Probability Distributions Related to Random Mappings

Harris, Bernard

doi:10.1214/aoms/1177705677

Cited by 213 publications

(164 citation statements)

References 4 publications

Supporting

Mentioning

128

Contrasting

Unclassified

Order By: Relevance

“…We call λ the period and µ the preperiod of the sequence (w k ). Under the assumption that w 0 ∈ R W and F is a random mapping, the expected values for µ and λ are close to π|W |/8 = 0.626... |W | ( [Har60]). A pair (w i , w j ) of two terms of the sequence is called a match if w i = w j and i < j.…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

On random walks for Pollard's rho method

Teske

2000

Math. Comp.

View full text Add to dashboard Cite

Abstract. 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 20% faster than before.

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…Let x = (λ + µ)/ |G|. In the case of a random random walk, the probability density function of x is given by f (x) = xe Har60]). This function belongs to a certain Weibull distribution; such distributions are extensively studied in reliability engineering (see, for example, [Kec93]).…”

Section: Reliability Considerationsmentioning

confidence: 99%

On random walks for Pollard's rho method

Teske

2000

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…Researchers have carried out extensive studies on the distribution of S λ as N → ∞. In particular, Harris proved that the mean value of S 0 is πN/2 [11], which is consistent with Theorem 1 as the number of the cycle nodes. After that, Mutafchiev [16] proved the following theorem as an extension of Harris's result.…”

Section: The Height Property Of a Node In A Functional Graphmentioning

confidence: 52%

Generic Universal Forgery Attack on Iterative Hash-Based MACs

Peyrin

Wang

2014

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. In this article, we study the security of iterative hash-based MACs, such as HMAC or NMAC, with regards to universal forgery attacks. Leveraging recent advances in the analysis of functional graphs built from the iteration of HMAC or NMAC, we exhibit the very first generic universal forgery attack against hash-based MACs. In particular, our work implies that the universal forgery resistance of an n-bit output HMAC construction is not 2 n queries as long believed by the community. The techniques we introduce extend the previous functional graphs-based attacks that only took in account the cycle structure or the collision probability: we show that one can extract much more meaningful secret information by also analyzing the distance of a node from the cycle of its component in the functional graph.

show abstract

“…A pair (X i , X j ) of two elements of the sequence is called a match if X i = X j where i ̸ = j. Under the assumption that an iteration function F : G → G behaves like a truly random mapping and the initial value X 0 is a randomly chosen group element, the expected number of evaluations before a match appears is E(µ + λ) = √ π|G|/2 [11]. Pollard rho method can be easily generalized to compute discrete logarithms in arbitrary finite abelian groups, such as in groups of points of elliptic curves over finite fields.…”

Section: Pollard Rho Methodsmentioning

confidence: 99%

Computing elliptic curve discrete logarithms with the negation map

Wang¹,

Zhang²

2012

Information Sciences

View full text Add to dashboard Cite

Abstract. It is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the Pollard rho method. However, the random walks defined on elliptic curve points equivalence class {±P } used by Pollard rho will always get trapped in fruitless cycles. We propose an efficient alternative approach to resolve fruitless cycles. Besides the theoretical analysis, we also examine the performance of the new algorithm in experiments with elliptic curve groups. The experiment results show that we can achieve the speedup by a factor extremely close to √ 2, which is the best performance one can achieve in theory, using the new algorithm with the negation map.

show abstract

Probability Distributions Related to Random Mappings

Cited by 213 publications

References 4 publications

On random walks for Pollard's rho method

On random walks for Pollard's rho method

Generic Universal Forgery Attack on Iterative Hash-Based MACs

Computing elliptic curve discrete logarithms with the negation map

Contact Info

Product

Resources

About