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

2014
*
Journal of Mathematical Cryptology
*

We prove

doi:10.1515/jmc-2012-0032
fatcat:nb3bcshajzct7b24zx3z2bow6a
*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. ...##
###
Random Walks Revisited: Extensions of Pollard's Rho Algorithm for Computing Multiple Discrete Logarithms
[chapter]

2001
*
Lecture Notes in Computer Science
*

This paper extends

doi:10.1007/3-540-45537-x_17
fatcat:adzsg6kjpbhs7na4b4ujokqq2a
*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. ...##
###
Spectral Analysis of Pollard Rho Collisions
[article]

2006
*
arXiv
*
pre-print

We show that

arXiv:math/0603727v2
fatcat:dhjeetqv35a3zkh7sfikclikji
*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. ...##
###
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

2007
*
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
*

We analyze a fairly standard idealization of Pollard's

doi:10.1109/focs.2007.38
dblp:conf/focs/BarakM07
fatcat:xkobb74ryzb6zpq3yvlsejiyjq
*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] . ...##
###
Non-degeneracy of Pollard Rho Collisions
[article]

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. ...

##
###
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log

2007
*
Foundations of Computer Science (FOCS), IEEE Annual Symposium on
*

We analyze a fairly standard idealization of Pollard's

doi:10.1109/focs.2007.4389494
fatcat:435sta3mj5e4fkvkumr3mes6q4
*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] . ...##
###
A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm

2010
*
The Annals of Applied Probability
*

As an application, we analyze Pollard's

doi:10.1214/09-aap625
fatcat:7zifzornsjc2rpfoiuqf4dd6pm
*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. ...##
###
Subset-Restricted Random Walks for Pollard rho Method on ${\mathbf{F}_{p^m}}$
[chapter]

2009
*
Lecture Notes in Computer Science
*

Hence our

doi:10.1007/978-3-642-00468-1_4
fatcat:ae33yhds3jfphijs3ogzq3zhpa
*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*...##
###
A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm
[chapter]

2008
*
Lecture Notes in Computer Science
*

As an application, we analyze Pollard's

doi:10.1007/978-3-540-79456-1_27
fatcat:nzgu6att2rathkm7g4i6n7mtky
*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. ...##
###
Recent progress on the elliptic curve discrete logarithm problem

2015
*
Designs, Codes and Cryptography
*

Keywords Elliptic curve

doi:10.1007/s10623-015-0146-7
fatcat:sh5w53c3hnbhzilz7jrht6fzmy
*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. ...##
###
Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm

2017
*
Advances in Mathematics of Communications
*

) and

doi:10.3934/amc.2017038
fatcat:xnlmk7eavzgnxayycn4zlt2tp4
*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. ...##
###
Speeding Up the Pollard Rho Method on Prime Fields
[chapter]

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). ...

##
###
Blockchain as cryptanalytic tool
[article]

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

dblp:journals/iacr/Lochter18
fatcat:n4m3hhgqjbgb3jq5jrms5fiwre
*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*. ...##
###
The Past, Evolving Present, and Future of the Discrete Logarithm
[chapter]

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*. ...

##
###
Computing discrete logarithms with the parallelized kangaroo method

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. ...

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