In this letter, we give a characterization for a generic construction of bent functions. This characterization enables us to obtain another efficient construction of bent functions and to give a positive answer on a problem of bent functions.

In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $sum_{uin mathbb{F}_2^n,vin mathbb{F}_2^m}mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.

This paper concentrates on a class of pseudorandom sequences generated by combining q-ary m-sequences and quadratic characters over a finite field of odd order, called binary generalized NTU sequences. It is shown that the relationship among the sub-sequences of binary generalized NTU sequences can be formulated as combinatorial structures called Hadamard designs. As a consequence, the combinatorial structures generalize the group structure discovered by Kodera et al. (IEICE Trans. Fundamentals, vol.E102-A, no.12, pp.1659-1667, 2019) and lead to a finite-geometric explanation for the investigated group structure.

In 2017, Tang et al. provided a complete characterization of generalized bent functions from ℤ2n to ℤq(q = 2m) in terms of their component functions (IEEE Trans. Inf. Theory. vol.63, no.7, pp.4668-4674). In this letter, for a general even q, we aim to provide some characterizations and more constructions of generalized bent functions with flexible coefficients. Firstly, we present some sufficient conditions for a generalized Boolean function with at most three terms to be gbent. Based on these results, we give a positive answer to a remaining question proposed by Hodžić in 2015. We also prove that the sufficient conditions are also necessary in some special cases. However, these sufficient conditions whether they are also necessary, in general, is left as an open problem. Secondly, from a uniform point of view, we provide a secondary construction of gbent function, which includes several known constructions as special cases.

A polyphase sequence set with orthogonality consisting complex elements with unit magnitude, can be expressed by a unitary matrix corresponding to the complex Hadamard matrix or the discrete Fourier transform (DFT) matrix, whose rows are orthogonal to each other. Its matched filter bank (MFB), which can simultaneously output the correlation between a received symbol and any sequence in the set, is effective for constructing communication systems flexibly. This paper discusses the compact design of the MFB of a polyphase sequence set, which can be applied to any sequence set generated by the given logic function. It is primarily focused on a ZCZ code with q-phase or more elements expressed as A(N=qn+s, M=qn-1, Zcz=qs(q-1)), where q, N, M and Zcz respectively denote, a positive integer, sequence period, family size, and a zero correlation zone, since the compact design of the MFB becomes difficult when Zcz is large. It is shown that the given logic function on the ring of integers modulo q generating the ZCZ code gives the matrix representation of the MFB that M-dimensional output vector can be represented by the product of the unitary matrix of order M and an M-dimensional input vector whose elements are written as the sum of elements of an N-dimensional input vector. Since the unitary matrix (complex Hadamard matrix) can be factorized into n-1 unitary matrices of order M with qM nonzero elements corresponding to fast unitary transform, a compact MFB with a minimum number of circuit elements can be designed. Its hardware complexity is reduced from O(MN) to O(qM log q M+N).

Hadamard matrix is defined as a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. In this work, we consider the similar matrix on finite field GF(p) where p is an odd prime. In such a matrix, every component is one of the integers on GF(p){0}, that is, {1,2,...,p-1}. Any additions and multiplications should be executed under modulo p. In this paper, a method to generate such matrices is proposed. In addition, the paper includes the applications to generate n-shift orthogonal sequences and complete complementary codes. The generated complete complementary code is a family of multi-valued sequences on GF(p){0}, where the number of sequence sets, the number of sequences in each sequence set and the sequence length depend on the various divisors of p-1. Such complete complementary codes with various parameters have not been proposed in previous studies.

The rotating element electric field vector (REV) method is a classical measurement technique for phased array calibration. Compared with other calibration methods, it requires only power measurements. Thus, the REV method is more reliable for operating phased array calibration systems. However, since the phase of each element must be rotated from 0 to 2π, the conventional REV method requires a large number of measurements. Moreover, the power of composite electric field vector doesn't vary significantly because only a single element's phase is rotated. Thus, it can be easily degraded by the receiver noise. A simplified REV method combined with Hadamard group division is proposed in this paper. In the proposed method, only power measurements are required. All the array elements are divided into different groups according to the group matrix derived from the normalized Hadamard matrix. The phases of all the elements in the same group are rotated at the same time, and the composite electric field vector of this group is obtained by the simplified REV method. Hence, the relative electric fields of all elements can be obtained by a matrix equation. Compared with the conventional REV method, the proposed method can not only reduce the number of measurements but also improve the measurement accuracy under the particular range of signal to noise ratio(SNR) at the receiver, especially under low and moderate SNRs.

To construct good DNA codes based on biologically motivated constraints, it is important that they have a large minimum Hamming distance and the number of GC-content is kept constant. Also, maximizing the number of codewords in a DNA code is required for given code length, minimum Hamming distance, and number of GC-content. In most previous works on the construction of DNA codes, quaternary constant weight codes were directly used because the alphabet of DNA strands is quaternary. In this paper, we propose new coding theoretic constructions of DNA codes based on the binary Hadamard matrix from a binary sequence with ideal autocorrelation. The proposed DNA codes have a greater number of codewords than or the equal number to existing DNA codes constructed from quaternary constant weight codes. In addition, it is numerically shown that for the case of codes with length 8 or 16, the number of codewords in the proposed DNA code sets is the largest with respect to the minimum reverse complementary Hamming distances, compared to all previously known results.

As a with-carry analog (based on modular arithmetic) of the usual Walsh-Hadamard transform (WHT), arithmetic Walsh transform (AWT) has been used to obtain analogs of some properties of Boolean functions which are important in the design and analysis of cryptosystems. The existence of nonzero linear structure of Boolean functions is an important criterion to measure the weakness of these functions in their cryptographic applications. In this paper, we find more analogs of linear structures of Boolean functions from AWT. For some classes of n-variable Boolean functions f, we find necessary and sufficient conditions for the existence of an invariant linear structure and a complementary linear structure 1n of f. We abstract out a sectionally linear relationship between AWT and WHT of n-variable balanced Boolean functions f with linear structure 1n. This result show that AWT can characterize cryptographic properties of these functions as long as WHT can. In addition, for a diagonal Boolean function f, a recent result by Carlet and Klapper says that the AWT of f can be expressed in terms of the AWT of a diagonal Boolean function of algebraic degree at most 3 in a larger number of variables. We provide for the result a complete and more modular proof which works for both even and odd weights (of the parameter c in the Corollary 19 by Carlet and Klapper (DCC 73(2): 299-318, 2014).

This letter presents a novel memory-based architecture for radix-2 fast Walsh-Hadamard-Fourier transform (FWFT) based on the constant geometry FWFT algorithm. It is composed of a multi-function Processing Engine, a conflict-free memory addressing scheme and an efficient twiddle factor generator. The address for memory access and the control signals for stride permutation are formulated in detail and the methods can be applied to other memory-based FFT-like architectures.

Biphase periodic sequences having elements +1 or -1 with the two-level autocorrelation function are desirable in communications and radars. However, in case of the biphase orthogonal periodic sequences, Turyn has conjectured that there exist only sequences with period 4, i.e., there exist the circulant Hadamard matrices for order 4 only. In this paper, it is described that the conjecture is proved to be true by means of the isomorphic mapping, the Chinese remainder theorem, the linear algebra, etc.

We propose a practical method that acquires dense light transports from unknown 3D objects by employing orthogonal illumination based on a Walsh-Hadamard matrix for relighting computation. We assume the presence of color crosstalk, which represents color mixing between projector pixels and camera pixels, and then describe the light transport matrix by using sets of the orthogonal illumination and the corresponding camera response. Our method handles not only direct reflection light but also global light radiated from the entire environment. Tests of the proposed method using real images show that orthogonal illumination is an effective way of acquiring accurate light transports from various 3D objects. We demonstrate a relighting test based on acquired light transports and confirm that our method outputs excellent relighting images that compare favorably with the actual images observed by the system.

In this paper, we propose a new structure for a compact matched filter bank for a mutually orthogonal zero-correlation zone (MO-ZCZ) sequence set consisting of ternary sequence pairs obtained by Hadamard and binary ZCZ sequence sets; this construction reduces the number of two-input adders and delay elements. The matched filter banks are implemented on a field-programmable gate array (FPGA) with 51,840 logic elements (LEs). The proposed matched filter bank for an MO-ZCZ sequence set of length 160 can be constructed by a circuit size that is about 8.6% that of a conventional matched filter bank.

An asymmetric zero correlation zone (A-ZCZ) sequence set is a type of ZCZ sequence set and consists of multiple sequence subsets. It is the most important property that is the cross-correlation function between arbitrary sequences belonging to different sequence subsets has quite a large zero-cross-correlation zone (ZCCZ). Our proposed A-ZCZ sequence sets can be constructed based on interleaved technique and orthogonality-preserving transformation by any perfect sequence of length P=Nq(2k+1) and Hadamard matrices of order T≥2, where N≥1, q≥1 and k≥1. If q=1, the novel sequence set is optimal ZCZ sequence set, which has parameters (TP,TN,2k+1) for all positive integers P=N(2k+1). The proposed A-ZCZ sequence sets have much larger ZCCZ, which are expected to be useful for designing spreading sequences for QS-CDMA systems.

A code shift keying (CSK) using pseudo-noise (PN) codes for optical wireless communications with intensity/modulation and direct detection (IM/DD) is considered. Since CSK has several PN codes, the data transmission rate and the bit error rate (BER) performance can be improved by increasing the number of PN codes. However, the conventional optical PN codes are not suitable for optical CSK with IM/DD because the ratio of the number of PN codes and the code length of PN code, M/L is smaller than 1/√L. In this paper, an optical CSK with a new PN code, which combines the generalized modified prime sequence code (GMPSC) and Hadamard code is analyzed. The new PN code can achieve M/L=1. Moreover, the BER performance and the data transmission rate of the CSK system with the new PN code are evaluated through theoretical analysis by taking the scintillation, background-noise, avalanche photodiode (APD) noise, thermal noise, and signal dependent noise into account. It is found that the CSK system with the new PN code outperforms the conventional optical CSK system.

There have been many matching pursuit algorithms (MPAs) which handle the sparse signal recovery problem, called compressed sensing (CS). In the MPAs, the correlation step makes a dominant computational complexity. In this paper, we propose a new fast correlation method for the MPA when we use partial Fourier sensing matrices and partial Hadamard sensing matrices which are widely used as the sensing matrix in CS. The proposed correlation method can be applied to almost all MPAs without causing any degradation of their recovery performance. Also, the proposed correlation method can reduce the computational complexity of the MPAs well even though there are restrictions depending on a used MPA and parameters.

This paper describes a unified VLSI architecture which can be applied to various types of transforms used in MPEG-2/4, H.264, VC-1, AVS and the emerging new video coding standard named HEVC (High Efficiency Video Coding). A novel design named configurable butterfly array (CBA) is also proposed to support both the forward transform and the inverse transform in this unified architecture. Hadamard transform or 4/8-point DCT/IDCT are used in traditional video coding standards while 16/32-point DCT/IDCT are newly introduced in HEVC. The proposed architecture can support all these transform types in a unified architecture. Two levels (architecture level and block level) of hardware sharing are adopted in this design. In the architecture level, the forward transform can share the hardware resource with the inverse transform. In the block level, the hardware for smaller size transform can be recursively reused by larger size transform. The multiplications of 4 or 8-point transform are implemented with Multiplierless MCM (Multiple Constant Multiplication). In order to reduce the hardware overhead, the multiplications of 16/32 point DCT are implemented with ICM (input-muxed constant multipliers) instead of MCM or regular multipliers. The proposed design is 51% more area efficient than previous work. To the author's knowledge, this is the first published work to support both forward and inverse 4/8/16/32-point integer transform for HEVC standard in a unified architecture.

We apply the Hadamard equivalence to all the binary matrices of the size mn and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class, and count and/or estimate the number of HR-minimals of size mn. Some properties and constructions of HR-minimals are investigated. Especially, we figure that the weight on an HR-minimal's second row plays an important role, and introduce the concept of Quasi-Hadamard matrices (QH matrices). We show that the row vectors of mn QH matrices form a set of m binary vectors of length n whose maximum pairwise absolute correlation is minimized over all such sets. Some properties, existence, and constructions of Quasi-orthogonal sequences are also discussed. We also give a relation of these with cyclic difference sets. We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.

Fidelity Range Extension (FRExt) (i.e. High Profile) was added to the H.264/AVC recommendation in the second version. One of the features included in FRExt is the Adaptive Block-size Transform (ABT). In order to conform to the FRExt, a Fractional Motion Estimation (FME) architecture is proposed to support the 88/44 adaptive Hadamard Transform (88/44 AHT). The 88/44 AHT circuit contributes to higher throughput and encoding performance. In order to increase the utilization of SATD (Sum of Absolute Transformed Difference) Generator (SG) in unit time, the proposed architecture employs two 8-pel interpolators (IP) to time-share one SG. These two IPs can work in turn to provide the available data continuously to the SG, which increases the data throughput and significantly reduces the cycles that are needed to process one Macroblock. Furthermore, this architecture also exploits the linear feature of Hadamard Transform to generate the quarter-pel SATD. This method could help to shorten the long datapath in the second-step of two-iteration FME algorithm. Finally, experimental results show that this architecture could be used in the applications requiring different performances by adjusting the supported modes and operation frequency. It can support the real-time encoding of the seven-mode 4 K2 K@24 fps or six-mode 4 K2 K@30 fps video sequences.

CLEFIA is a 128-bit block cipher proposed by Shirai et al. in 2007. On its saturation attack, Tsunoo et al. reported peculiar saturation characteristics in 2010. They formulated some hypotheses on the existence of the characteristics with no proof. In this paper we have theoretically proved their hypotheses. In their attack scenario, we show that the mod-2 distribution is a code word of Extended Hamming code, and then proof is given by using the property of Hadamard transform.