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Factorization of a 768bit RSA modulus
, 2010
"... This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA. ..."
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This paper reports on the factorization of the 768bit number RSA768 by the number field sieve factoring method and discusses some implications for RSA.
Analysis of Bernstein's Factorization Circuit
, 2002
"... In [1], Bernstein proposed a circuitbased implementation of the matrix step of the number field sieve factorization algorithm. These circuits o er an asymptotic cost reduction under the measure "construction cost × run time". We evaluate the cost of these circuits, in agreement ..."
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Cited by 16 (2 self)
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In [1], Bernstein proposed a circuitbased implementation of the matrix step of the number field sieve factorization algorithm. These circuits o er an asymptotic cost reduction under the measure "construction cost &times; run time". We evaluate the cost of these circuits, in agreement with [1], but argue that compared to previously known methods these circuits can factor integers that are 1.17 times larger, rather than 3.01 as claimed (and even this, only under the nonstandard cost measure).
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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Cited by 3 (0 self)
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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.
Reconfigurable hardware implementation of mesh routing in number field sieve factorization”, Special Purpose Hardware for Attacking Cryptographic Systems
 in the Number Field Sieve Factorization,” Proc. Field Programmable Technology Conf. (FPT’04
, 2005
"... Factorization of large numbers has been a constant source of interest in cryptanalysis. The fastest known algorithm for factoring large numbers is the Number Field Sieve (NFS). The two most time consuming phases of NFS are Sieving and Matrix Step. In this paper, we propose an efficient way of implem ..."
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Cited by 2 (2 self)
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Factorization of large numbers has been a constant source of interest in cryptanalysis. The fastest known algorithm for factoring large numbers is the Number Field Sieve (NFS). The two most time consuming phases of NFS are Sieving and Matrix Step. In this paper, we propose an efficient way of implementing the Matrix step in reconfigurable hardware. Our solution is based on the MeshRouting method proposed by Lenstra et al. We determine the practical size of a partial mesh that can fit in one FPGA device, Xilinx Virtex II XC2V8000. We further extrapolate the computation time for the case of a square systolic array of FPGAs for 512bit and 1024bit numbers' factorization. We demonstrate that for practical sizes of numbers used in cryptography, 1024 bits, the Matrix Step of factorization can be performed using 1024 Virtex II FPGAs in about 27 days. 1.
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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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.
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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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
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"... Constructing trapdoor primes for the proposed DSS A. K. LenstraConstructing trapdoor primes for the proposed DSS Abstract. In this note we present a method to construct primes p that satisfy the requirements of the recently proposed Digital Signature Standard, but for which the discrete logarithm pr ..."
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Constructing trapdoor primes for the proposed DSS A. K. LenstraConstructing trapdoor primes for the proposed DSS Abstract. In this note we present a method to construct primes p that satisfy the requirements of the recently proposed Digital Signature Standard, but for which the discrete logarithm problem in F can be expected to be easier than for general primes. Constructing trapdoor primes for the proposed DSS Recently, the National Institute of Standards and Technology (NIST) published a proposal for a Digital Signature Standard (DSS) [6]. In this proposed standard a Digital Signature Algorithm (DSA) is specified that would allow users to digitally sign data in such a way that unauthorized modifications to the data can be detected, and such that the identity of the signer of the data can be authenticated. For this purpose each signer has a public key consisting of four integers p, q, g and y, and a private key x that satisfy the following conditions: (i) p is a prime with 2511 <p < 2512; (ii) q is a prime divisor of p — 1 with 2 ’ < q < 2160;