Permutation Properties Research Articles

On Some Permutation Trinomials in Characteristic Three

In this paper, we determine the permutation properties of the polynomial x^3+x^q+2−x^4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x^4q−1 +x^2q+1 ±x^3. In particular, we first show that x^3 +x^q+2 −x^4q−1 permutes F_q^2 with q = 3^m if and only if m is odd. This enables us to show that the sufficient condition in [35, Theorem 4] is also necessary. Next, we prove that x^4q−1 + x^2q+1 − x^3 permutes F_q^2 with q = 3^m if and only if m is congruent to 0 modulo 4. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x^4q−1 +x^2q+1 +x^3 and show that it is never a permutation polynomial of F_q^2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.

Functional repeated measures analysis of variance and its application

This paper is inspired by medical studies in which the same patients with multiple sclerosis are examined at several successive visits (doctor’s appointments) and described by fractional anisotropy tract profiles, which can be represented as f unctions. Since the observations for each patient are dependent random processes, they follow a repeated measures design for functional data. To compare the results for different visits, we thus consider functional repeated measures analysis of variance. For this purpose, a pointwise test statistic is constructed by adapting the classical test statistic for one-way repeated measures analysis of variance to the functional data framework. By integrating and taking the supremum of the pointwise test statistic, we create two global test statistics. In addition to verifying the general null hypothesis of the equality of mean functions corresponding to different objects, we also propose a simple method for post hoc analysis. We illustrate the finite sample properties of permutation and bootstrap testing procedures in an extensive simulation study. Finally, we analyze a real data example in detail. All methods are implemented in the R package rmfanova, available on CRAN.

Permutation-adapted complete and independent basis for atomic cluster expansion descriptors

Atomic cluster expansion (ACE) methods provide a systematic way to describe particle local environments of arbitrary body order. For practical applications it is often required that the basis of cluster functions be symmetrized with respect to rotations and permutations. Existing methodologies yield sets of symmetrized functions that are over-complete. These methodologies thus require an additional numerical procedure, such as singular value decomposition (SVD), to eliminate redundant functions. In this work, it is shown that analytical linear relationships for subsets of cluster functions may be derived using recursion and permutation properties of generalized Wigner symbols. From these relationships, subsets (blocks) of cluster functions can be selected such that, within each block, functions are guaranteed to be linearly independent. It is conjectured that this block-wise independent set of permutation-adapted rotation and permutation invariant (PA-RPI) functions forms a complete, independent basis for ACE. Along with the first analytical proofs of block-wise linear dependence of ACE cluster functions and other theoretical arguments, numerical results are offered to demonstrate this. The utility of the method is demonstrated in the development of an ACE interatomic potential for tantalum. Using the new basis functions in combination with Bayesian compressive sensing sparse regression, some high degree descriptors are observed to persist and help achieve high-accuracy models.

Geometric Algebra Jordan-Wigner Transformation for Quantum Simulation.

Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan-Wigner transformation (JWT) and the Bravyi-Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework.

Constructing permutation polynomials from permutation polynomials of subfields

In this paper we study the permutational property of polynomials of the form f(L(x))+k(L(x))⋅M(x)∈Fqn[x] over the finite field Fqn, where L,M∈Fq[x] are q-linearized polynomials and k∈Fqn[x] satisfies a generic condition. We specialize in the case where L(x) is the linearized q-associate of gt,a(x)=(xn−1)/(xt−a), t is a divisor of n and a∈Fq satisfies an/t=1. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of Fqn from permutations of Fqt, by simply solving a system of independent equations of the form Trqn/qt(δi−1ai)=ci, where the ai's are the coefficients of f. In fact, the same method can be applied to construct complete mappings of Fqn from complete mappings of Fq.

B\"{a}cklund transformations as integrable discretization. The geometric approach

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that B\"{a}cklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

Constructive aspects of Riemann’s permutation theorem for series

Abstract The notions of permutable and weak-permutable convergence of a series $\sum _{n=1}^{\infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD- $\mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $\mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $\mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.

A class of permutation quadrinomials over finite fields

Let denote the finite field of order q. In this paper, we investigate the permutation property of the quadrinomial over , where q is odd and . More precisely, we obtain the necessary and sufficient conditions for to be a permutation polynomial over .

Several classes of permutation pentanomials with the form [formula omitted] over [formula omitted

In this paper, we study the permutation property of pentanomials with the form xrh(xpm−1) over Fp2m, where p∈{2,3}. More precisely, based on some seventh-degree and eighth-degree irreducible pentanomials over F2, we present eight classes of permutation pentanomials over F22m by determining the solutions of some equations with low degrees. In addition, based on the investigation of algebraic curves associated with fractional polynomials over finite fields, eight classes of permutation pentanomials over F32m are discovered by choosing some seventh-degree irreducible pentanomials over F3. Finally, several classes of permutation pentanomials and heptanomials over F22m and F32m are derived from known permutation polynomials on μ2m+1 and μ3m+1, respectively, where μd is the set of d-th roots of unity.

Automatic Preimage Attack Framework on Ascon Using a Linearize-and-Guess Approach

Ascon is the final winner of the lightweight cryptography standardization competition (2018 − 2023). In this paper, we focus on preimage attacks against round-reduced Ascon. The preimage attack framework, utilizing the linear structure with the allocating model, was initially proposed by Guo et al. at ASIACRYPT 2016 and subsequently improved by Li et al. at EUROCRYPT 2019, demonstrating high effectiveness in breaking the preimage resistance of Keccak. In this paper, we extend this preimage attack framework to Ascon from two aspects. Firstly, we propose a linearize-and-guess approach by analyzing the algebraic properties of the Ascon permutation. As a result, the complexity of finding a preimage for 2-round Ascon-Xof with a 64-bit hash value can be significantly reduced from 239 guesses to 227.56 guesses. To support the effectiveness of our approach, we find an actual preimage of all ‘0’ hash in practical time. Secondly, we develop a SAT-based automatic preimage attack framework using the linearize-and-guess approach, which is efficient to search for the optimal structures exhaustively. Consequently, we present the best theoretical preimage attacks on 3-round and 4-round Ascon-Xof so far.

A class of permutation polynomials over the group ring F_pC_p^n

Let p be a prime number and F_pC_p^n denote the group ring of a cyclic group of order pn over Fp. We study the permutation property of the polynomial a_0 + a_1x + a_px^p + · · · + a_p^n x^(p^n) with coefficients a_i ∈ F_pC_p^n where i = 0, 1, p, . . . , p^n. Necessary and sufficient conditions on the coefficients have been obtained so that it becomes a permutation polynomial.

Strong watermark numbers encoded as reducible permutation graphs against edge modification attacks

Software watermarking is a defense technique used to prevent or discourage software piracy by embedding a signature in the code. In (Discrete Applied Mathematics 250 (2018) 145–164), a software watermarking system is presented which encodes an integer number w (i.e., a watermark) as a reducible permutation flow-graph F [ π ∗ ] embeddable in the code through the use of a self-inverting permutation π ∗ . In this work, we theoretically investigate this watermarking system and exploit structural properties of the self-inverting permutation π ∗ encoding the watermark in order to prove its resilience to edge-modification attacks on the flow-graph F [ π ∗ ]. Based on the minimum number of edge modifications needed to be applied on F [ π ∗ ] so that a different watermark can be extracted from the resulting graph, we give a characterization of the watermarks as strong, intermediate or weak and provide good recommendations for the choices of watermark.

More constructions of n-cycle permutations

n-cycle permutations with small n have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was proposed, the permuting property of several forms of polynomials has been studied. In this paper, characterizations of several types of n-cycle permutations are investigated. Three criteria for n-cycle permutations of the form xh(λ(x)), h(ψ(x))φ(x)+g(ψ(x)) and g(xqi−x+δ)+bx with general n are provided. We demonstrate these criteria by providing explicit constructions. For the form of xrh(xs), several new explicit triple-cycle permutations are also provided. Finally, we also consider triple-cycle permutations of the form xt+cTrqm/q(xs) and provide one explicit construction. Many of our constructions are both new in the n-cycle property and the permutation property.

Classification of some quadrinomials over finite fields of odd characteristic

In this paper, we completely determine all necessary and sufficient conditions such that the polynomial f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, is a permutation quadrinomial of Fq2 over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where char(Fq)=2 and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3+axq+2+bx2q+1+cx3q, where char(Fq)=3,5 and a,b,c∈Fq⁎ and proposed some new classes of permutation quadrinomials of Fq2.In particular, in this paper we classify all permutation polynomials of Fq2 of the form f(x)=x3+axq+2+bx2q+1+cx3q, where a,b,c∈Fq⁎, over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials.

New Key-Independent Structural Properties of AES-Like Permutations

The Internet of Things (IoT) technology makes our lives very simple and convenient by interacting with sensors/devices around the world and using the smart data collected from them. However, IoT devices typically have a resource-constrained architecture, rendering them vulnerable to cyberattacks. The advent of lightweight cryptography provides an opportunity to meet the challenges of IoT. With the promotion of 5G (5th generation wireless systems) technology, the amount of data in IoT devices is bound to grow rapidly. The design and analysis of lightweight block cipher is still a hot issue that needs to be solved in the coming period of time, as it responds to the strong push of many national governments to adopt IoT systems in the management of public affairs. MANTIS is a new tweakable block cipher suitable for IoT with the goal of low-latency implementations and it has drawn lots of attention in the form of prior cryptanalysis. This work first reveals a novel property of MANTIS. The characteristic is established under the condition that there is a certain equivalence relation between the input pairs in a particular subspace and it is evidenced by the first introduction of a key-independent distinguisher for 6-round MANTIS. We obtain that the number of input plaintext pairs in the same equivalence class is always divisible by 8. Then, we demonstrate a general and comprehensive proof as why it has to exist. Additionally, we have successfully verified the validity of the distinguisher. Only 216 chosen plaintexts and 222 table lookups computational cost are required to guarantee that the success probability exceeds 99%. Moreover, we discover that the same kind of property holds for other AES-like permutations with the example of the lightweight hash function PHOTON. Finally, we put forward some future explorations in this promising field.

A Proposal of a New Chaotic Map for Application in the Image Encryption Domain

Several chaos-based image encryption schemes have been proposed in the last decade. Each encryption scheme has pros and cons regarding its speed, complexity, and security. This paper proposes a new chaotic map called Power-Chaotic Map (PCM). Characteristics of the proposed PCM, such as chaotic behaviour, randomness, sensitivity, and s-unimodality, are investigated. As an application of the proposed chaotic map, an image encryption scheme is proposed to encrypt greyscale and text images. The proposed three-phase image encryption scheme performs a series of substitution and permutation operations. The Pixel-Level phase utilises the PCM’s generated keystreams to perform the substitution operation of image pixels. The Row-Level phase permutates, via a proposed pseudorandom number generator, pixel locations of each row and then shuffles row locations. Finally, the Column-Level phase performs a substitution operation on pixels of each column. Performance of the proposed PCM-based image encryption scheme is investigated through histogram analysis, statistical correlation analysis, key sensitivity, encryption performance of text images, and permutation and substitution properties. Experimental results indicate that the PCM has a wider range of chaotic behaviour than well-known one-dimensional maps, meets the s-unimodality property, has high sensitivity, and generates keystreams with random-like behaviour. Furthermore, results indicate that the PCM-based image encryption scheme provides high encryption security for text images, high key sensitivity, immunity against brute-force attacks, strong statistical correlation results, strong encryption performance, and low computational complexity.

Haystack hunting hints and locker room communication

AbstractWe want to efficiently find a specific object in a large unstructured set, which we model by a random ‐permutation, and we have to do it by revealing just a single element. Clearly, without any help this task is hopeless and the best one can do is to select the element at random, and achieve the success probability . Can we do better with some small amount of advice about the permutation, even without knowing the target object? We show that by providing advice of just one integer in , one can improve the success probability considerably, by a factor. We study this and related problems, and show asymptotically matching upper and lower bounds for their optimal probability of success. Our analysis relies on a close relationship of such problems to some intrinsic properties of random permutations related to the rencontres number.

Construction of Permutation Polynomials Using Additive and Multiplicative Characters

Permutation is a natural phenomenon useful for understanding and explaining the structural and functional behavior of objects or concepts. The mathematical formulation of permutation behavior can be readily achieved by permutation polynomials. Permutation polynomials are constructed by suitably modifying linearized polynomials and associated affine polynomials with the help of additive characters, multiplicative characters, and special types of Trace functions for the polynomials in one and more than one indeterminates. The permutation properties of the obtained polynomials are verified using the AGW criterion.

The seeds of EFT double copy

We explore the double copy of effective field theories (EFTs), in the recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator seeds with trace-like permutation properties. This construction has the advantage of being easily applicable to any multiplicity, which we exemplify up to 6-point. It employs the linear map between color factors formed by single traces of generators and by products of the structure constants, which also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

The socle tableau as a dual version of the Littlewood–Richardson tableau

Like the LR-tableau, a socle tableau is given as a skew diagram with certain entries. Unlike in the LR-tableau, the entries in the socle tableau are weakly increasing in each row, strictly increasing in each column and satisfy a modified lattice permutation property. In the study of embeddings of a subgroup in a finite abelian p $p$ -group, socle tableaux occur as isomorphism invariants, they are given by the socle series of the subgroup. We show that each socle tableau can be realized by some embedding. Moreover, the socle tableau of an embedding and the LR-tableau of the dual embedding determine each other — a correspondence which appears to be related to the tableau switching studied by Benkart, Sottile and Stroomer. We illustrate how the entries in the socle tableau position the object within the Auslander–Reiten quiver. Like the LR-tableau, the socle tableau determines the irreducible component of the object in representation space.