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.

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.

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 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.

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.

A class of balanced semi-bent functions with an even number of variables is proposed. It is shown that they include one subclass of semi-bent functions with maximum algebraic degrees. Furthermore, an example of semi-bent functions in a small field is given by using the zeros of some Kloosterman sums. Based on the result given by S.Kim et al., an example of infinite families of semi-bent functions is also obtained.

This paper presents an integer discrete cosine transform (IntDCT) with only dyadic values such as k/2n (k, n∈ in N). Although some conventional IntDCTs have been proposed, they are not suitable for lossless-to-lossy image coding in low-bit-word-length (coefficients) due to the degradation of the frequency decomposition performance in the system. First, the proposed M-channel lossless Walsh-Hadamard transform (LWHT) can be constructed by only (log2M)-bit-word-length and has structural regularity. Then, our 8-channel IntDCT via LWHT keeps good coding performance even if low-bit-word-length is used because LWHT, which is main part of IntDCT, can be implemented by only 3-bit-word-length. Finally, the validity of our method is proved by showing the results of lossless-to-lossy image coding in low-bit-word-length.

In this paper, the previous definition of the Reverse Jacket matrix (RJM) is revised and generalized. In particular, it is shown that the inverse of the RJM can be obtained easily by a constructive approach similar to that used for the RJM itself. As new results, some useful properties of RJMs, such as commutativity and the Hamiltonian symmetry appearing in half the blocks of a RJM, are shown, and also 1-D fast Reverse Jacket transform (FRJT) is presented. The algorithm of the FRJT is remarkably efficient than that of the center-weighted Hadamard transform (CWHT). The FRJT is extended in terms of the Kronecker products of the Hadamard matrix. The 1-D FRJT is applied to the discrete Fourier transform (DFT) with order 4, and the N-point DFT can be expressed in terms of matrix decomposition by using 4 4 FRJT.

In this article, it is shown that Unified Complex Hadamard Transform (UCHT) can be derived from Walsh functions and through direct matrix operation. Unique properties of UCHT are analyzed. Recursive relations through Kronecker product can be applied to the basic matrices to obtain higher dimensions. These relations are the basis for the flow diagram of a constant-geometry iterative VLSI hardware architecture. New Normalized Complex Hadamard Transform (NCHT) matrices are introduced which are another class of complex Hadamard matrices. Relations of UCHT and NCHT with other discrete transforms are discussed.

The channel noise in OFDM systems affects the accuracy of channel estimation, deteriorating the performance of equalization. We present a novel algorithm with MMSE (Minimum Mean Square Error) channel estimation based on Hadamard Transform, to mitigate the effects of noise. The performance of the proposed algorithm is proved to be better than that with LS (Least Square) estimation, and very close to that with the MMSE estimation based on Fourier Transform, while the computation required is pretty small due to the use of Hadamard Transform.

This paper proposes a high-resolution CMOS image sensor, which has Hadamard transform function. This Hadamard transform circuit consists of two base generators, an array of pixel circuits, and analog-to-digital converters. In spite of simple composition, a base generator outputs a variety of bases, a pixel circuit calculates a two-dimensional base from one-dimensional bases and outputs values to common line for current addition, and analog-to-digital converter converts current value to digital value and stabilize a common line voltage for elimination of parasitic capacitance. We simulated these circuit elements and optimized using SPICE. Basic operations of this Hadamard transform circuit are also confirmed by simulation. A 256 256 pixel test chip was designed in 4.73 mm 4.73 mm area with 0.35 µm CMOS technology. A fill factor of this chip is 42% and dynamic range is 55.6 [dB]. Functions of this chip are Hadamard transform, Harr transform, projection, obtaining center of gravity, and so on.

The UCHT (Unified Complex Hadamard Transform) has been proposed as a new family of spreading sequences for DS-SSMA systems recently. In this Letter, the periodic autocorrelation (PAC) properties of the Unified Complex Hadamard Transform (UCHT) sequences are analyzed. Upper bounds for the out-of-phase PAC are derived for two groups of the UCHT sequences, namely the HSP-UCHT and the NHSP-UCHT sequences (the later is a more general representation of the well-known Walsh-Hadamard (WH) sequences). A comparison of the two bounds is performed. It turns out that the HSP-UCHT sequences have a lower upper bound for the out-of-phase PAC. This makes the HSP-UCHT sequences more effective than the WH sequences in combating multipath effect for DS-SSMA systems.

A circuit that carries out an Hadamard transform of an input image using the pulse width modulation technique is proposed. The proposed circuit architecture realizes the function of an Hadamard transform with a full-size pixel image. A test chip that we designed and fabricated integrates 64 64 pixels in a 4.9 mm 4.9 mm area, with 0.35 µm CMOS technology. The functional operation and linearity of this chip are measured. An image processing application utilizing this chip is demonstrated.

This paper discusses factorization of bent function type complex Hadamard matrices of order pn with a prime p. It is shown that any bent function type complex Hadamard matrix has symmetrical factorization, which can be expressed by the product of n matrices of order pn with pn+1 non-zero elements, a matrix of order pn with pn non-zero ones, and the n matrices, at most. As its application, a correlator for M-ary spread spectrum communications is successfully given, which can be simply constructed by the same circuits with reduced multiplicators, before and behind.

In this paper, comparison of various orthogonal transforms in Wiener filtering is discussed. The study involves the family of discrete orthogonal transforms called Complex Hadamard Transform, which has been recently introduced by the same authors. Basic definitions, properties and transformation kernel of Complex Hadamard Transform are also shown.

In this paper, we consider the selection of analysis filters used in the delayless subband adaptive digital filter (SBADF) and propose to use simple analysis filters to reduce the computational complexity. The coefficients of filters are determined using the components of the first order Hadamard matrix. Because coefficients of Hadamard matrix are either 1 or -1, we can analyze signals without multiplication. Moreover, the conditions for convergence of the proposed method is considered. It is shown by computer simulations that the proposed method can converge to the Wiener filter.

In recent years, the digitalization of transmission links, such as optical fibre cables, satellite links, and terrestrial microwave links, has been progressed rapidly in many countries. In addition, many types of digital studio equipment have been developed and TV programs can be produced or edited without any picture quality degradation by using such equipment, for example, digital VTR. A high-efficiency bit-reduction coding system is the most promising and effective means for this situation in terms of reducing the cost of digital transmission of TV programs with high picture quality. Considering this background, a new digital coding system has been developed, which makes it possible to transmit up to 4 NTSC TV programs simultaneously over a single DS3 45Mbps link including two high quality sound channels and one 64kbps ancillary data channel for each TV program. The principal bit-reduction technique employed is 2 dimensional intraframe WHT (Walsh Hadamard Transform) coding, which gives higher coding performance for composite TV signals than DCT (Discrete Cosine Transform) coding. In order to attain high picture quality at around 8Mbps for 4 channel transmission, a 3 dimensional adaptive quantization cube which reflects human visual perception sufficiently is employed in the intraframe WHT coding scheme. The hardware has been made compact like a home use VTR. In this paper, first, the algorithm of the coding scheme developed for the coding system is presented, and then the system configuration and its basic coding performance are described.