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Recent progress in graph pebbling
 Graph Theory Notes N. Y
"... The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1 ..."
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The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999. 2 1
An Application of Graph Pebbling . . .
, 2008
"... A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening ..."
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A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdős and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zerosum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (gk) G k=1 of G  elements of a finite abelian group G there is a nonempty subsequence (gk)k∈K such that ∑ k∈K gk = 0G and ∑ k∈K 1/gk  ≤ 1, where g  is the order of the element g ∈ G.
and
, 2004
"... A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening ..."
Abstract
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A sequence of elements of a finite group G is called a zerosum sequence if it sums to the identity of G. The study of zerosum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdős and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zerosum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (gk) G k=1 of G  elements of a finite abelian group G there is a nonempty subsequence (gk)k∈K such that ∑ k∈K gk = 0G and ∑ k∈K 1/gk  ≤ 1, where g  is the order of the element g ∈ G.
J. Aust. Math. Soc. 94 (2013), 268–275 doi:10.1017/S1446788712000547 CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
, 2013
"... We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m). ..."
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We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).
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