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Speeding Up Pollard's Rho Method For Computing Discrete Logarithms
, 1998
"... . In Pollard's rho method, an iterating function f is used to define a sequence (y i ) by y i+1 = f(y i ) for i = 0; 1; 2; : : : , with some starting value y 0 . In this paper, we define and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their pe ..."
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Cited by 44 (7 self)
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. In Pollard's rho method, an iterating function f is used to define a sequence (y i ) by y i+1 = f(y i ) for i = 0; 1; 2; : : : , with some starting value y 0 . In this paper, we define and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their performances in experiments with elliptic curve groups. Our experiments show that one of our newly defined functions is expected to reduce the number of steps by a factor of approximately 0:8, in comparison with Pollard's originally used function, and we show that this holds independently of the size of the group order. For group orders large enough such that the run time for precomputation can be neglected, this means a realtime speedup of more than 1:2. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. Given an element h in G, we wish to find the least nonnegative number x such that g x = h. This problem is the discre...
Random walks on finite groups
 Encyclopaedia of Mathematical Sciences
, 2004
"... Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time ..."
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Cited by 20 (2 self)
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Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cutoff phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great
Random Lazy Random Walks on Arbitrary Finite Groups
 J. Theoret. probab
, 2000
"... This paper considers "lazy" random walks supported on a random subset of k elements of a finite group G with order n. If k = da log 2 ne where a ? 1 is constant, then most such walks take no more than a multiple of log 2 n steps to get close to uniformly distributed on G. If k = log 2 n + f(n) whe ..."
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Cited by 3 (2 self)
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This paper considers "lazy" random walks supported on a random subset of k elements of a finite group G with order n. If k = da log 2 ne where a ? 1 is constant, then most such walks take no more than a multiple of log 2 n steps to get close to uniformly distributed on G. If k = log 2 n + f(n) where f(n) ! 1 and f(n)= log 2 n ! 0 as n ! 1, then most such walks take no more than a multiple of (log 2 n) ln(log 2 n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdos and R'enyi and of Pak. Key words: Random walks, finite groups, uniform distribution. 1
A SIGNED GENERALIZATION OF THE BERNOULLI–LAPLACE DIFFUSION MODEL
, 2000
"... We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (Z2 ≀ Sn)/(Sr × Sn−r). Specifically, for r not too far from n/2, we determine that, to first order in n, 1 4n l ..."
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Cited by 1 (0 self)
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We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (Z2 ≀ Sn)/(Sr × Sn−r). Specifically, for r not too far from n/2, we determine that, to first order in n, 1 4n logn steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the “cutoff phenomenon.” 1. Introduction. Consider the classical Bernoulli–Laplace model for the diffusion of gases through a membrane, in which at each step two randomly chosen balls from different urns are switched. How many steps does it take for this process to achieve nearrandomness? This question was answered by Diaconis and Shahshahani (1987). Suppose that the balls also have charges and that, at each step, the two balls are not only switched, but their
Generating Random Vectors in (Z/pZ) d Via an Affine Random Process
, 2008
"... This paper considers some random processes of the form Xn+1 = TXn + Bn (mod p) where Bn and Xn are random variables over (Z/pZ) d and T is a fixed dxd integer matrix which is invertible over the complex numbers. For a particular distribution for Bn, this paper improves results of Asci to show that i ..."
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This paper considers some random processes of the form Xn+1 = TXn + Bn (mod p) where Bn and Xn are random variables over (Z/pZ) d and T is a fixed dxd integer matrix which is invertible over the complex numbers. For a particular distribution for Bn, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p) 2 steps suffice to make Xn close to uniformly distributed where X0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p b steps are needed for Xn to get close to uniform where b is a value which may depend on T and X0 is the zero vector. 1 1
A Survey of Results on Random Random Walks on Finite Groups
, 2008
"... A number of papers have examined various aspects of “random random walks ” on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some ..."
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Cited by 1 (0 self)
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A number of papers have examined various aspects of “random random walks ” on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this work. This article also provides a number of exercises. Some exercises involve straightforward computations; others involve proving details in proofs or extending results proved in the article. This article also describes some problems for further study. 1
COLLISION BOUNDS FOR THE ADDITIVE POLLARD RHO ALGORITHM FOR SOLVING DISCRETE LOGARITHMS
"... Abstract. We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group G. Unlike the setting studied by Kim et al. we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem. Our bounds ..."
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Abstract. We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group G. Unlike the setting studied by Kim et al. we consider additive walks: the setting used in practice to solve the elliptic curve discrete logarithm problem. Our bounds differ from the birthday bound O ( √ G) by a factor of √ log G  and are based on mixing time estimates for random walks on finite abelian groups due to Hildebrand. 1.