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Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms

Joppe W. Bos, Alina Dudeanu, Dimitar Jetchev
2014 Journal of Mathematical Cryptology  
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.  ...  Acknowledgements We are grateful to Arjen Lenstra for the careful reading of the draft and for the numerous helpful comments and discussions.  ... 
doi:10.1515/jmc-2012-0032 fatcat:nb3bcshajzct7b24zx3z2bow6a

Random Walks Revisited: Extensions of Pollard's Rho Algorithm for Computing Multiple Discrete Logarithms [chapter]

Fabian Kuhn, René Struik
2001 Lecture Notes in Computer Science  
This paper extends the analysis of Pollard's rho algorithm for solving a single instance of the discrete logarithm problem in a finite cyclic group G to the case of solving more than one instance of the  ...  discrete logarithm problem in the same group G.  ...  As already explained, the time for finding a discrete logarithm with parallel Pollard-rho can be divided in two phases, the time until a collision occurs and the time needed for its detection.  ... 
doi:10.1007/3-540-45537-x_17 fatcat:adzsg6kjpbhs7na4b4ujokqq2a

Spectral Analysis of Pollard Rho Collisions [article]

Stephen D. Miller, Ramarathnam Venkatesan
2006 arXiv   pre-print
We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3).  ...  This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho graph, and is close to the conjectured optimal bound of O(sqrt(n)).  ...  Acknowledgements: the authors wish to thank R. Balasubramanian, Michael Ben-Or, Noam Elkies, David Jao, László Lovász, and Prasad Tetali for helpful discussions and comments.  ... 
arXiv:math/0603727v2 fatcat:dhjeetqv35a3zkh7sfikclikji

Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

Jeong Han Kim, Ravi Montenegro, Prasad Tetali
2007 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)  
We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G.  ...  It is found that, with high probability, a collision occurs in O(√(|G| |G| |G|)) steps, not far from the widely conjectured value of Θ(√(|G|)).  ...  Pollard suggested algorithms to help solve both factoring large integers [11] and the discrete logarithm problem [12] .  ... 
doi:10.1109/focs.2007.38 dblp:conf/focs/BarakM07 fatcat:xkobb74ryzb6zpq3yvlsejiyjq

Non-degeneracy of Pollard Rho Collisions [article]

Stephen D. Miller, Ramarathnam Venkatesan
2008 arXiv   pre-print
The Pollard Rho algorithm is a widely used algorithm for solving discrete logarithms on general cyclic groups, including elliptic curves.  ...  In this paper we show that for n satisfying a mild arithmetic condition, the collisions guaranteed by these results are nondegenerate with high probability: that is, the Pollard Rho algorithm successfully  ...  Introduction The Pollard ρ algorithm is, to date, the leading algorithm for solving discrete logarithm problems on general groups, including elliptic curves. The algorithm can be stated as follows.  ... 
arXiv:0808.0469v2 fatcat:pc46mdblbne6xcxfotqyn2gmuy

Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

Jeong Han Kim, Ravi Montenegro, Prasad Tetali
2007 Foundations of Computer Science (FOCS), IEEE Annual Symposium on  
We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G.  ...  It is found that, with high probability, a collision occurs in O( |G| log |G| log log |G|) steps, not far from the widely conjectured value of Θ( |G|).  ...  Pollard suggested algorithms to help solve both factoring large integers [11] and the discrete logarithm problem [12] .  ... 
doi:10.1109/focs.2007.4389494 fatcat:435sta3mj5e4fkvkumr3mes6q4

A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

Jeong Han Kim, Ravi Montenegro, Yuval Peres, Prasad Tetali
2010 The Annals of Applied Probability  
As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G and find that if the partition in the algorithm is given by a random oracle, then with high  ...  These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from G.  ...  Acknowledgments The authors thank S. Kijima, S. Miller, I. Mironov, R. Venkatesan and D. Wilson for several helpful discussions.  ... 
doi:10.1214/09-aap625 fatcat:7zifzornsjc2rpfoiuqf4dd6pm

Subset-Restricted Random Walks for Pollard rho Method on ${\mathbf{F}_{p^m}}$ [chapter]

Minkyu Kim, Jung Hee Cheon, Jin Hong
2009 Lecture Notes in Computer Science  
Hence our algorithm suggests that the order of subgroups, on which the pairing-based cryptosystems rely, needs to be increased by a factor of approximately m. keywords: discrete logarithm problem, pairing  ...  In this paper, we propose a variant of the Pollard rho method.  ...  We remark that, while we have achieved complexity lower than the straightforward application of Pollard rho, this does not conflict with the complexity lower bound known [18] for generic algorithms solving  ... 
doi:10.1007/978-3-642-00468-1_4 fatcat:ae33yhds3jfphijs3ogzq3zhpa

A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm [chapter]

Jeong Han Kim, Ravi Montenegro, Yuval Peres, Prasad Tetali
2008 Lecture Notes in Computer Science  
As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G and find that if the partition in the algorithm is given by a random oracle, then with high  ...  These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from G.  ...  Our intent in generalizing the Birthday Paradox was to bound the collision time of the Pollard Rho algorithm for discrete logarithm. As such, we briefly introduce the algorithm here.  ... 
doi:10.1007/978-3-540-79456-1_27 fatcat:nzgu6att2rathkm7g4i6n7mtky

Recent progress on the elliptic curve discrete logarithm problem

Steven D. Galbraith, Pierrick Gaudry
2015 Designs, Codes and Cryptography  
Keywords Elliptic curve discrete logarithm problem (ECDLP) · Summation polynomials · Pollard rho · Index calculus Introduction Let E be an elliptic curve over a finite field Fq, where q = p n and p is  ...  We survey recent work on the elliptic curve discrete logarithm problem. In particular we review index calculus algorithms using summation polynomials, and claims about their complexity.  ...  The second author also thanks Maike Massierer, Pierre-Jean Spaenlehauer and Vanessa Vitse for various discussions on the topic.  ... 
doi:10.1007/s10623-015-0146-7 fatcat:sh5w53c3hnbhzilz7jrht6fzmy

Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm

Fangguo Zhang, Ping Wang, Steven Galbraith
2017 Advances in Mathematics of Communications  
) and the Pollard rho method (for probabilistic algorithms).  ...  The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step-giant-step algorithm (BSGS) or Pollard rho.  ...  Acknowledgements The authors thank Siouxsie Wiles for assistance with the graphs.  ... 
doi:10.3934/amc.2017038 fatcat:xnlmk7eavzgnxayycn4zlt2tp4

Speeding Up the Pollard Rho Method on Prime Fields [chapter]

Jung Hee Cheon, Jin Hong, Minkyu Kim
2008 Lecture Notes in Computer Science  
The r-adding walk is an iterating function used with the Pollard rho algorithm and is known to require less iterations than Pollard's original iterating function in reaching a collision.  ...  In practice, our rudimentary implementation of the proposed method increased the speed of Pollard rho with r-adding walks by a factor of more than 10 for 1024-bit random primes p.  ...  This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant (No. R01-2008-000-11287-0).  ... 
doi:10.1007/978-3-540-89255-7_29 fatcat:45segiuulnclrhu575no7kuxti

Blockchain as cryptanalytic tool [article]

Manfred Lochter
2018 IACR Cryptology ePrint Archive  
We show how to substitute hashing by performing multiplications on Elliptic Curves in order to find distinguished points that can then be used to solve the discrete logarithm problem on a chosen curve.  ...  In principle all cryptanalytic algorithms that use Rabin's idea of distinguished points can be used in blockchain based attacks. Similar ideas can be used for the number field sieve.  ...  This choice of a small T does not influence the performance of the parallel Pollard-Rho algorithm.  ... 
dblp:journals/iacr/Lochter18 fatcat:n4m3hhgqjbgb3jq5jrms5fiwre

The Past, Evolving Present, and Future of the Discrete Logarithm [chapter]

Antoine Joux, Andrew Odlyzko, Cécile Pierrot
2014 Open Problems in Mathematics and Computational Science  
The first practical public key cryptosystem ever published, the Diffie-Hellman key exchange algorithm, relies for its security on the assumption that discrete logarithms are hard to compute.  ...  Since the introduction of the Diffie-Hellman key exchange more than three decades ago, there have been substantial algorithmic advances in the computation of discrete logarithms.  ...  Recall that as in Pollard Rho, we wish to find a collision of f in order to compute the desired discrete logarithm.  ... 
doi:10.1007/978-3-319-10683-0_2 fatcat:ykp3rhh2lnhn5ml4otrrfszoii

Computing discrete logarithms with the parallelized kangaroo method

Edlyn Teske
2003 Discrete Applied Mathematics  
The Pollard kangaroo method computes discrete logarithms in arbitrary cyclic groups.  ...  This makes the kangaroo method the most powerful method to solve the discrete logarithm problem in this situation.  ...  The author is grateful to the Mathematical Sciences Research Institute (MSRI) in Berkeley, CA for its hospitality for a month in the Fall 2000 during its program on Algorithmic Number Theory.  ... 
doi:10.1016/s0166-218x(02)00590-5 fatcat:jktt4mumnfbvnnu6dtbatof6qy
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