27 research outputs found
A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm
We show a Birthday Paradox for self-intersections of Markov chains with
uniform stationary distribution. As an application, we analyze Pollard's Rho
algorithm for finding the discrete logarithm in a cyclic group and find
that if the partition in the algorithm is given by a random oracle, then with
high probability a collision occurs in steps. Moreover,
for the parallelized distinguished points algorithm on processors we find
that steps suffices. 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 .Comment: Published in at http://dx.doi.org/10.1214/09-AAP625 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
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 steps,
not far from the widely conjectured value of . This
improves upon a recent result of Miller--Venkatesan which showed an upper bound
of . Our proof is based on analyzing an appropriate
nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|,
and showing that the mixing time of the corresponding walk is
Collision bounds for the additive Pollard rho algorithm for solving discrete logarithms
We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group . 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 (||) by a factor of log|| and are based on mixing time estimates for random walks on finite abelian groups due to Dou and Hildebran
Collision Bounds for the Additive Pollard Rho Algorithm for Solving Discrete Logarithms
We prove collision bounds for the Pollard rho algorithm to solve the discrete logarithm problem in a general cyclic group . 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 by a factor of and are based on mixing time estimates for random walks on finite abelian groups due to Hildebrand
Stopping time signatures for some algorithms in cryptography
We consider the normalized distribution of the overall running times of some
cryptographic algorithms, and what information they reveal about the
algorithms. Recent work of Deift, Menon, Olver, Pfrang, and Trogdon has shown
that certain numerical algorithms applied to large random matrices exhibit a
characteristic distribution of running times, which depends only on the
algorithm but are independent of the choice of probability distributions for
the matrices. Different algorithms often exhibit different running time
distributions, and so the histograms for these running time distributions
provide a time-signature for the algorithms, making it possible, in many cases,
to distinguish one algorithm from another. In this paper we extend this
analysis to cryptographic algorithms, and present examples of such algorithms
with time-signatures that are indistinguishable, and others with
time-signatures that are clearly distinct.Comment: 20 page
Collision Times in Multicolor Urn Models and Sequential Graph Coloring With Applications to Discrete Logarithms
Consider an urn model where at each step one of colors is sampled
according to some probability distribution and a ball of that color is placed
in an urn. The distribution of assigning balls to urns may depend on the color
of the ball. Collisions occur when a ball is placed in an urn which already
contains a ball of different color. Equivalently, this can be viewed as
sequentially coloring a complete -partite graph wherein a collision
corresponds to the appearance of a monochromatic edge. Using a Poisson
embedding technique, the limiting distribution of the first collision time is
determined and the possible limits are explicitly described. Joint distribution
of successive collision times and multi-fold collision times are also derived.
The results can be used to obtain the limiting distributions of running times
in various birthday problem based algorithms for solving the discrete logarithm
problem, generalizing previous results which only consider expected running
times. Asymptotic distributions of the time of appearance of a monochromatic
edge are also obtained for other graphs.Comment: Minor revision. 35 pages, 2 figures. To appear in Annals of Applied
Probabilit