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

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]**James Arney and Edward A. Bender,*Random mappings with constraints on coalescence and number of origins*, Pacific J. Math.**103**(1982), no. 2, 269–294. MR**705228****[BM]**S.R. Blackburn and S. Murphy,*The number of partitions in Pollard rho*, Private communication, May 1998.**[BP81]**Richard P. Brent and John M. Pollard,*Factorization of the eighth Fermat number*, Math. Comp.**36**(1981), no. 154, 627–630. MR**606520**, 10.1090/S0025-5718-1981-0606520-5**[DH97]**Jack J. Dai and Martin V. Hildebrand,*Random random walks on the integers mod 𝑛*, Statist. Probab. Lett.**35**(1997), no. 4, 371–379. MR**1483024**, 10.1016/S0167-7152(97)00035-7**[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]**Bernard Harris,*Probability distributions related to random mappings*, Ann. Math. Statist.**31**(1960), 1045–1062. MR**0119227****[Hil94]**Martin Hildebrand,*Random walks supported on random points of 𝐙/𝐧𝐙*, Probab. Theory Related Fields**100**(1994), no. 2, 191–203. MR**1296428**, 10.1007/BF01199265**[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]**Donald E. Knuth,*The art of computer programming. Volume 3*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing. MR**0445948****[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]**Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone,*Handbook of applied cryptography*, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1997. With a foreword by Ronald L. Rivest. MR**1412797****[PH78]**Stephen C. Pohlig and Martin E. Hellman,*An improved algorithm for computing logarithms over 𝐺𝐹(𝑝) and its cryptographic significance*, IEEE Trans. Information Theory**IT-24**(1978), no. 1, 106–110. MR**0484737****[Pol]**J. M. Pollard, Private communications, March 1998, March 1999.**[Pol75]**J. M. Pollard,*A Monte Carlo method for factorization*, Nordisk Tidskr. Informationsbehandling (BIT)**15**(1975), no. 3, 331–334. MR**0392798****[Pol78]**J. M. Pollard,*Monte Carlo methods for index computation (𝑚𝑜𝑑𝑝)*, Math. Comp.**32**(1978), no. 143, 918–924. MR**0491431**, 10.1090/S0025-5718-1978-0491431-9**[SL84]**C.-P. Schnorr and H. W. Lenstra Jr.,*A Monte Carlo factoring algorithm with linear storage*, Math. Comp.**43**(1984), no. 167, 289–311. MR**744939**, 10.1090/S0025-5718-1984-0744939-5**[SSY82]**Robert Sedgewick, Thomas G. Szymanski, and Andrew C. Yao,*The complexity of finding cycles in periodic functions*, SIAM J. Comput.**11**(1982), no. 2, 376–390. MR**652911**, 10.1137/0211030**[Tes98a]**E. Teske,*New algorithms for finite abelian groups*, Ph.D. thesis, Technische Universität Darmstadt, Germany, 1998, Shaker Verlag, Aachen.**[Tes98b]**Edlyn Teske,*A space efficient algorithm for group structure computation*, Math. Comp.**67**(1998), no. 224, 1637–1663. MR**1474658**, 10.1090/S0025-5718-98-00968-5**[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