or (1980)
Probabilistic algorithms in finite fields22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
E. Kaltofen (1985)
Sparse Hensel Lifting
A. Yao (1982)
Theory and application of trapdoor functions23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
C. Pomerance, J. Selfridge, S. Wagstaff (1980)
The pseudoprimes to 25⋅10⁹Mathematics of Computation, 35
H. Lauwerier (1986)
4. Two-dimensional iterative maps
P. Beame, Stephen Cook, H. Hoover (1984)
Log Depth Circuits for Division and Related Problems
D. Lehmer (1930)
An Extended Theory of Lucas' FunctionsAnnals of Mathematics, 31
By Pollard (1978)
Monte Carlo methods for index computation ()Mathematics of Computation, 32
R. Rivest, A. Shamir, L. Adleman (1978)
A method for obtaining digital signatures and public-key cryptosystemsCommun. ACM, 26
G. Miller (1975)
Riemann's Hypothesis and tests for primalityProceedings of the seventh annual ACM symposium on Theory of computing
E. Canfield, P. Erdös, C. Pomerance (1983)
On a problem of Oppenheim concerning “factorisatio numerorum”Journal of Number Theory, 17
de Bruijn (1951)
The asymptotic behaviour of a function occuring in the theory of primesJournal of the Indian Mathematical Society, 15
A. Lenstra (1987)
Fast and rigorous factorization under the generalized Riemann hypothesis, 91
R. Brent, J. Pollard (1981)
Factorization of the eighth Fermat numberMathematics of Computation, 36
H. Davenport (1967)
Multiplicative Number Theory
J. Gathen (1987)
Computing Powers in ParallelSIAM J. Comput., 16
P. Deligne (1974)
La conjecture de Weil. IPublications Mathématiques de l'Institut des Hautes Études Scientifiques, 43
R. Brent (1979)
On the zeros of the Riemann zeta function in the critical stripMathematics of Computation, 33
E. Berlekamp (1984)
Algebraic coding theory
Wang Wei (1991)
On the least prime in an arithmetic progressionActa Mathematica Sinica, 7
J. Sklansky (1960)
Conditional-Sum Addition LogicIRE Trans. Electron. Comput., 9
D. Coppersmith, S. Winograd (1987)
Matrix multiplication via arithmetic progressionsProceedings of the nineteenth annual ACM symposium on Theory of computing
D. Lehmann (1982)
On Primality TestsSIAM J. Comput., 11
P. Erdös, C. Pomerance (1986)
On the number of false witnesses for a composite numberMathematics of Computation, 46
L. Adleman (1980)
On distinguishing prime numbers from composite numbers21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
D. Lehmer (1938)
Euclid's Algorithm for Large NumbersAmerican Mathematical Monthly, 45
S. Graham, C. Ringrose (1990)
Lower Bounds for Least Quadratic Non-Residues
D. Coppersmith (1984)
Fast evaluation of logarithms in fields of characteristic twoIEEE Trans. Inf. Theory, 30
A. Odlyzko (1987)
On the distribution of spacings between zeros of the zeta functionMathematics of Computation, 48
K. McCurley (1984)
Explicit estimates for the error term in the prime number theorem for arithmetic progressionsMathematics of Computation, 42
R. Dedekind
Abriss einer Theorie der höhern Congruenzen in Bezug auf einen reellen Primzahl-Modulus.Journal für die reine und angewandte Mathematik (Crelles Journal), 1857
J. Pintz, W. Steiger, E. Szemerédi (1989)
Infinite sets of primes with fast primality tests and quick generation of large primesMathematics of Computation, 53
D. S., J. Rosser, L. Schoenfeld (1962)
Approximate formulas for some functions of prime numbersIllinois Journal of Mathematics, 6
Michael Morrison, J. Brillhart (1975)
A method of factoring and the factorization ofMathematics of Computation, 29
L. Adleman, H. Lenstra (1986)
Finding irreducible polynomials over finite fields
E. Bach (1988)
How to Generate Factored Random NumbersSIAM J. Comput., 17
D. Knuth (1997)
The art of computer programming: V.2.: Seminumerica algorithms
This paper is a report on algorithms to solve problems in number theory. I believe the most interesting such problems to be those from elementary number theory whose complexity is still unknown. For this reason, I concentrate on methods to test primality, to find the prime factors of numbers, and to solve equations in various finite groups, rings, and fields. These problems have the attractive feature that they are easily stated and frequently can be solved by algorithms that are easy to implement. However, the intuition behind these algorithms and the methods used to analyze them are often anything but elementary; for this reason I describe not only the algorithms but also the underlying mathematics. The present review is not an exhaustive survey of the area, and neces sarily reflects my own biases and interests. For example, the inclusion of algorithms for polynomials over finite fields may seem inappropriate. However, the polynomials in one variable over a finite field have a well studied and attractive arithmetic, and it seemed interesting to contrast algorithms for this domain with algorithms for the integers. At other points, the subject impinges on algebraic number theory, but I omit the details of such
Annual Review of Computer Science – Annual Reviews
Published: Jun 1, 1990
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.