On random walks for Pollard's rho method

Author:
Edlyn Teske

Journal:
Math. Comp. **70** (2001), 809-825

MSC (2000):
Primary 11Y16; Secondary 65C05, 94A60

DOI:
https://doi.org/10.1090/S0025-5718-00-01213-8

Published electronically:
February 18, 2000

MathSciNet review:
1697652

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about faster than before.

**[AB82]**J. Arney and E. A. Bender,*Random mappings with constraints on coalescence and number of origins*, Pacific Journal of Mathematics**103**(1982), 269-294. MR**84h:05110****[BM]**S.R. Blackburn and S. Murphy,*The number of partitions in Pollard rho*, Private communication, May 1998.**[BP81]**R.P. Brent and J.M. Pollard,*Factorization of the eighth Fermat number*, Mathematics of Computation**36**(1981), 627-630. MR**83h:10014****[DH97]**J. J. Dai and M. V. Hildebrand,*Random random walks on the integers*, Statistics and Probability Letters**35**(1997), 371-379. MR**99f:60128****[GLV]**R. Gallant, R. Lambert, and S. Vanstone,*Improving the parallelized Pollard lambda search on binary anomalous curves*, to appear in*Mathematics of Computation*.**[Har60]**B. Harris,*Probability distributions related to random mappings*, Annals of Math. Statistics**31**(1960), 1045-1062. MR**22:9993****[Hil94]**M. V. Hildebrand,*Random walks supported on the random points of*, Probability Theory and Related Fields**100**(1994), 191-203. MR**95j:60015****[HV]**J. Horwitz and R. Venkatesan,*Random Cayley graphs and the discrete log*, Preprint, 1998.**[Kec93]**D. B. Kececioglu,*Reliability and life testing handbook*, vol. 1, Prentice Hall, Englewood Cliffs, New Jersey, 1993.**[Knu73]**D. E. Knuth,*The art of computer programming. volume 3: Sorting and searching*, Addison-Wesley, Reading, Massachusetts, 1973. MR**56:4281****[LiD97]**LiDIA Group, Technische Universität Darmstadt,*LiDIA - a library for computational number theory, version 1.3*, 1997, Available from http://www.informatik.tu-darmstadt.de/TI/LiDIA.**[MvOV96]**A. Menezes, P. van Oorschot, and S. A. Vanstone,*Handbook of applied cryptography*, CRC Press, 1996. MR**99g:94015****[PH78]**S. C. Pohlig and M. E. Hellman,*An improved algorithm for computing logarithms over and its cryptographic significance*, IEEE-Transactions on Information Theory**24**(1978), 106-110. MR**58:4617****[Pol]**J. M. Pollard, Private communications, March 1998, March 1999.**[Pol75]**J. M. Pollard,*A Monte Carlo method for factorization*, BIT**15**(1975), no. 3, 331-335. MR**52:13611****[Pol78]**J. M. Pollard,*Monte Carlo methods for index computation (mod )*, Mathematics of Computation**32**(1978), no. 143, 918-924. MR**58:10684****[SL84]**C. P. Schnorr and H. W. Lenstra, Jr.,*A Monte Carlo factoring algorithm with linear storage*, Mathematics of Computation**43**(1984), no. 167, 289-311. MR**85d:11106****[SSY82]**R. Sedgewick, T. G. Szymanski, and A. C. Yao,*The complexity of finding cycles in periodic functions*, SIAM J. Computing**11**(1982), no. 2, 376-390. MR**83f:68045****[Tes98a]**E. Teske,*New algorithms for finite abelian groups*, Ph.D. thesis, Technische Universität Darmstadt, Germany, 1998, Shaker Verlag, Aachen.**[Tes98b]**E. Teske,*A space efficient algorithm for group structure computation*, Mathematics of Computation**67**(1998), 1637-1663. MR**99a:11146****[Tes98c]**E. Teske,*Speeding up Pollard's rho method for computing discrete logarithms*, Algorithmic Number Theory Seminar ANTS-III, Lecture Notes in Computer Science, vol. 1423, Springer-Verlag, 1998, pp. 541-554.**[vOW99]**P. C. van Oorschot and M. J. Wiener,*Parallel collision search with cryptanalytic applications*, Journal of Cryptology**12**(1999), 1-28. MR**99i:49054****[WZ98]**M. Wiener and R. Zuccerato,*Faster attacks on elliptic curve cryptosystems*, Proceedings of SAC, Workshop on Selected Areas in Cryptography, Lecture Notes in Computer Science, 1998.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11Y16,
65C05,
94A60

Retrieve articles in all journals with MSC (2000): 11Y16, 65C05, 94A60

Additional Information

**Edlyn Teske**

Affiliation:
University of Waterloo, Department of Combinatorics and Optimization, Waterloo, Ontario, Canada N2L 3G1

Email:
eteske@cacr.math.uwaterloo.ca

DOI:
https://doi.org/10.1090/S0025-5718-00-01213-8

Keywords:
Pollard's rho method,
discrete logarithm,
random walks in groups

Received by editor(s):
February 23, 1999

Received by editor(s) in revised form:
May 24, 1999

Published electronically:
February 18, 2000

Article copyright:
© Copyright 2000
American Mathematical Society