Four primality testing algorithms

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or ‘very probably’ prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.

[1]  François Morain,et al.  Primality Proving Using Elliptic Curves: An Update , 1998, ANTS.

[2]  Ming-Deh A. Huang,et al.  Primality Testing and Abelian Varieties over Finite Fields , 1992 .

[3]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[4]  Gary L. Miller.  Riemann's Hypothesis and Tests for Primality , 1976, J. Comput. Syst. Sci..

[5]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[6]  Preda Mihailescu,et al.  Cyclotomy Primality Proving - Recent Developments , 1998, ANTS.

[7]  Faculteit der Wiskunde en Natuurwetenschappen,et al.  Divisors in residue classes , 1983 .

[8]  Hendrik W. Lenstra,et al.  Primality testing algorithms [after Adleman, Rumely and Williams] , 1981 .

[9]  M. Rabin.  Probabilistic algorithm for testing primality , 1980 .

[10]  Leonard M. Adleman,et al.  On distinguishing prime numbers from composite numbers , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[11]  R. Schoof.  Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p , 1985 .

[12]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[13]  Joe Kilian,et al.  Almost all primes can be quickly certified , 1986, STOC '86.

[14]  Hendrik W. Lenstra,et al.  Miller's Primality Test , 1979, Inf. Process. Lett..

[15]  Hendrik W. Lenstra,et al.  Primality Testing with Gaussian Periods , 2002, FSTTCS.

[16]  L. Washington.  Introduction to Cyclotomic Fields , 1982 .

[17]  H. W. Lenstra,et al.  Factoring integers with elliptic curves , 1987 .

[18]  E. Bach.  Explicit bounds for primality testing and related problems , 1990 .

[19]  François Morain.  Implementing the asymptotically fast version of the elliptic curve primality proving algorithm , 2007, Math. Comput..

[20]  Andreas Enge,et al.  The complexity of class polynomial computation via floating point approximations , 2006, Math. Comput..

[21]  A. Atkin,et al.  ELLIPTIC CURVES AND PRIMALITY PROVING , 1993 .

[22]  P. Stevenhagen,et al.  Computational Class Field Theory , 2008, 0802.3843.