scholarly journals Constructions of Beyond-Birthday Secure PRFs from Random Permutations, Revisited

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1296
Author(s):  
Jiehui Nan ◽  
Ping Zhang ◽  
Honggang Hu
Keyword(s):  
Slight Modification ◽  
Random Permutation ◽  
Variable Length ◽  
Random Permutations ◽  
Pseudorandom Functions ◽  
New Construction ◽  
Beyond Birthday Bound

In CRYPTO 2019, Chen et al. showed how to construct pseudorandom functions (PRFs) from random permutations (RPs), and they gave one beyond-birthday secure construction from sum of Even-Mansour, namely SoEM22 in the single-key setting. In this paper, we improve their work by proving the multi-key security of SoEM22, and further tweaking SoEM22 but still preserving beyond birthday bound (BBB) security. Furthermore, we use only one random permutation to construct parallelizable and succinct beyond-birthday secure PRFs in the multi-key setting, and then tweak this new construction. Moreover, with a slight modification of our constructions of tweakable PRFs, two parallelizable nonce based MACs for variable length messages are obtained.

Generation of the symmetric group Sn2

2021 ◽  
pp. 2150107
Author(s):  
Carlos Zequeira Sánchez ◽  
Evaristo José Madarro Capó ◽  
Guillermo Sosa-Gómez
Keyword(s):  
Symmetric Group ◽  
Random Element ◽  
Matrix Representation ◽  
Random Permutation ◽  
Random Permutations ◽  
Dimension Formula ◽  
Indispensable Tool

In various scenarios today, the generation of random permutations has become an indispensable tool. Since random permutation of dimension [Formula: see text] is a random element of the symmetric group [Formula: see text], it is necessary to have algorithms capable of generating any permutation. This work demonstrates that it is possible to generate the symmetric group [Formula: see text] by shifting the components of a particular matrix representation of each permutation.

scholarly journals Random permutations fix a worst case for cyclic coordinate descent

2018 ◽  
Vol 39 (3) ◽  
pp. 1246-1275 ◽  
Author(s):  
Ching-pei Lee ◽  
Stephen J Wright
Keyword(s):  
Nonlinear Function ◽  
Quadratic Function ◽  
Random Permutation ◽  
Coordinate Descent ◽  
Random Permutations ◽  
Worst Case ◽  
Computational Performance ◽  
Technical Report ◽  
Cyclic Coordinate Descent ◽  
Better Than

Abstract Variants of the coordinate descent approach for minimizing a nonlinear function are distinguished in part by the order in which coordinates are considered for relaxation. Three common orderings are cyclic (CCD), in which we cycle through the components of $x$ in order; randomized (RCD), in which the component to update is selected randomly and independently at each iteration; and random-permutations cyclic (RPCD), which differs from CCD only in that a random permutation is applied to the variables at the start of each cycle. Known convergence guarantees are weaker for CCD and RPCD than for RCD, though in most practical cases, computational performance is similar among all these variants. There is a certain type of quadratic function for which CCD is significantly slower than for RCD; a recent paper by Sun & Ye (2016, Worst-case complexity of cyclic coordinate descent: $O(n^2)$ gap with randomized version. Technical Report. Stanford, CA: Department of Management Science and Engineering, Stanford University. arXiv:1604.07130) has explored the poor behavior of CCD on functions of this type. The RPCD approach performs well on these functions, even better than RCD in a certain regime. This paper explains the good behavior of RPCD with a tight analysis.

APPLYING BUCKET RANDOM PERMUTATIONS TO STATIONARY SEQUENCES WITH LONG-RANGE DEPENDENCE

Fractals ◽  
2007 ◽  
Vol 15 (02) ◽  
pp. 105-126 ◽  
Author(s):  
YINGCHUN ZHOU ◽  
MURAD S. TAQQU
Keyword(s):  
Long Range ◽  
Short Range ◽  
Random Permutation ◽  
Stationary Sequence ◽  
Long Range Dependence ◽  
Finite Variance ◽  
Dependence Structure ◽  
Random Permutations ◽  
Medium Range ◽  
The Time Domain

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

scholarly journals On the semantic security of cellular automata based pseudo-random permutation using results from the Luby-Rackoff construction

2015 ◽  
Vol 15 (1) ◽  
pp. 21
Author(s):  
Kamel Mohammed Faraoun
Keyword(s):  
Cellular Automata ◽  
Block Cipher ◽  
Random Permutation ◽  
Block Ciphers ◽  
Random Permutations ◽  
Semantic Security ◽  
Transition Rules ◽  
Reversible Cellular Automata ◽  
Symmetric Block ◽  
Number Of Iterations

This paper proposes a semantically secure construction of pseudo-random permutations using second-order reversible cellular automata. We show that the proposed construction is equivalent to the Luby-Rackoff model if it is built using non-uniform transition rules, and we prove that the construction is strongly secure if an adequate number of iterations is performed. Moreover, a corresponding symmetric block cipher is constructed and analysed experimentally in comparison with popular ciphers. Obtained results approve robustness and efficacy of the construction, while achieved performances overcome those of some existing block ciphers.

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

A Generalization of the Erdős–Turán Law for the Order of Random Permutation

2012 ◽  
Vol 21 (5) ◽  
pp. 715-733 ◽  
Author(s):  
ALEXANDER GNEDIN ◽  
ALEXANDER IKSANOV ◽  
ALEXANDER MARYNYCH
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Random Permutation ◽  
Unit Interval ◽  
Random Permutations ◽  
Cycle Structure ◽  
Limit Laws ◽  
Cycle Lengths

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

scholarly journals Separation Probabilities for Products of Permutations

2013 ◽  
Vol 23 (2) ◽  
pp. 201-222 ◽  
Author(s):  
OLIVIER BERNARDI ◽  
ROSENA R. X. DU ◽  
ALEJANDRO H. MORALES ◽  
RICHARD P. STANLEY
Keyword(s):  
Exact Formula ◽  
Random Permutation ◽  
Random Permutations ◽  
Mixing Properties

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements 1,2,. . .,k are in distinct cycles of the random permutation of {1,2,. . .,n} obtained as a product of two uniformly random n-cycles.

scholarly journals Patterns in Random Permutations Avoiding the Pattern 132

2016 ◽  
Vol 26 (1) ◽  
pp. 24-51 ◽  
Author(s):  
SVANTE JANSON
Keyword(s):  
Limit Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Brownian Excursion

We consider a random permutation drawn from the set of 132-avoiding permutations of lengthnand show that the number of occurrences of another pattern σ has a limit distribution, after scaling bynλ(σ)/2, where λ(σ) is the length of σ plus the number of descents. The limit is not normal, and can be expressed as a functional of a Brownian excursion. Moments can be found by recursion.

scholarly journals Local extrema in random permutations and the structure of longest alternating subsequences

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Dan Romik
Keyword(s):  
Joint Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Local Minima ◽  
New Approach ◽  
Successive Minimum ◽  
Local Extrema ◽  
Nous Montrons ◽  
Dimensional Distribution ◽  
International Audience

International audience Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$. Pour une permutation aléatoire d'ordre $n$, on désigne par $\textbf{as}_n$ la longueur maximale d'une de ses sous-suites alternantes. Stanley a étudié la distribution de $\textbf{as}_n$ en utilisant des méthodes algébriques, et il a démontré en particulier que $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ et $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. A partir du résultat de Stanley on peut montrer qu'après changement d'échelle, $\textbf{as}_n$ converge vers la distribution normale. Nous présentons ici une approche nouvelle pour l'étude de $\textbf{as}_n$, en la reliant à la suite des extrema locaux d'une permutation aléatoire, dont nous montrons qu'elle constitue une sous-suite alternante maximale "canonique''. En utilisant cette relation, nous prouvons à nouveau les résultats mentionnés ci-dessus d'une façon plus probabiliste et transparente. En plus, nous prouvons un résultat asymptotique sur la distribution limite des paires formées d'un minimum et d'un maximum locaux consécutifs.

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