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Truly Low-Space Element Distinctness and Subset Sum via Pseudorandom Hash Functions [article]

Lijie Chen, Ce Jin, R. Ryan Williams, Hongxun Wu
2021 arXiv   pre-print
We consider low-space algorithms for the classic Element Distinctness problem: given an array of n input integers with O(log n) bit-length, decide whether or not all elements are pairwise distinct.  ...  As a corollary, we also obtain a poly(n)-space O^*(2^0.86n)-time randomized algorithm for the Subset Sum problem, removing the random oracles required in the algorithm of Bansal, Garg, Nederlof, and Vyas  ...  Pseudorandom Hash Functions, the Dependency Tree, and the Indexing Scheme Next we describe our construction of pseudorandom hash functions h based on iterative restrictions.  ... 
arXiv:2111.01759v1 fatcat:j7gqly754rgndlcv3mdmwj6niy

Pseudorandom Bit Generators That Fool Modular Sums [chapter]

Shachar Lovett, Omer Reingold, Luca Trevisan, Salil Vadhan
2009 Lecture Notes in Computer Science  
We note that even for the case M = 3 the best previously known constructions were generators fooling general bounded-space computations, and required O(log 2 n) seed length.  ...  Our generalization handles a product of two distinct graphs with distinct bounds on their expansion.  ...  Thus, in each construction, we shall present two generators: one that is pseudorandom against low-weight sums, and one that is pseudorandom against highweight sums.  ... 
doi:10.1007/978-3-642-03685-9_46 fatcat:xafdsn5p2fcadib7fd7ayrd2ey

On the Round Security of Symmetric-Key Cryptographic Primitives [chapter]

Zulfikar Ramzan, Leonid Reyzin
2000 Lecture Notes in Computer Science  
A similar result can be obtained for message authentication codes based on universal hash functions.  ...  functions.  ...  The authors would like to thank Ron Rivest and Salil Vadhan for helpful discussions, and the anonymous referees for many detailed suggestions.  ... 
doi:10.1007/3-540-44598-6_24 fatcat:4gd5uluiu5cibhb5voh52fvhpu

Optimal streaming and tracking distinct elements with high probability [chapter]

Jarosław Błasiok
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
of distinct elements in the input.  ...  This settles completely the space complexity of the distinct elements problem with respect to all standard parameters.  ...  The author is especially grateful to Jelani Nelson for many inspiring and helpful discussions and comments.  ... 
doi:10.1137/1.9781611975031.156 dblp:conf/soda/Blasiok18 fatcat:iiak7lreingindwtupl4ueygvi

Locally Computable UOWHF with Linear Shrinkage

Benny Applebaum, Yoni Moses
2016 Journal of Cryptology  
We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs) H : {0, 1} n → {0, 1} m .  ...  Our construction is based on the one-wayness of "random" local functions -a variant of an assumption made by Goldreich (ECCC 2000).  ...  We thank Uri Feige and Danny Vilenchik for valuable discussions.  ... 
doi:10.1007/s00145-016-9232-x fatcat:o6n6pjucuremla3d2csjrnihty

Pseudorandomness via the Discrete Fourier Transform

Parikshit Gopalan, Daniel M. Kane, Raghu Meka
2018 SIAM journal on computing (Print)  
We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs.  ...  We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the  ...  Taking the pointwise sum of such generators modulo m gives a family of hash functions that is both δ-biased and k-wise independent generated from a seed of length s = O(log(n/δ) + k log(nm)). A.  ... 
doi:10.1137/16m1062132 fatcat:rzvoaq2fkzf3xatrjqrx5v6k5e

Perfect hashing

Zbigniew J. Czech, George Havas, Bohdan S. Majewski
1997 Theoretical Computer Science  
Poor pseudorandom functions that do not yield distinct triples when used with sets of several hundred keys.  ...  The number of distinct subsets of U of size n, when the order of elements counts, is U! ___ =n! u (u -n)!  ...  Notation index This appendix describes the notation and symbols that are used consistently throughout the work.  ... 
doi:10.1016/s0304-3975(96)00146-6 fatcat:htaph24frffzjevl2q6kxz467e

On the distribution of the number of roots of polynomials and explicit weak designs

Tzvika Hartman, Ran Raz
2003 Random structures & algorithms (Print)  
These constructions are explicit in the sense that they require time and space polynomial in the number of subsets.  ...  However, the constructions require time and space polynomial in the number of subsets even when needed to output only one specific subset out of the collection.  ...  ACKNOWLEDGMENTS We would like to thank Omer Reingold and Ronen Shaltiel for some fruitful discussions. We also gratefully thank Salil Vadhan and the anonymous referees for their invaluable comments.  ... 
doi:10.1002/rsa.10095 fatcat:joei6ebkefhetbuikkm5kxdrsa

SPRING: Fast Pseudorandom Functions from Rounded Ring Products [chapter]

Abhishek Banerjee, Hai Brenner, Gaëtan Leurent, Chris Peikert, Alon Rosen
2015 Lecture Notes in Computer Science  
Recently, Banerjee, Peikert and Rosen (EUROCRYPT 2012) proposed new theoretical pseudorandom function candidates based on "rounded products" in certain polynomial rings, which have rigorously provable  ...  In this work we give two concrete and practically efficient instantiations of the BPR design, which we call SPRING, for "subset-product with rounding over a ring."  ...  Informally, over the choice of a random (and secret) key that is used for all inputs, a PRF cannot be efficiently distinguished from a truly random function via adaptive oracle (i.e., "black-box") access  ... 
doi:10.1007/978-3-662-46706-0_3 fatcat:dkweldpsoffixdfttdglsuljha

Locally Computable UOWHF with Linear Shrinkage [chapter]

Benny Applebaum, Yoni Moses
2013 Lecture Notes in Computer Science  
We study the problem of constructing locally computable Universal One-Way Hash Functions (UOWHFs) H : {0, 1} n → {0, 1} m .  ...  to UOWHFs [21, 24, 15] are inherently non-local as they employ primitives such as k-wise independent hash functions which cannot be computed locally. 2 When applied to local functions, the AIK compiler  ...  We thank Uri Feige and Danny Vilenchik for valuable discussions.  ... 
doi:10.1007/978-3-642-38348-9_29 fatcat:zhgfefzzyjdela747zuhtzr46e

Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs

László Babai, Noam Nisant, Márió Szegedy
1992 Journal of computer and system sciences (Print)  
We give a tight time-space trade-off of the form TS = @(n'), for general, k-head Turing machines; the bounds hold for a function that can be computed in linear time and constant space by a k + l-head Turing  ...  The . time-space and related trade-off results mentioned above are not affected by this development.  ...  Recently Nisan [Ni2] constructed stronger pseudorandom generators based on universal families of hash functions.  ... 
doi:10.1016/0022-0000(92)90047-m fatcat:e4mxtxumwjhh3btjtgng2itrdy

Multi-Input Correlation-Intractable Hash Functions via Shift-Hiding [article]

Alex Lombardi, Vinod Vaikuntanathan
2020 IACR Cryptology ePrint Archive  
We construct single-input CI hash functions from indistinguishability obfuscation (iO) and one-way permutations.  ...  We give a conceptually simple and generic construction of single-input CI hash functions from shift-hiding shiftable functions (Peikert and Shiehian, PKC 2018) satisfying an additional one-wayness property  ...  Acknowledgements We thank an anonymous reviewer for pointing out that the [PS19] hash function can likely also be shown to satisfy multi-input CI for shifted sum relations.  ... 
dblp:journals/iacr/LombardiV20a fatcat:73wpvsfkfbcvpch4ikbwrtu6je

Pseudorandomness for Approximate Counting and Sampling

Ronen Shaltiel, Christopher Umans
2006 Computational Complexity  
We use the "boosting" theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM.  ...  An (n, s, )-discrepancy set is a subset T ⊆ {0, 1} n with the property that for all Boolean circuits C of size at most s: Pr x C(x) = 1 − Pr t∈T C(t) = 1 ≤ . cc 15 (2007) Pseudorandomness for approx. counting  ...  We also thank the anonymous referees for numerous helpful comments and suggestions.  ... 
doi:10.1007/s00037-007-0218-9 fatcat:cfndygmw6jc2lhb6obaznpzlfu

Pseudorandomness via the discrete Fourier transform [article]

Parikshit Gopalan, Daniel Kane, Raghu Meka
2015 arXiv   pre-print
We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs.  ...  We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the  ...  For n, m, δ > 0 we say that a family of hash functions H = {h : [n] → [m]} is δ-biased if for any r ≤ n distinct indices i 1 , i 2 , . . . , i r ∈ [n] and j 1 , . . . , j r ∈ [m], Pr h∈uH [h(i 1 ) = j  ... 
arXiv:1506.04350v2 fatcat:itgcwf3nmjh7fnfganlvgi3vwu

The SPHINCS+ Signature Framework

Daniel J. Bernstein, Andreas Hülsing, Stefan Kölbl, Ruben Niederhagen, Joost Rijneveld, Peter Schwabe
2019 Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security - CCS '19  
Our second main contribution is the introduction of tweakable hash functions and a demonstration how they allow for a unified security analysis of hash-based signature schemes.  ...  KEYWORDS Post-quantum cryptography, SPHINCS, hash-based signatures, stateless, tweakable hash functions, NIST PQC, exact security of [10] made some choices.  ...  Pseudorandom functions and the message digest.  ... 
doi:10.1145/3319535.3363229 dblp:conf/ccs/BernsteinHKNRS19 fatcat:izvoarblrjgndd7fzildws7gny
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