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Parallel Solution of Sparse Linear Systems Defined over GF(p)
"... Introduction The security of modern publickey cryptography is usually based on the presumed hardness of problems such as factoring integers or computing discrete logarithms. The Number Field Sieve [19] (NFS) and Function Field Sieve [1] (FFS) oer two examples of algorithms that can attack these pr ..."
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Introduction The security of modern publickey cryptography is usually based on the presumed hardness of problems such as factoring integers or computing discrete logarithms. The Number Field Sieve [19] (NFS) and Function Field Sieve [1] (FFS) oer two examples of algorithms that can attack these problems. Such algorithms are generally speci ed in two phases. The rst phase, sometimes called the sieving step, aims to collect many relations that represent small items of information about the problem one is trying to solve. This phase is easy to parallelise since one can generate the relations independently. It is therefore attractive for distributed, Internet based collaborative computation [26]. The second phase of processing, sometimes called the matrix step, aims to collect the relations and combine them into a single linear system which, when solved, allows one to eciently compute answers to the original problem. Ecient implementation of the matrix step is challenging since the li
Solving Discrete Logarithms in SmoothOrder Groups with CUDA 1
"... This paper chronicles our experiences using CUDA to implement a parallelized variant of Pollard’s rho algorithm to solve discrete logarithms in groups with cryptographically large moduli but smooth order using commodity GPUs. We first discuss some key design constraints imposed by modern GPU archite ..."
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This paper chronicles our experiences using CUDA to implement a parallelized variant of Pollard’s rho algorithm to solve discrete logarithms in groups with cryptographically large moduli but smooth order using commodity GPUs. We first discuss some key design constraints imposed by modern GPU architectures and the CUDA framework, and then explain how we were able to implement efficient arbitraryprecision modular multiplication within these constraints. Our implementation can execute roughly 51.9 million 768bit modular multiplications per second — or a whopping 840 million 192bit modular multiplications per second — on a single Nvidia Tesla M2050 GPU card, which is a notable improvement over all previous results on comparable hardware. We leverage this fast modular multiplication in our implementation of the parallel rho algorithm, which can solve discrete logarithms modulo a 1536bit RSA number with a 2 55smooth totient in less than two minutes. We conclude the paper by discussing implications to discrete logarithmbased cryptosystems, and by pointing out how efficient implementations of parallel rho (or related algorithms) lead to trapdoor discrete logarithm groups; we also point out two potential cryptographic applications for the latter. Our code is written in C for CUDA and PTX; it is open source and freely available for download online. 1
Integer Factorization
, 1994
"... 6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer f ..."
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6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer factorisation algorithms. Parallel algorithms We would hope that an algorithm which required time T 1 on a computer with one processor could be implemented to run in time T P ¸ T 1 =P on a computer with P independent processors. This is not always the case, since it may be impossible to use all P processors effectively. However, it is true for many integer factorisation algorithms, provided that P is not too large. Integer factorization algorithms There are many algorithms for finding a nontrivial fac
Integer Factorization Summary
, 1994
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorization algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. However, the problem of integer factorization still appears difficult, both in a practical sense (for numbers of more than about 80 decimal digits), and in a theoretical sense (because none of the algorithms run in polynomial time). We outline several recent integer factorization algorithms, including the elliptic curve algorithm (ECM), the multiple polynomial quadratic sieve (MPQS), and the special/general number field sieve (NFS), give examples of their use, and mention some applications. Public key cryptography Large primes have at least one practical application – they can be used to construct public key
Integer Factorisation on the AP1000
, 1995
"... We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard "rho " method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vect ..."
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We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard "rho " method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures.
Factorisation of Large Integers on some Vector and Parallel Computers
 The Australian National University TRCS9501
, 1995
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Update 1 to Factorizations of a...
, 1994
"... In an earlier Report (NMR9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 ..."
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In an earlier Report (NMR9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 80 . In this Report we update the original tables. The factorizations are now complete for n 58, and there are no composite numbers smaller than 10 86 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 1104. Keywords and Phrases: Factor Tables. Appeared as Report NMR94??, Centrum voor Wiskunde en Informatica, Amsterdam, September 1994, 46 pp. Only the front matter is given here. Copyright c fl 1994, the authors. rpb134u1 typeset using T E X 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, i...
Update 1 to: Factorizations of a
, 1994
"... In an earlier Report (NMR9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 ..."
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In an earlier Report (NMR9212, June 1992), two of us gave tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfied a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations were complete for n 46, and the tables contained no composite numbers smaller than 10 80 . In this Report we update the original tables. The factorizations are now complete for n 58, and there are no composite numbers smaller than 10 86 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 1104. Keywords and Phrases: Factor Tables. 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n elements, and the multiplicative group of F has a n \Gamma 1 elements. Also, for prime a the sum of divisors of a n is oe(a n ) = (a n+1 ...
Factorizations of a^n±1, 13 ≤ a < 100
, 1992
"... As an extension of the "Cunningham" tables, we present tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfy a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations are complete for n 46, and the tables contain no composite numbers smaller than 10 80 . ..."
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As an extension of the "Cunningham" tables, we present tables of factorizations of a n \Sigma 1 for 13 a ! 100. The exponents n satisfy a n ! 10 255 if a ! 30, and n 100 if a 30. The factorizations are complete for n 46, and the tables contain no composite numbers smaller than 10 80 . 1991 Mathematics Subject Classification: Primary 11A25; Secondary 1104. Keywords and Phrases: Factor Tables. Appeared as Report NMR9212, Centrum voor Wiskunde en Informatica, Amsterdam, June 1992, 368 pp. Only the front matter is given here. Copyright c fl 1992, the authors. rpb134 typeset using T E X 1. Introduction For many years there has been an interest in the prime factors of numbers of the form a n \Sigma 1, where a is a moderately small integer (the base) and n is a positive exponent. Such numbers often arise. For example, if a is prime then there is a finite field F with a n elements, and the multiplicative group of F has a n \Gamma 1 elements. Also, for prime a the sum ...