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Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
Nagaraj, Density of Carmichael numbers with three prime factors
 Math.Comp.66 (1997), 1705–1708. MR 98d:11110
"... Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1. ..."
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Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1.
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
Breaking a Cryptographic Protocol with
"... Abstract. The MillerRabin pseudo primality test is widely used in cryptographic libraries, because of its apparent simplicity. But the test is not always correctly implemented. For example the pseudo primality test in GNU Crypto 1.1.0 uses a fixed set of bases. This paper shows how this flaw can be ..."
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Abstract. The MillerRabin pseudo primality test is widely used in cryptographic libraries, because of its apparent simplicity. But the test is not always correctly implemented. For example the pseudo primality test in GNU Crypto 1.1.0 uses a fixed set of bases. This paper shows how this flaw can be exploited to break the SRP implementation in GNU Crypto. The attack is demonstrated by explicitly constructing pseudoprimes that satisfy the parameter checks in SRP and that allow a dictionary attack. This dictionary attack would not be possible if the pseudo primality test were correctly implemented. Often important details are overlooked in implementations of cryptographic protocols until specific attacks have been demonstrated. The goal of the paper is to demonstrate the need to implement pseudo primality tests carefully. This is done by describing a concrete attack against GNU Crypto 1.1.0. The pseudo primality test of this library is incorrect. It performs a trial division and a MillerRabin test with a fixed set of bases. Because the bases are known in advance an