The generalized two-dimensional fractional sine transform (2D-FrST) is a powerful analytic instrument for the spectral analysis of bivariate signals that arise in multi-scale supply-chain dynamics, hybrid-vehicle systems and resilient logistics networks. In this paper we state and rigorously prove Modulation Theorem-2, which expresses the 2D-FrST of a function modulated by a product of cosines (or by mixed sine–cosine factors) as a linear combination of four fractional sine or cosine transforms evaluated at frequency points shifted by the modulation frequencies. We further establish the Shifting Property, showing that a spatial translation of the original function maps, under the 2D-FrST, into a linear combination of fractional sine and cosine transforms of a phase-modulated version of the function. Both results are derived by means of elementary product-to-sum and angle-addition identities together with a careful accounting of the quadratic-phase factors that appear in the fractional kernel. The theoretical developments are illustrated by concrete applications to artificial-intelligence pattern mining and blockchain-based immutable logging, thereby enhancing transparency, traceability and resilience of digital supply-chain ecosystems. All results are placed in the broader context of the FKF-transform framework and related spectral methods recently introduced for multi-echelon lead-time analysis and secure logistics networks. Keywords— generalized two-dimensional fractional sine transform; modulation theorem; shifting property; fractional cosine transform; AI–blockchain integration; supply-chain resilience; FKF transform; spectral analysis; transparency; traceability.
Recently, designing symmetric primitives for applications in cryptographic protocols including multi-party computation, fully homomorphic encryption, and zero-knowledge proofs has become an important research topic. Among many such new symmetric schemes, a power function over a large finite field$\mathbb{F}_{q}$is commonly used. In this paper, we revisit the algebraic degree's growth for a substitution-permutation network (SPN) cipher over$\mathbb{F}_{2^{n}}(n\geq 3)$, whose S-box is defined as a composition of a power function$P(x)=x^{2^{d}+1}$where$d\geq 1$with a polynomial$A(x)=a_{0}+ \sum\limits_{w=1}^{W}a_{w}x^{2^{\beta_{w}}}$where$a_{i}\in \mathbb{F}_{2^{n}}$for$0\leq i\leq W$and$a_{w}\neq 0$for$1\leq w\leq W$. We propose a new coefficient grouping technique, which is based on our new description of the monomials that will probably appear in the state. Specifically, we propose a new measure to find proper$(\beta_{1},\beta_{2}, \ldots,\beta_{W})$for the algebraic degree's fastest growth and a new method to compute the algebraic degree's upper bound for arbitrary$A(x)$. Especially for Chaghri, which was presented at ACM CCS 2022, we obtained a tighter upper bound on the algebraic degree.
With the continuous development of blockchain technology, an increasing number of scholars have begun to consider the harm of data leakage during on-chain transactions and the requirement for privacy data protection. Zero-knowledge range proof, as a cryptographic technology, can perform legitimacy verification of data while hiding private data, effectively realizing the protection of private data on the blockchain, so it is increasingly used to protect blockchain privacy. The mainstream construction methods for range proofs can be mainly divided into two categories: n-ary decomposition and square decomposition. This paper introduces and analyzes the advantages and disadvantages of these construction methods in detail. Then, based on these two methods, a zero-knowledge range proof scheme based on multibit split square decomposition (ZKRPMSSD) is proposed, which requires no trusted third-party setting and can achieve range proofs for arbitrary ranges. The proposed ZKRPMSSD scheme processes the original data based on the multibit split idea, and the acquisition method of secret value components is optimized so that the acquisition of components does not depend on the scale of the original problem. Additionally, the algorithms for proof generation and verification in the ZKRPMSSD scheme are redesigned based on the \(\Sigma\) protocol and Pedersen commitments, effectively reducing the computational cost of the proof generation and verification process. Finally, typical n-ary decomposition and square decomposition zero-knowledge range proof construction schemes are taken for comparative analysis. Under 256-bit security and the same problem scale, experimental results indicate that ZKRPMSSD has advantages in proof and verification time costs.
Open access
Cryptography and Data Security
Digital Filter Design and Implementation
Advanced Steganography and Watermarking Techniques
Zero-knowledge protocols (ZKPs) allow a party to prove the validation of secret information to some other party without revealing any information about the secret itself. Appropriate, effective, and efficient use of cryptographic ZKPs contributes to many novel advances in real-world privacy-preserving frameworks. One of the most important type of cryptographic ZKPs is the zero-knowledge range proofs (ZKRPs). Such proofs have wide range of applications such as anonymous credentials, cryptocurrencies, e-cash schemes etc. In many ZKRPs the secret is represented in binary then committed via a suitable commitment scheme. Though there exist different base approaches on bilinear paring-based and RSA-like based constructions, to our knowledge there is no study on investigating the discrete logarithm-based constructions. In this study, we focus on a range proof construction produced by Mao in 1998. This protocol contains a bit commitment scheme with an OR-construction. We investigate the effect of different base approach on Mao's range proof and compare the efficiency of these basis approaches. To this end, we have extended Mao's range proof to base-3 with a modified OR-proof. We derive the number of computations in modulo exponentiations and the cost of the number of integers exchanged between parties. Then, we have generalized these costs for the base-u construction. Here, we mainly show that comparing with other base approaches, the base-3 approach consistently provides approximately 12% efficiency in computation cost and 10% efficiency in communication cost. We implemented the base-3 protocol and demonstrated that the results are consistent with our theoretical computations.
Advanced Steganography and Watermarking Techniques
For an oscillator that is periodically swept in frequency between some upper and lower bound, the output amplitude may easily be made constant and therefore known with a high degree of certainty. The instantaneous frequency exists only at a point in time and therefore possesses a zero probability of existing at any point. This thesis deals with the development of a method for interchanging the probability density functions of amplitude and frequency so that the latter becomes known with certainty while the former is known only to the extent that it is within a certain range. The method developed makes practical the use of the fast tuned voltage controlled oscillator as the local oscillator in a frequency scanning superheterodyne receiver. Exact frequency is expressed by a digital word of finite bit length that, in actuality, expresses the value of a quantized amplitude variable whose quantized value represents a precise frequency. Because of the interrelationship of amplitude, frequency, and time through the Fourier Transform, functions of these variables are also interrelated suggesting the possibility that the original certainty of amplitude information may be traded with the original uncertainty of frequency information. The success of the method presented makes use of the precise knowledge of the frequencies of the sidebands generated by the angle modulation process rather than make direct use of the instantaneous frequency. After mathematical development, a design example addresses the actual frequency range in the microwave region where the scanning superheterodyne receiver finds military application. To demonstrate the concept of precise frequency control with words of finite length, a practical frequency model is designed and constructed by scaling megahertz to hertz. Extensive use is made of monolithic waveform generators, balanced mixers, and operational amplifiers used as active filters and time domain summers. All assemblies within the model have practical microwave counterparts. Time and frequency domain waveforms are observed at virtually every major point of the model corresponding to the functional block interfaces and are compared with the mathematical predictions. The ultimate goal of precise frequency selection as a function of an imprecise independent variable is also obtained with the aid of a spectrum analyzer and dual trace oscilloscope. The causes of less than optimum signal level separation of adjacent discrete frequencies are analyzed in a qualitative manner. Reasons for the ineffectiveness of a quantitative critique are also presented. Experimental results, however, are demonstrated proof of the feasibility of the concept of exchanging probability density functions of related variables and that refinement is the only ingredient missing to render the fast scan VCO a useful local oscillator.
The problem of suppressing zero-input limit cycles in coupled-form state-space digital filters when the quantization is performed after the multiplication is addressed. To the extent of the authors' knowledge, no proof has been presented that the parasitic oscillations are suppressed for poles anywhere in the unit circle, under that condition. Some authors have addressed this subject, but their results constrain the poles to a bounded region inside the unit circle. With the objective of exploring this topic further, the authors present a proof that for poles whose angle is either 0, /spl plusmn/45, /spl plusmn/90, /spl plusmn/135 or 180 degrees and whose radius is lower than one, the second-order state-space coupled-form digital filter is free of zero-input limit cycles when quantizers placed just after the multipliers implement magnitude truncation.