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For a random function with |Y|=N and |X|=aN, the probability that a random y∈Y is a j-node is q^{(aN)}(j)≈(1/e^a)(a^j/j!), and the probability that f(x) for random x∈X is a j-node is r^{(aN)}(j)≈j q^{(aN)}(j).
“For y $ ←Y , the probability of y to be a j-node, denoted by q (a N ) : Z ≥0 → [0, 1], is q (a N ) ( j) ≈ 1 e a • a j j! . (5 ) For x $ ←X , the probability of f (x) to be a j-node, denoted by r (a N ) : N → [0, 1], is r (a N ) ( j) ≈ j • q (a N ) ( j). (6 )”
From Time–space complexity of quantum search algorithms in symmetric cryptanalysis: applying to AES and SHA-2 (2018) · cites this paper as [29]
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Quantum Time–Space Tradeoff for Finding Multiple Collision Pairs
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Low-Gate Quantum Golden Collision Finding
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Functional Graph Revisited: Updates on (Second) Preimage Attacks on Hash Combiners
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On random mappings and random permutations
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- Authors
- Flajolet, P., & Odlyzko, A. M.
- Affiliations
- Institut national de recherche en sciences et technologies du numérique (FR)
- AT&T (United States) (US)
- Venue
- Lecture Notes in Computer Science
- Publisher
- Springer Berlin Heidelberg
- Published
- 2007-11-15
- Pages
- 329-354
- OpenAlex
- W2226303357
- Cited by
- 274
- References
- 42
- Citations by year
- 2026: 1 · 2025: 4 · 2024: 5 · 2023: 7 · 2022: 6 · 2021: 6 · 2020: 9 · 2019: 15 · 2018: 12 · 2017: 11 · 2016: 15 · 2015: 15 · 2014: 17 · 2013: 11 · 2012: 13
- FWCI
- 12.13
- Type
- conference paper
- Language
- en
- Open access
- green
- inria.hal.science
- Licence
- other-oa
- Locations
- Springer eBooks (publishedVersion)
- HAL (submittedVersion, other-oa)
- Topics
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Computer Science › Artificial Intelligence › Algorithms and Data Compression
- Keywords
- random maps, finite sets, generating functions, singularity analysis, characteristic parameters, symbolic computation, random number generation, algorithm analysis
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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