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A Survey of Modern Integer Factorization Algorithms
 CWI Quarterly
, 1994
"... Introduction An integer n ? 1 is said to be a prime number (or simply prime) if the only divisors of n are \Sigma1 and \Sigman. There are infinitely many prime numbers, the first four being 2, 3, 5, and 7. If n ? 1 and n is not prime, then n is said to be composite. The integer 1 is neither prime ..."
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Introduction An integer n ? 1 is said to be a prime number (or simply prime) if the only divisors of n are \Sigma1 and \Sigman. There are infinitely many prime numbers, the first four being 2, 3, 5, and 7. If n ? 1 and n is not prime, then n is said to be composite. The integer 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a finite (perhaps empty) product of prime numbers, and that this factorization is unique except for the ordering of the factors. Table 1.1 has some sample factorizations. 1990 = 2 \Delta 5 \Delta 199 1995 = 3 \Delta 5 \Delta 7 \Delta 19 2000 = 2 4 \Delta 5 3 2005 = 5 \Delta 401
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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Cited by 2 (0 self)
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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.
The Magic Words Are Squeamish Ossifrage (Extended Abstract)
"... We describe the computation which resulted in the title of this paper. Furthermore, we give an analysis of the data collected during this computation. From these data, we derive the important observation that in the final stages, the progress of the double large prime variation of the quadratic siev ..."
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We describe the computation which resulted in the title of this paper. Furthermore, we give an analysis of the data collected during this computation. From these data, we derive the important observation that in the final stages, the progress of the double large prime variation of the quadratic sieve integer factoring algorithm can more effectively be approximated by a quartic function of the time spent, than by the more familiar quadratic function. We also present, as an update to [15], some of our experiences with the management of a large computation distributed over the Internet. Based on this experience, we give some realistic estimates of the current readily available computational power of the Internet. We conclude that commonlyused 512bit RSA moduli are vulnerable to any organization prepared to spend a few million dollars and to wait a few months.
//eprint.iacr.org/2009/082. The Case for Quantum Key Distribution
, 2009
"... Quantum key distribution (QKD) promises secure key agreement by using quantum mechanical systems. We argue that QKD will be an important part of future cryptographic infrastructures. It can provide longterm confidentiality for encrypted information without reliance on computational assumptions. Alt ..."
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Quantum key distribution (QKD) promises secure key agreement by using quantum mechanical systems. We argue that QKD will be an important part of future cryptographic infrastructures. It can provide longterm confidentiality for encrypted information without reliance on computational assumptions. Although QKD still requires authentication to prevent maninthemiddle attacks, it can make use of either informationtheoretically secure symmetric key authentication or computationally secure public key authentication: even when using public key authentication, we argue that QKD still offers stronger security than classical key agreement. 1
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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Cited by 1 (1 self)
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Computing Science The Magic Words Are Squeamish Ossifrage
"... . bu are given two integers, a and b, and ' asked to compute their product, ab = c. An algorithm for this task is taught in the early primary grades. For those of us who were day dreaming in class that day, a computer implemen tation of the algorithm yields an answer in micro seconds, even if a ..."
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. bu are given two integers, a and b, and ' asked to compute their product, ab = c. An algorithm for this task is taught in the early primary grades. For those of us who were day dreaming in class that day, a computer implemen tation of the algorithm yields an answer in micro seconds, even if a and b are rather large numbers, say 60 or 70 decimal digits. Now suppose you are given the number c and asked to discover the two factors a and b, which you may assume are prime numbers (that is, they have no factors of their own, apart from 1 and themselves). This is a much harder assignment. If a and b are in the 60digit range, so that c has more than 120 digits, finding the factors is definitely not elementaryschool homework. The dramatic asymmetry between multiplica tion and factorization is the basis of an important cryptographic system: the RSA publickey cryptosystem, named for the initials of its inventors,
The Case for Quantum Key Distribution Douglas Stebila1,2, Michele Mosca1,2,3 1,3,4 ∗
, 2009
"... Quantum key distribution (QKD) promises secure key agreement by using quantum mechanical systems. We argue that QKD will be an important part of future cryptographic infrastructures. It can provide longterm confidentiality for encrypted information without reliance on computational assumptions. Alt ..."
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Quantum key distribution (QKD) promises secure key agreement by using quantum mechanical systems. We argue that QKD will be an important part of future cryptographic infrastructures. It can provide longterm confidentiality for encrypted information without reliance on computational assumptions. Although QKD still requires authentication to prevent maninthemiddle attacks, it can make use of either informationtheoretically secure symmetric key authentication or computationally secure public key authentication: even when using public key authentication, we argue that QKD still offers stronger security than classical key agreement. 1
Integer Factoring
"... Abstract. 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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Abstract. Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
Acknowledgements
, 2002
"... In the first place, I owe a great debt of gratitude to Prof. Jan Peřina, who brought me to physics and whose memorable lectures showed me into the realm of quantum optics. It has certainly been an exceptional privilege to meet and work with such a personality. I wish to acknowledge his advice, suppo ..."
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In the first place, I owe a great debt of gratitude to Prof. Jan Peřina, who brought me to physics and whose memorable lectures showed me into the realm of quantum optics. It has certainly been an exceptional privilege to meet and work with such a personality. I wish to acknowledge his advice, support and comments during my PhD study as well as his careful reading of the manuscript of the Thesis. I am also deeply indebted to my coworkers Miloslav Dušek, Ondřej Haderka and Robert Myška without who the quantum key distribution and quantum identification experiments could have never succeeded. I am grateful to Prof. Malvin C. Teich for giving me the opportunity to work at