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The Generation of Random Numbers That Are Probably Prime
 Journal of Cryptology
, 1988
"... In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing _only_ on composite numbers that can actually be split in expected polynomia ..."
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In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing _only_ on composite numbers that can actually be split in expected polynomial time. Therefore, factoring would be easy if Rabin's test systematically failed with a 25% probability on each composite integer (which, of course, it does not). The second observation is more fundamental because is it _not_ restricted to primality testing: it has consequences for the entire field of probabilistic algorithms. The failure probability when using a probabilistic algorithm for the purpose of testing some property is compared with that when using it for the purpose of obtaining a random element hopefully having this property. More specifically, we investigate the question of how reliable Rabin's test is when used to _generate_ a random integer that is probably prime, rather than to _test_ a specific integer for primality.
Key words: factorization, false witnesses, primality testing, probabilistic algorithms, Rabin's test.
MO419 – Probabilistic Algorithms – Flávio K. Miyazawa – IC/UNICAMP 2010 A survey on Probabilistic Algorithms to Primality Test
"... One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or ..."
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One of the longstanding problems in using encryption to encode messages is that the recipient of the message needs to know the key in order to decrypt the message. Clearly we somehow have to get the key to the participants so they can use it. We can’t send the key to them without encrypting *it*, or someone might “eavesdrop ” and get it. But this puts us in an infinite loop: the
FLINT Fast Library for Number Theory
, 2011
"... FLINT is a C library of functions for doing number theory. It is highly optimised and can be compiled on numerous platforms. FLINT also has the aim of providing support for multicore and multiprocessor computer architectures, though we do not yet provide this facility.
FLINT is currently maintained ..."
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FLINT is a C library of functions for doing number theory. It is highly optimised and can be compiled on numerous platforms. FLINT also has the aim of providing support for multicore and multiprocessor computer architectures, though we do not yet provide this facility.
FLINT is currently maintained by William Hart of Warwick University in the UK. Its main authors are William Hart, Sebastian Pancratz, Fredrik Johannson, Andy Novocin and David Harvey (no longer active).
FLINT 2 and following should compile on any machine with GCC and a standard GNU toolchain, however it is specially optimized for x86 (32 and 64 bit) machines. As of version 2.0 FLINT required GCC version 2.96 or later, MPIR 2.1.1 or later and MPFR 3.0.0 or later.
FLINT is supplied as a set of modules, fmpz, fmpz_poly, etc., each of which can be linked to a C program making use of their functionality. All of the functions in FLINT have a corresponding test function provided in an appropriately named test le. For example, the function fmpz_poly_add located in fmpz_poly/add.c has test code in the le fmpz_poly/test/tadd.c.
SOME INTERESTING SUBSEQUENCES OF THE FIBONACCI AND LUCAS PSEUDOPRIMES
, 1994
"... In this paper, certain interesting sequences of positive integers are investigated. As will be demonstrated, these are subsequences of the Fibonacci and Lucas pseudoprimes, as they have been defined in the author's previous papers ([2], [3], [4], [9]). Indeed, it will be shown that the elements of t ..."
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In this paper, certain interesting sequences of positive integers are investigated. As will be demonstrated, these are subsequences of the Fibonacci and Lucas pseudoprimes, as they have been defined in the author's previous papers ([2], [3], [4], [9]). Indeed, it will be shown that the elements of two of these subsequences are strong Lucas pseudoprimes and EulerLucas pseudoprimes.