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An Implementation of the Number Field Sieve
 EXPERIMENTAL MATHEMATICS
, 1996
"... This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implem ..."
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Cited by 14 (0 self)
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This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of 12^151 1.
Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware
"... A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been perform ..."
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Cited by 12 (2 self)
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A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been performed, and a substantial improvement has been demonstrated in terms of both the execution time and the areatime product. The ECM architecture has been ported across five different families of FPGA devices in order to select the family with the best performance to cost ratio. A timing comparison with the highly optimized software implementation, GMPECM, has been performed. Our results indicate that lowcost families of FPGAs, such as Spartan3 and Spartan3E, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio. 1.
Euclidean rings of algebraic integers
 Canad. J. Math
"... Abstract. Let K be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothes ..."
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Cited by 7 (4 self)
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Abstract. Let K be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally. 1
HighPerformance Integer Factoring with Reconfigurable Devices
 INTERNATIONAL CONFERENCE ON FIELD PROGRAMMABLE LOGIC AND APPLICATIONS
, 2010
"... We present a novel FPGAbased implementation of the Elliptic Curve Method (ECM) for the factorization of mediumsized composite integers. More precisely, we demonstrate an ECM implementation capable to determine prime factors of up to 2,424 151bit integers per second using a single Xilinx Virtex4 ..."
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We present a novel FPGAbased implementation of the Elliptic Curve Method (ECM) for the factorization of mediumsized composite integers. More precisely, we demonstrate an ECM implementation capable to determine prime factors of up to 2,424 151bit integers per second using a single Xilinx Virtex4 SX35 FPGA. Using this implementation on a cluster like the COPACOBANA is beneficial for attacking cryptographic primitives like the wellknown RSA cryptosystem with advanced methods such as the Number Field Sieve (NFS). To provide this vast number of integer factorizations per FPGA, we make use of the available DSP blocks on each Virtex4 device to accelerate lowlevel arithmetic computations. This methodology allows the development of a timearea efficient design that runs 24 ECM cores in parallel, implementing both phase 1 and phase 2 of the ECM. Moreover, our design is fully scalable and supports composite integers in the range from 66 to 236 bits without any significant modifications to the hardware. Compared to the implementation by Gaj et al., who reported an ECM design for the same Virtex4 platform, our improved architecture provides an advanced costperformance ratio which is better by a factor of 37.
Evaluation Report on the Factoring Problem
 Society International Conference, September N., (1981C).  23rd IEEE
, 2001
"... This document is an evaluation of the factoring problem, as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the factoring problem ..."
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This document is an evaluation of the factoring problem, as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the factoring problem
Information and computation: . . .
, 2002
"... Quantum theory has found a new field of application in the realm of information and computation during recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpa ..."
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Quantum theory has found a new field of application in the realm of information and computation during recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some notable aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as examples of the impact of quanta on the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement made possible in information processing by a quantum computer. Finally the authors provide some examples of current experimental realizations for quantum computers and future prospects
Algebraic Number Theory
, 2009
"... 2. Number fields........................................ 9 3. Norms, traces and discriminants.............................. 15 4. Rings of integers....................................... 20 ..."
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2. Number fields........................................ 9 3. Norms, traces and discriminants.............................. 15 4. Rings of integers....................................... 20
Chapter 9 Mathematical Models in PublicKey Cryptology
"... Chapter 8 has described several of the classical models of cryptography in which the decryption key was the same as or easily derivable from the encryption key. This meant that the corresponding encryption and decryption algorithms were closely related in the sense that one could be easily deduced f ..."
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Chapter 8 has described several of the classical models of cryptography in which the decryption key was the same as or easily derivable from the encryption key. This meant that the corresponding encryption and decryption algorithms were closely related in the sense that one could be easily deduced from the other. Such cryptographic systems are called symmetrickey or conventional systems, and their security relies exclusively on the secrecy of the keys. Other examples of privatekey systems are the Data Encryption Standard (DES) [24] and IDEA [12], in which users of the system who share a secret key can communicate securely over an unsecure channel. In all of the privatekey systems, two users who wish to correspond must have a common key before the communication starts, and in practice, establishing a common secret key can be expensive, difficult, and sometimes nearly impossible, especially in a large network where the users need not know each other. In 1976, Diffie and Hellman [7] introduced a revolutionary new concept