Recently, trace representation of a class of balanced quaternary sequences of period p from the classical cyclotomic classes was given by Yang et al. (Cryptogr. Commun.,15 (2023): 921-940). In this letter, based on the generalized cyclotomic classes, we define a class of balanced quaternary sequences of period pn, where p = ef + 1 is an odd prime number and satisfies e ≡ 0 (mod 4). Furthermore, we calculate the defining polynomial of these sequences and obtain the formula for determining their trace representations over ℤ4, by which the linear complexity of these sequences over ℤ4 can be determined.

Permutation polynomials have important applications in cryptography, coding theory and combinatorial designs. In this letter, we construct four classes of permutation polynomials over 𝔽2n × 𝔽2n, where 𝔽2n is the finite field with 2n elements.

A PBN is well known as a mathematical model of complex network systems such as gene regulatory networks. In Boolean networks, interactions between nodes (e.g., genes) are modeled by Boolean functions. In PBNs, Boolean functions are switched probabilistically. In this paper, for a PBN, a simplified representation that is effective in analysis and control is proposed. First, after a polynomial representation of a PBN is briefly explained, a simplified representation is derived. Here, the steady-state value of the expected value of the state is focused, and is characterized by a minimum feedback vertex set of an interaction graph expressing interactions between nodes. Next, using this representation, input selection and stabilization are discussed. Finally, the proposed method is demonstrated by a biological example.

We propose how to homomorphically evaluate arbitrary univariate and bivariate integer functions such as division. A prior work proposed by Okada et al. (WISTP'18) uses polynomial evaluations such that the scheme is still compatible with the SIMD operations in BFV and BGV schemes, and is implemented with the input domain ℤ257. However, the scheme of Okada et al. requires the quadratic numbers of plaintext-ciphertext multiplications and ciphertext-ciphertext additions in the input domain size, and although these operations are more lightweight than the ciphertext-ciphertext multiplication, the quadratic complexity makes handling larger inputs quite inefficient. In this work, first we improve the prior work and also propose a new approach that exploits the packing method to handle the larger input domain size instead of enabling the SIMD operation, thus making it possible to work with the larger input domain size, e.g., ℤ215 in a reasonably efficient way. In addition, we show how to slightly extend the input domain size to ℤ216 with a relatively moderate overhead. Further we show another approach to handling the larger input domain size by using two ciphertexts to encrypt one integer plaintext and applying our techniques for uni/bivariate function evaluation. We implement the prior work of Okada et al., our improved version of Okada et al., and our new scheme in PALISADE with the input domain ℤ215, and confirm that the estimated run-times of the prior work and our improved version of the prior work are still about 117 days and 59 days respectively while our new scheme can be computed in 307 seconds.

Distributed computing is one of the powerful solutions for computational tasks that need the massive size of dataset. Lagrange coded computing (LCC), proposed by Yu et al. [15], realizes private and secure distributed computing under the existence of stragglers, malicious workers, and colluding workers by using an encoding polynomial. Since the encoding polynomial depends on a dataset, it must be updated every arrival of new dataset. Therefore, it is necessary to employ efficient algorithm to construct the encoding polynomial. In this paper, we propose Newton coded computing (NCC) which is based on Newton interpolation to construct the encoding polynomial. Let K, L, and T be the number of data, the length of each data, and the number of colluding workers, respectively. Then, the computational complexity for construction of an encoding polynomial is improved from O(L(K+T)log 2(K+T)log log (K+T)) for LCC to O(L(K+T)log (K+T)) for the proposed method. Furthermore, by applying the proposed method, the computational complexity for updating the encoding polynomial is improved from O(L(K+T)log 2(K+T)log log (K+T)) for LCC to O(L) for the proposed method.

Today's electronic devices must meet many requirements, such as those related to performance, limits to the radiated electromagnetic field, size, etc. For such a design, the requirement is to have a solution that simultaneously meets multiple objectives that sometimes include conflicting requirements. In addition, it is also necessary to consider uncertain parameters. This paper proposes a new combination of statistical analysis using the Polynomial Chaos (PC) method for dealing with the random and multi-objective satisfactory design using the Preference Set-based Design (PSD) method. The application in this paper is an Electromagnetic Interference (EMI) filter for a practical case, which includes plural element parameters and uncertain parameters, which are resistors at the source and load, and the performances of the attenuation characteristics. The PC method generates simulation data with high enough accuracy and good computational efficiency, and these data are used as initial data for the meta-modeling of the PSD method. The design parameters of the EMI filter, which satisfy required performances, are obtained in a range by the PSD method. The authors demonstrate the validity of the proposed method. The results show that applying a multi-objective design method using PSD with a statistical method using PC to handle the uncertain problem can be applied to electromagnetic designs to reduce the time and cost of product development.

We show that every polynomial threshold function that sign-represents the ODD-MAXBITn function has total absolute weight 2Ω(n1/3). The bound is tight up to a logarithmic factor in the exponent.

A formal graph system (FGS for short) is a logic program consisting of definite clauses whose arguments are graph patterns instead of first-order terms. The definite clauses are referred to as graph rewriting rules. An FGS is shown to be a useful unifying framework for learning graph languages. In this paper, we show the polynomial-time PAC learnability of a subclass of FGS languages defined by parameterized hereditary FGSs with bounded degree, from the viewpoint of computational learning theory. That is, we consider VH-FGSLk,Δ(m, s, t, r, w, d) as the class of FGS languages consisting of graphs of treewidth at most k and of maximum degree at most Δ which is defined by variable-hereditary FGSs consisting of m graph rewriting rules having TGP patterns as arguments. The parameters s, t, and r denote the maximum numbers of variables, atoms in the body, and arguments of each predicate symbol of each graph rewriting rule in an FGS, respectively. The parameters w and d denote the maximum number of vertices of each hyperedge and the maximum degree of each vertex of TGP patterns in each graph rewriting rule in an FGS, respectively. VH-FGSLk,Δ(m, s, t, r, w, d) has infinitely many languages even if all the parameters are bounded by constants. Then we prove that the class VH-FGSLk,Δ(m, s, t, r, w, d) is polynomial-time PAC learnable if all m, s, t, r, w, d, Δ are constants except for k.

The problem of Isomorphism of Polynomials (IP problem) is known to be important to study the security of multivariate public key cryptosystems, one of the major candidates of post-quantum cryptography, against key recovery attacks. In these years, several schemes based on the IP problem itself or its generalization have been proposed. At PQCrypto 2020, Santoso introduced a generalization of the problem of Isomorphism of Polynomials, called the problem of Blockwise Isomorphism of Polynomials (BIP problem), and proposed a new Diffie-Hellman type encryption scheme based on this problem with Circulant matrices (BIPC problem). Quite recently, Ikematsu et al. proposed an attack called the linear stack attack to recover an equivalent key of Santoso's encryption scheme. While this attack reduced the security of the scheme, it does not contribute to solving the BIPC problem itself. In the present paper, we describe how to solve the BIPC problem directly by simplifying the BIPC problem due to the conjugation property of circulant matrices. In fact, we experimentally solved the BIPC problem with the parameter, which has 256 bit security by Santoso's security analysis and has 72.7bit security against the linear stack attack, by about 10 minutes.

With the advent of the Internet of Things (IoT), short packet transmissions will dominate the future wireless communication. However, traditional coherent demodulation and channel estimation schemes require large pilot overhead, which may be highly inefficient for short packets in multipath fading scenarios. This paper proposes a novel pilot-free short packet structure based on the association of modulation on conjugate-reciprocal zeros (MOCZ) and tail-biting convolutional codes (TBCC), where a noncoherent demodulation and decoding scheme is designed without the channel state information (CSI) at the transceivers. We provide a construction method of constellation sets and demodulation rule for M-ary MOCZ. By deriving low complexity log-likelihood ratios (LLR) for M-ary MOCZ, we offer a reasonable balance between energy and bandwidth efficiency for joint coding and modulation scheme. Simulation results show that our proposed scheme can attain significant performance and throughput gains compared to the pilot-based coherent modulation scheme over multipath fading channels.

Deep unfolding is a promising deep-learning technique, whose network architecture is based on expanding the recursive structure of existing iterative algorithms. Although deep unfolding realizes convergence acceleration, its theoretical aspects have not been revealed yet. This study details the theoretical analysis of the convergence acceleration in deep-unfolded gradient descent (DUGD) whose trainable parameters are step sizes. We propose a plausible interpretation of the learned step-size parameters in DUGD by introducing the principle of Chebyshev steps derived from Chebyshev polynomials. The use of Chebyshev steps in gradient descent (GD) enables us to bound the spectral radius of a matrix governing the convergence speed of GD, leading to a tight upper bound on the convergence rate. Numerical results show that Chebyshev steps numerically explain the learned step-size parameters in DUGD well.

The differential uniformity, the boomerang uniformity, and the extended Walsh spectrum etc are important parameters to evaluate the security of S (substitution)-box. In this paper, we introduce efficient formulas to compute these cryptographic parameters of permutation polynomials of the form xrh(x(2n-1)/d) over a finite field of q=2n elements, where r is a positive integer and d is a positive divisor of 2n-1. The computational cost of those formulas is proportional to d. We investigate differentially 4-uniform permutation polynomials of the form xrh(x(2n-1)/3) and compute the boomerang spectrum and the extended Walsh spectrum of them using the suggested formulas when 6≤n≤12 is even, where d=3 is the smallest nontrivial d for even n. We also investigate the differential uniformity of some permutation polynomials introduced in some recent papers for the case d=2n/2+1.

A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The K-PATH VERTEX COVER RECONFIGURATION (K-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of K-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k=2, known as the VERTEX COVER RECONFIGURATION (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes can be extended for K-PVCR. In particular, we prove a complexity dichotomy for K-PVCR on general graphs: on those whose maximum degree is three (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is two (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for K-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

The problem of enumerating connected induced subgraphs of a given graph is classical and studied well. It is known that connected induced subgraphs can be enumerated in constant time for each subgraph. In this paper, we focus on highly connected induced subgraphs. The most major concept of connectivity on graphs is vertex connectivity. For vertex connectivity, some enumeration problem settings and enumeration algorithms have been proposed, such as k-vertex connected spanning subgraphs. In this paper, we focus on another major concept of graph connectivity, edge-connectivity. This is motivated by the problem of finding evacuation routes in road networks. In evacuation routes, edge-connectivity is important, since highly edge-connected subgraphs ensure multiple routes between two vertices. In this paper, we consider the problem of enumerating 2-edge-connected induced subgraphs of a given graph. We present an algorithm that enumerates 2-edge-connected induced subgraphs of an input graph G with n vertices and m edges. Our algorithm enumerates all the 2-edge-connected induced subgraphs in O(n3m|SG|) time, where SG is the set of the 2-edge-connected induced subgraphs of G. Moreover, by slightly modifying the algorithm, we have a O(n3m)-delay enumeration algorithm for 2-edge-connected induced subgraphs.

We introduce the r-ary sequence with period 2p2 derived from Euler quotients modulo 2p (p is an odd prime) where r is an odd prime divisor of (p-1). Then based on the cyclotomic theory and the theory of trace function in finite fields, we give the trace representation of the proposed sequence by determining the corresponding defining polynomial. Our results will be help for the implementation and the pseudo-random properties analysis of the sequences.

Permutation polynomials over finite fields have been widely studied due to their important applications in mathematics and cryptography. In recent years, 2-to-1 mappings over finite fields were proposed to build almost perfect nonlinear functions, bent functions, and the semi-bent functions. In this paper, we generalize the 2-to-1 mappings to m-to-1 mappings, including their construction methods. Some applications of m-to-1 mappings are also discussed.

Non-linear behavioral models play a key role in designing digital pre-distorters (DPDs) for non-linear power amplifiers (NLPAs). In general, more complex behavioral models have better capability, but they should be converted into simpler versions to assist implementation. In this paper, a conversion from a complex fifth order inverse of a parallel Wiener (PRW) model to a simpler memory polynomial (MP) model is developed by using frequency domain expressions. In the developed conversion, parameters of the converted MP model are calculated from those of original fifth order inverse and frequency domain statistics of the transmit signal. Since the frequency domain statistics of the transmit signal can be precalculated, the developed conversion is deterministic, unlike the conventional conversion that identifies a converted model from lengthy input and output data. Computer simulations are conducted to confirm that conversion error is sufficiently small and the converted MP model offers equivalent pre-distortion to the original fifth order inverse.

A construction method of self-orthogonal and self-dual quasi-cyclic codes is shown which relies on factorization of modulus polynomials for cyclicity in this study. The smaller-size generator polynomial matrices are used instead of the generator matrices as linear codes. An algorithm based on Chinese remainder theorem finds the generator polynomial matrix on the original modulus from the ones constructed on each factor. This method enables us to efficiently construct and search these codes when factoring modulus polynomials into reciprocal polynomials.

Permutation polynomials and their compositional inverses are crucial for construction of Maiorana-McFarland bent functions and their dual functions, which have the optimal nonlinearity for resisting against the linear attack on block ciphers and on stream ciphers. In this letter, we give the explicit compositional inverse of the permutation binomial $f(z)=z^{2^{r}+2}+alpha zinmathbb{F}_{2^{2r}}[z]$. Based on that, we obtain the dual of monomial bent function $f(x)={ m Tr}_1^{4r}(x^{2^{2r}+2^{r+1}+1})$. Our result suggests that the dual of f is not a monomial any more, and it is not always EA-equivalent to f.

The unattended malicious nodes pose great security threats to the integrity of the IoT sensor networks. However, preventions such as cryptography and authentication are difficult to be deployed in resource constrained IoT sensor nodes with low processing capabilities and short power supply. To tackle these malicious sensor nodes, in this study, the trust computing method is applied into the IoT sensor networks as a light weight security mechanism, and based on the theory of Chebyshev Polynomials for the approximation of time series, the trust data sequence generated by each sensor node is linearized and treated as a time series for malicious node detection. The proposed method is evaluated against existing schemes using several simulations and the results demonstrate that our method can better deal with malicious nodes resulting in higher correct packet delivery rate.