scholarly journals A Random Permutation Model Arising in Chemistry

2008 ◽  
Vol 45 (04) ◽  
pp. 1060-1070
Author(s):  
Mark Brown ◽  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Random Permutation ◽  
Permutation Model ◽  
Single Cluster ◽  
Random Permutation Model

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.

scholarly journals A Random Permutation Model Arising in Chemistry

2008 ◽  
Vol 45 (4) ◽  
pp. 1060-1070 ◽  
Author(s):  
Mark Brown ◽  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Random Permutation ◽  
Permutation Model ◽  
Single Cluster ◽  
Random Permutation Model

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.

Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model

1996 ◽  
Vol 5 (4) ◽  
pp. 351-371 ◽  
Author(s):  
Robert P. Dobrow ◽  
James Allen Fill
Keyword(s):  
Probability Model ◽  
Mass Function ◽  
Random Permutation ◽  
Search Tree ◽  
Probability Mass Function ◽  
Search Trees ◽  
Asymptotic Expressions ◽  
Minimum Probability ◽  
Permutation Model ◽  
Random Permutation Model

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

scholarly journals On the Resilience of Even-Mansour to Invariant Permutations

2021 ◽  
Author(s):  
Bart Mennink ◽  
Samuel Neves
Keyword(s):  
Invariant Subspace ◽  
Block Cipher ◽  
Random Permutation ◽  
Cryptographic Primitives ◽  
Distinguishing Attack ◽  
Cryptographic Primitive ◽  
Permutation Model ◽  
Random Permutation Model

AbstractSymmetric cryptographic primitives are often exposed to invariances: deterministic relations between plaintexts and ciphertexts that propagate through the primitive. Recent invariant subspace attacks have shown that these can be a serious issue. One way to mitigate invariant subspace attacks is at the primitive level, namely by proper use of round constants (Beierle et al., CRYPTO 2017). In this work, we investigate how to thwart invariance exploitation at the mode level, namely by assuring that a mode never evaluates its underlying primitive under any invariance. We first formalize the use of invariant cryptographic permutations from a security perspective, and analyze the Even-Mansour block cipher construction. We further demonstrate how the model composes, and apply it to the keyed sponge construction. The security analyses exactly pinpoint how the presence of linear invariances affects the bounds compared with analyses in the random permutation model. As such, they give an exact indication how invariances can be exploited. From a practical side, we apply the derived security bounds to the case where the Even-Mansour construction is instantiated with the 512-bit ChaCha permutation, and derive a distinguishing attack against Even-Mansour-ChaCha in $$2^{128}$$ 2 128 queries, faster than the birthday bound. Comparable results are derived for instantiation using the 200-bit Keccak permutation without round constants (attack in $$2^{50}$$ 2 50 queries), the 1024-bit CubeHash permutation (attack in $$2^{256}$$ 2 256 queries), and the 384-bit Gimli permutation without round constants (attack in $$2^{96}$$ 2 96 queries). The attacks do not invalidate the security of the permutations themselves, but rather they demonstrate the tightness of our bounds and confirm that care should be taken when employing a cryptographic primitive that has nontrivial linear invariances.

scholarly journals Almost sure asymptotics for the random binary search tree

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Matthew Roberts
Keyword(s):  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Saturation Level ◽  
Binary Search Tree ◽  
International Audience ◽  
Permutation Model ◽  
Random Permutation Model

International audience We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.

scholarly journals Security of Symmetric Primitives against Key-Correlated Attacks

2019 ◽  
pp. 193-230
Author(s):  
Aisling Connolly ◽  
Pooya Farshim ◽  
Georg Fuchsbauer
Keyword(s):  
Natural Transformation ◽  
Random Oracle Model ◽  
Random Permutation ◽  
Random Oracle ◽  
Dependent Data ◽  
Authenticated Encryption ◽  
Ideal Cipher ◽  
Permutation Model ◽  
Ideal Cipher Model ◽  
The Ideal

We study the security of symmetric primitives against key-correlated attacks (KCA), whereby an adversary can arbitrarily correlate keys, messages, and ciphertexts. Security against KCA is required whenever a primitive should securely encrypt key-dependent data, even when it is used under related keys. KCA is a strengthening of the previously considered notions of related-key attack (RKA) and key-dependent message (KDM) security. This strengthening is strict, as we show that 2-round Even–Mansour fails to be KCA secure even though it is both RKA and KDM secure. We provide feasibility results in the ideal-cipher model for KCAs and show that 3-round Even–Mansour is KCA secure under key offsets in the random-permutation model. We also give a natural transformation that converts any authenticated encryption scheme to a KCA-secure one in the random-oracle model. Conceptually, our results allow for a unified treatment of RKA and KDM security in idealized models of computation.

scholarly journals CTET+: A Beyond-Birthday-Bound Secure Tweakable Enciphering Scheme Using a Single Pseudorandom Permutation

2021 ◽  
pp. 1-35
Author(s):  
Benoît Cogliati ◽  
Jordan Ethan ◽  
Virginie Lallemand ◽  
Byeonghak Lee ◽  
Jooyoung Lee ◽  
...  
Keyword(s):  
Block Cipher ◽  
Random Permutation ◽  
Pseudorandom Permutation ◽  
Linear Layer ◽  
Non Linear ◽  
Single Block ◽  
Permutation Model ◽  
Beyond Birthday Bound ◽  
Disk Encryption ◽  
Block Sizes

In this work, we propose a construction of 2-round tweakable substitutionpermutation networks using a single secret S-box. This construction is based on non-linear permutation layers using independent round keys, and achieves security beyond the birthday bound in the random permutation model. When instantiated with an n-bit block cipher with ωn-bit keys, the resulting tweakable block cipher, dubbed CTET+, can be viewed as a tweakable enciphering scheme that encrypts ωκ-bit messages for any integer ω ≥ 2 using 5n + κ-bit keys and n-bit tweaks, providing 2n/3-bit security.Compared to the 2-round non-linear SPN analyzed in [CDK+18], we both minimize it by requiring a single permutation, and weaken the requirements on the middle linear layer, allowing better performance. As a result, CTET+ becomes the first tweakable enciphering scheme that provides beyond-birthday-bound security using a single permutation, while its efficiency is still comparable to existing schemes including AES-XTS, EME, XCB and TET. Furthermore, we propose a new tweakable enciphering scheme, dubbed AES6-CTET+, which is an actual instantiation of CTET+ using a reduced round AES block cipher as the underlying secret S-box. Extensivecryptanalysis of this algorithm allows us to claim 127 bits of security.Such tweakable enciphering schemes with huge block sizes become desirable in the context of disk encryption, since processing a whole sector as a single block significantly worsens the granularity for attackers when compared to, for example, AES-XTS, which treats every 16-byte block on the disk independently. Besides, as a huge amount of data is being stored and encrypted at rest under many different keys in clouds, beyond-birthday-bound security will most likely become necessary in the short term.

Asymptotics with remainder term for moments of the total cycle number of random A-permutation

2021 ◽  
Vol 31 (1) ◽  
pp. 51-60
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Remainder Term ◽  
Random Permutation ◽  
Natural Numbers ◽  
Cycle Number ◽  
Fixed Set ◽  
Cycle Lengths

Abstract Dedicated to the memory of Alexander Ivanovich Pavlov. We consider the set of n-permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A-permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

scholarly journals Fe3 Cluster Anchored on the C2N Monolayer for Efficient Electrochemical Nitrogen Fixation

Catalysts ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 974
Author(s):  
Bing Han ◽  
Haihong Meng ◽  
Fengyu Li ◽  
Jingxiang Zhao
Keyword(s):  
Nitrogen Fixation ◽  
First Principles ◽  
Catalytic Performance ◽  
Reduction Reaction ◽  
First Principles Calculations ◽  
Ammonia Gas ◽  
Nitrogen Reduction ◽  
Single Cluster ◽  
Excellent Catalytic Performance ◽  
Production Of Ammonia

Under the current double challenge of energy and the environment, an effective nitrogen reduction reaction (NRR) has become a very urgent need. However, the largest production of ammonia gas today is carried out by the Haber–Bosch process, which has many disadvantages, among which energy consumption and air pollution are typical. As the best alternative procedure, electrochemistry has received extensive attention. In this paper, a catalyst loaded with Fe3 clusters on the two-dimensional material C2N (Fe3@C2N) is proposed to achieve effective electrochemical NRR, and our first-principles calculations reveal that the stable Fe3@C2N exhibits excellent catalytic performance for electrochemical nitrogen fixation with a limiting potential of 0.57 eV, while also suppressing the major competing hydrogen evolution reaction. Our findings will open a new door for the development of non-precious single-cluster catalysts for effective nitrogen reduction reactions.

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