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Efficiency and Security of Cryptosystems Based on Number Theory
, 1996
"... , 44 equivalent, 48 admissible, 19 associated, 48 binary addition chain, 45 binary method, 43, 63 Carmichael function, 4 Carmichael number, 16, 29 Chinese Remainder Theorem, 5 complex extension, 3 conjugate, 3 CRT, 5 Dickson polynomials, 11 doubling step, 63 dual, 48 Fermat test, 15, 16 graph reduce ..."
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, 44 equivalent, 48 admissible, 19 associated, 48 binary addition chain, 45 binary method, 43, 63 Carmichael function, 4 Carmichael number, 16, 29 Chinese Remainder Theorem, 5 complex extension, 3 conjugate, 3 CRT, 5 Dickson polynomials, 11 doubling step, 63 dual, 48 Fermat test, 15, 16 graph reduced, 48 group of units, 3 indegree, 45 Jacobi symbol, 6 Legendre symbol, 5 Lucas chain, 62 composite, 63 degenerate, 63 simple, 63 Lucas sequence, 8 Mathematica, 23, 41 MillerRabin test, 18 norm, 3 order of a group element, 7 outdegree, 45 Pocklington, 25 probable prime, 15 pseudoprimality, 2 BIBLIOGRAPHY 85 [R'ed48] L. R'edei. Uber eindeutig umkehrbare Polynome in endlichen Korpern. Acta Sci. Math., 11:7176, 194648. [Rie85] H. Riesel. Prime Numbers and Computer Methods for Factorization. Birkhauser, 1985. [RLS + 93] R. A. Rueppel, A. K. Lenstra, M. E. Smid, K. S. McCurley, Y. Desmedt, A. Odlyzko, and P. Landrock. Panel
Primality Testing Revisited
, 1992
"... . Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm. Introduction It is customary ..."
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. Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm. Introduction It is customary in computer algebra to use the algorithm presented by Rabin [1980] to determine if numbers are prime (and primes are needed throughout algebraic algorithms). As is well known, a single iteration of Rabin's algorithm, applied to the number N , has probability at most 0.25 of reporting "N is probably prime", when in fact N is composite. For most N , the probability is much less than 0.25. Here, "probability" refers to the fact that Rabin's algorithm begins with the choice of a "random" seed x, not congruent to 0 modulo N . In practice, however, true randomness is hard to achieve, and computer algebra systems often use a fixed set of x  for example Axiom release 1 uses the set f3; 5; 7; 11;...
The RabinMonier theorem for Lucas pseudoprimes
 Math. Comp
, 1997
"... Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1. ..."
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Abstract. We give bounds on the number of pairs (P, Q)with0≤P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q). 1.
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
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