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On Generalized Carmichael Numbers
, 2000
"... . For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several spe ..."
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Cited by 2 (2 self)
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. For arbitrary integers k 2 Z we investigate the set C k of the generalized Carmichael numbers, i.e. the natural numbers n ? maxf1; 1 \Gamma kg such that the equation a n+k j a mod n holds for all a 2 N. We give a characterization of these generalized Carmichael numbers and discuss several special cases. In particular, we prove that C 1 is finite and that C k is infinite, whenever 1 \Gamma k ? 1 is squarefree. We also discuss generalized Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers n which satisfy the equation a n j a mod n only for a = 2, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares. 1 Introduction: Historical Background On October 18th, 1640, Pierre de Fermat wrote in a letter to Bernard Frenicle de Bessy that if p is a prime number, then p divides a p\Gamma1 \Gamma 1 for all in...
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
FERMAT’S COMPOSITENESS TEST
"... The most naive method of seeing if an integer m> 1 is composite is trial division up to √ m. This method will only prove m is composite if we find a divisor of m. A slight improvement on trial division is the following: pick a random a with 1 ≤ a ≤ m − 1 and compute (a, m) by Euclid’s algorithm. ..."
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The most naive method of seeing if an integer m> 1 is composite is trial division up to √ m. This method will only prove m is composite if we find a divisor of m. A slight improvement on trial division is the following: pick a random a with 1 ≤ a ≤ m − 1 and compute (a, m) by Euclid’s algorithm. If (a, m)> 1 then m is certifiably composite, and in fact (a, m) will be a nontrivial factor of m. If (a, m) = 1, pick another a at random and try again. We call this the gcd test for compositeness.
ON THE NUMBER OF PRIME DIVISORS OF
, 2001
"... The classical Carmichael numbers are well known in number theory. These numbers were introduced independently by Korselt in [8] and Carmichael in [2] and since then they have been the subject of intensive study. The reader may find extensive but not exhaustive lists of references in [5, Sect. A13], ..."
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The classical Carmichael numbers are well known in number theory. These numbers were introduced independently by Korselt in [8] and Carmichael in [2] and since then they have been the subject of intensive study. The reader may find extensive but not exhaustive lists of references in [5, Sect. A13], [11, Ch. 2, Sec. IX].
unknown title
, 711
"... Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implement ..."
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Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding threeprime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.
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
PRIMES AND QUADRATIC RECIPROCITY
"... Abstract. We discuss number theory with the ultimate goal of understanding quadratic reciprocity. We begin by discussing Fermat’s Little Theorem, the Chinese Remainder Theorem, and Carmichael numbers. Then we define the Legendre symbol and prove Gauss’s Lemma. Finally, using Gauss’s Lemma we prove t ..."
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Abstract. We discuss number theory with the ultimate goal of understanding quadratic reciprocity. We begin by discussing Fermat’s Little Theorem, the Chinese Remainder Theorem, and Carmichael numbers. Then we define the Legendre symbol and prove Gauss’s Lemma. Finally, using Gauss’s Lemma we prove the Law of Quadratic Reciprocity.