@article{damgard1993:average_case_error,
    author = {Ivan Damgård and Peter Landrock and Carl Pomerance},
    title = {Average Case Error Estimates for the Strong Probable Prime Test},
    journal = {Mathematics of Computation},
    year = {1993},
    volume = {61},
    number = {203},
    pages = {177-194},
    doi = {10.2307/2152945},
    issn = {00255718, 10886842},
    url = {http://www.jstor.org/stable/2152945},
    abstract = {Consider a procedure that chooses k-bit odd numbers independently and from the uniform distribution, subjects each number to t independent iterations of the strong probable prime test (Miller-Rabin test) with randomly chosen bases, and outputs the first number found that passes all t tests. Let p<sub>k, t</sub> denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for p<sub>k, t</sub> for various choices of k, t and obtain clean explicit functions that bound p<sub>k, t</sub> for certain infinite classes of k, t. For example, we show p<sub>100, 10</sub> ≤ 2<sup>-44</sup>, p<sub>300, 5</sub> ≤ 2<sup>-60</sup>, p<sub>600, 1</sub> ≤ 2<sup>-75</sup>, and <tex-math>$p_{k, 1} \leq k^2 4^{2 - \sqrt k}$</tex-math> for all <tex-math>$k \qeq 2$</tex-math>. In addition, we characterize the worst-case numbers with unusually many "false witnesses" and give an upper bound on their distribution that is probably close to best possible.},
    publisher = {American Mathematical Society}
}

@book{davenport1988:computer_algebra,
    author = {Davenport, J. H. and Siret, Y. and Tournier, E.},
    title = {Computer Algebra, Systems and Algorithms for Algebraic
             Computation},
    year = {1988},
    isbn = {0-122-04230-1},
    publisher = {Academic Press},
    address = {New York, NY, USA}
}

@InProceedings{davenport1992:primality_test_revisited,
    author = {J. H. Davenport},
    title = {Primality Testing Revisited},
    booktitle = {Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC'92)},
    year = {1992},
    publisher = {ACM Press},
    editor = {P. S. Wang},
    pages = {123--129}
}


@book{gathen1999:modern_computer_algebra,
    author = {von zur Gathen, Joachim and Gerhard, Jürgen},
    title = {Modern Computer Algebra},
    year = {1999},
    isbn = {0-521-64176-4},
    publisher = {Cambridge University Press},
    address = {New York, NY, USA},
}

@book{gathen2003:modern_computer_algebra,
    author = {{\noop{Gathen}{von zur Gathen}}, Joachim and Gerhard, Jürgen},
    title = {Modern Computer Algebra},
    year = {2003},
    isbn = {0-521-82646-2},
    edition = {2},
    publisher = {Cambridge University Press},
    address = {New York, NY, USA},
} 

@book{hardy1945:introduction_theory_numbers,
    author = {G. H. Hardy and E. M. Wright},
    title = {An Introduction to the Theory of Numbers},
    year = {1945},
    publisher = {Oxford University Press}
}

@book{knuth1997:acp_seminumerical_algorithms,
    author = {Knuth, Donald E.},
    title = {The Art of Computer Programming, Volume 2 (3rd Ed.): Seminumerical Algorithms},
    year = {1997},
    isbn = {0-201-89684-2},
    publisher = {Addison-Wesley Longman Publishing Co., Inc.},
    address = {Boston, MA, USA},
} 

@book{pollard1975:theory_algebraic_numbers,
    author = {H. Pollard and H. G. Diamond},
    title = {The Theory of Algebraic Numbers},
    year = {1975},
    publisher = {Wiley},
    address = {New York}
}

@article {pomerance1980:pseudoprimes,
    author = {Pomerance, Carl and Selfridge, J. L. and Wagstaff, Jr., Samuel S.},
    title = {The Pseudoprimes to {$25\cdot 10^{9}$}},
    journal = {Mathematics of Computation},
    year = {1980},
    volume = {35},
    number = {151},
    pages = {1003--1026},
    issn = {0025-5718},
    doi = {10.2307/2006210}
}

@article{rabin1980:probabilistic_algorithm_testing,
    author = "Michael O Rabin",
    title = "Probabilistic algorithm for testing primality",
    journal = "Journal of Number Theory",
    volume = "12",
    number = "1",
    pages = "128 -- 138",
    year = "1980",
    issn = "0022-314X",
    doi = "http://dx.doi.org/10.1016/0022-314X(80)90084-0",
    url = "http://www.sciencedirect.com/science/article/pii/0022314X80900840",
    abstract = "We present a practical probabilistic algorithm for testing large numbers of arbitrary form for primality. The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error. The algorithm was used to generate large numbers asserted to be primes of arbitrary and special forms, including very large numbers asserted to be twin primes. Theoretical foundations as well as details of implementation and experimental results are given."
}
